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WO2018187604A1 - Methods and systems for enhanced data-centric scalar multiplicative homomorphic encryption systems using geometric algebra - Google Patents

Methods and systems for enhanced data-centric scalar multiplicative homomorphic encryption systems using geometric algebra Download PDF

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Publication number
WO2018187604A1
WO2018187604A1 PCT/US2018/026305 US2018026305W WO2018187604A1 WO 2018187604 A1 WO2018187604 A1 WO 2018187604A1 US 2018026305 W US2018026305 W US 2018026305W WO 2018187604 A1 WO2018187604 A1 WO 2018187604A1
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WO
WIPO (PCT)
Prior art keywords
multivector
scalar
computing device
cryptotext
shared secret
Prior art date
Application number
PCT/US2018/026305
Other languages
French (fr)
Inventor
Carlos A. Paz De Araujo
David W. HONORIO ARAUJO DA SILVA
Gregory B. Jones
Original Assignee
X-Logos, LLC
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Filing date
Publication date
Priority claimed from US15/667,325 external-priority patent/US10728227B2/en
Application filed by X-Logos, LLC filed Critical X-Logos, LLC
Publication of WO2018187604A1 publication Critical patent/WO2018187604A1/en

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Classifications

    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
    • H04L9/00—Cryptographic mechanisms or cryptographic arrangements for secret or secure communications; Network security protocols
    • H04L9/008—Cryptographic mechanisms or cryptographic arrangements for secret or secure communications; Network security protocols involving homomorphic encryption

Definitions

  • Modern encryption employs mathematical techniques that manipulate positive integers or binary bits.
  • Asymmetric encryption such as RSA (Rivest-Shamir-Adleman) relies on number theoretic one-way functions that are predictably difficult to factor and can be made more difficult with an ever increasing size of the encryption keys.
  • Symmetric encry tion such as DES (Data Encryption Standard) and AES (Advanced Encryption Standard), uses bit manipulations within registers to shuffle the cryptotext to increase "diffusion" as well as register based operations with a shared key to increase "confusion.” Diffusion and confusion are measures for the increase in statistical entropy on the data payload being transmitted.
  • Diffusion is generally thought of as complicating the mathematical process of generating unencrypted (plain text) data from the encrypted (cryptotext) data, thus, making it difficult to discover the encryption key of the encryption process by spreading the influence of each piece of the unencrypted (plain) data across several pieces of the encrypted (cryptotext) data.
  • an encryption system that has a high degree of diffusion will typically change several characters of the encrypted (cryptotext) data for the change of a single character in the unencrypted (plain) data making it difficult for an attacker to identify changes in the unencrypted (plain) data.
  • Confusion is generally thought of as obscuring the relationship between the unencrypted (plain) data and the encrypted (cryptotext) data.
  • an encryption system that has a high degree of confusion would entail a process that drastically changes the unencrypted (plain) data into the encrypted (cryptotext) data in a way that, even when an attacker knows the operation of the encryption method (such as the public standards of RSA, DES, and/or AES), it is still difficult to deduce the encryption key.
  • Homomorphic Encryption is a form of encryption that allows computations to be carried out on cipher text as it is encrypted without decrypting the cipher text that generates an encrypted result which, when decrypted, matches the result of operations performed on the unencrypted plaintext
  • homomorphism comes from the ancient Greek language: ⁇ (homos) meaning “same” and ⁇ (morphe) meaning "form” or “shape.”
  • homomorphism may have different definitions depending on the field of use. In mathematics, for example, homomorphism may be considered a transformation of a first set into a second set where the relationship between the elements of the first set are preserved in the relationship of the elements of the second set.
  • a map / between sets A and B is a homomorphism of A into B if where "op" is the respective group operation defining the relationship between A and B.
  • homomorphism may be a structure-preserving map between two algebraic structures such as groups, rings, or vector spaces. Isomorphisms, automorphisms, and endomorphisms are typically considered special types of homomorphisms. Among other more specific definitions of homomorphism, algebra homomorphism may be considered a homomorphism that preserves the algebra structure between two sets.
  • An embodiment of the present invention may comprise a method for performing homomorphic scalar multiplication on a cryptotext encrypted data representation of a corresponding plain text data value and an unencrypted scalar data value, the method comprising: distributing by a source computing device a numeric message data value (M ) into coefficients of a message multivector ) representing the numeric message data value ( M ) in accord with a homomorphic preserving mathematical relationship between an unencrypted numeric data value and multivector coefficients representing the unencrypted numeric data value that is known to the source computing device and a destination computing device; distributing by the source computing device a shared secret numeric value (S s ) into coefficients of a shared secret multivector in accord with a shared secret coefficient
  • the source computing device and the destination computing device, such that the shared secret numeric value (S s ) is known or knowable to the source computing device and the destination computing device, but is kept secret from other devices not intended to have access to the numeric message data including an intermediary computing system; encrypting by the source computing device a cryptotext multivector ) as an encryption function of at least one Geometric Algebra geometric product operation on the message multivector ) and the shared secret multivector ; sending by the source
  • the computing device the cryptotext multivector ) to the intermediary computing system; receiving by the intermediary computing system the cryptotext multivector ( sent by the
  • An embodiment of the present invention may further comprise a method for encrypting a numeric message data value (M) on a source computing device in order to transfer a cryptotext multivector ( ) encrypted representation of the numeric message data value (M) to an intermediary computing system that will perform homomorphic scalar multiplication of the cryptotext multivector ) and an unencrypted scalar data value (V) and
  • the method comprising: distributing by the source computing device the numeric message data value (M) into coefficients of a message multivector ( ) in accord with a homomorphic preserving mathematical relationship between an unencrypted numeric data value and multivector coefficients representing the unencrypted numeric data value that is known to the source computing device and the destination computing device; distributing by the source computing device a shared secret numeric value (S s ) into coefficients of a shared secret multivector ) in accord with a shared secret coefficient distribution algorithm that is known to the source computing device and the destination computing device, the shared secret numeric value (S s ) being known or knowable to the source computing device and the destination computing device, but is kept secret from other devices not intended to have access to the numeric message data including the intermediary computing system; encrypting by the source computing device the cryptotext multivector (C) as an encryption function of at least one Geometric Algebra geometric product operation on
  • An embodiment of the present invention may further comprise a method for performing homomorphic scalar multiplication on an intermediary computer system of a cryptotext multivector (C) encrypted data representation of a corresponding plain text numeric data value received from a source computing device and an unencrypted scalar data value (V) and delivering a homomorphic scalar multiplicative result cryptotext multivector (SMRC) to a destination computing device, the method comprising: receiving by the intermediary computing system the cryptotext multivector (C) sent by the source computing device; multiplying by the intermediary computing system the unencrypted scalar data value (V) and the cryptotext multivector (C) using scalar-vector multiplication in order to obtain a scalar multiplicative result cryptotext multivector (SMRC); and sending by the intermediary computing system the scalar multiplicative result cryptotext multivector (SMRC) to the destination computing device.
  • C cryptotext multivector
  • V unencrypted scal
  • An embodiment of the present invention may further comprise a method for decrypting a scalar multiplicative result cryptotext multivector (SMRC) on a destination computing device received from an intermediary computing system that performed homomorphic scalar multiplication of a cryptotext multivector (C) originated from a source computing device and an unencrypted scalar data value (V), the method comprising:
  • the destination computing device receives by the destination computing device the scalar multiplicative result cryptotext multivector (SMRC) sent by the intermediary computing system; distributing by the source computing device a shared secret numeric value (S s ) into a shared secret multivector in accord with a shared secret coefficient distribution algorithm that is known to the source computing device and the destination computing device, the shared secret numeric value (S s ) being known or knowable to the source computing device and the destination computing device, but is kept secret from other devices not intended to have access to the numeric message data including the intermediary computing system; decrypting by the destination computing device the scalar multiplicative result cryptotext multivector (SMRC) as a decryption function of at least one Geometric Algebra geometric product operation on the scalar multiplicative result cryptotext multivector (SMRC) and an inverse of the shared secret multivector (S s ) into a scalar multiplicative result multivector (SMR) such that the decryption function provides a
  • An embodiment of the present invention may further comprise a scalar multiplicative homomorphic Enhanced Data-Centric Encryption (EDCE) system for scalar multiplicative homomorphic multiplication of a cryptotext encrypted data representation of a corresponding plain text data value and an unencrypted scalar data value
  • the scalar multiplicative homomorphic EDCE system comprising: a source computing device, wherein the source computing device further comprises: a source numeric message distribution subsystem that distributes a numeric message data value (M) into coefficients of a message multivector (M ) representing the numeric message data value (M) in accord with a homomorphic preserving mathematical relationship between an unencrypted numeric data value and multivector coefficients representing the unencrypted numeric data value that is known to the source computing device and a destination computing device; a source numeric shared secret distribution subsystem that distributes a shared secret numeric value (S s ) into coefficients of a shared secret multive
  • An embodiment of the present invention may further comprise a scalar multiplicative homomorphic Enhanced Data-Centric Encryption (EDCE) system source computing device for encrypting a numeric message data value (M) in order to transfer a cryptotext multivector (C) encrypted representation of the numeric message data value (M) to an intermediary computing system that will perform homomorphic scalar multiplication of the cryptotext multivector (C) and an unencrypted scalar data value (V) and deliver a result of the homomorphic scalar multiplication to a destination computing device
  • the scalar multiplicative homomorphic EDCE system source computing device comprising: a source numeric message distribution subsystem that distributes the numeric message data value (M) into coefficients of a message multivector (M) in accord with a homomorphic preserving mathematical relationship between an unencrypted numeric data value and multivector coefficients representing the unencrypted numeric data value that is known to the source computing device
  • An embodiment of the present invention may further comprise a scalar multiplicative homomorphic Enhanced Data-Centric Encryption (EDCE) system
  • EDCE Enhanced Data-Centric Encryption
  • the intermediary computing system for performing homomorphic scalar multiplication of a cryptotext multivector (C) encrypted data representation of a corresponding plain text numeric data value received from a source computing device and an unencrypted scalar data value (V) and delivering a homomorphic scalar multiplicative result cryptotext multivector (SMRC) to a destination computing device
  • the scalar multiplicative homomorphic EDCE system intermediary computing system comprising: an intermediary receive subsystem that receives the cryptotext multivector (C) sent by the source computing device; an intermediary homomorphic scalar multiplication subsystem that multiplies the unencrypted scalar data value (V) and the cryptotext multivector (C) using scalar-vector multiplication in order to obtain a scalar multiplicative result cryptotext multivector (SMRC); and an intermeidary send subsystem that sends the scalar multiplicative result cryptotext multivector (SMRC) to the destination
  • An embodiment of the present invention may further comprise a scalar multiplicative homomorphic Enhanced Data-Centric Encryption (EDCE) system destination computing device for decrypting a scalar multiplicative result cryptotext multivector (SMRC) received from an intermediary computing system that performed homomorphic scalar multiplication of a cryptotext multivector (C) originated from a source computing device and an unencrypted scalar data value (V), the scalar multiplicative homomorphic EDCE system destination computing device comprising: a destination receive subsystem that receives the scalar multiplicative result cryptotext multivector (SMRC) sent by the intermediary computing system; a destination numeric shared secret distribution subsystem that distributes a shared secret numeric value (S s ) into a shared secret multivector (S s ) in accord with a shared secret coefficient distribution algorithm that is known to the source computing device and the destination computing device, the shared secret numeric value (S s ) being known or know
  • FIG. 1 is a block diagram of the hardware implementation for a core encryption embodiment (i.e., a core Enhanced Data-Centric Encryption— EDCE— embodiment).
  • a core encryption embodiment i.e., a core Enhanced Data-Centric Encryption— EDCE— embodiment.
  • FIG. 2 is a flow chart of the general operation for a core encryption
  • FIG. 3 A is a flow chart of the source computing device symmetric key operation for a core encryption embodiment.
  • FIG. 3B is a flow chart of the destination computing device symmetric key operation for a core encryption embodiment.
  • FIG. 4A is a flow chart of the source computing device symmetric key and cryptotext masking operation for a core encryption embodiment.
  • FIG. 4B is a flow chart of the destination computing device symmetric key and cryptotext masking operation for an encryption embodiment.
  • FIG. 5 is a flow chart of a core encryption embodiment for the EDCE encryption/decryption by using a geometric product "sandwich.”
  • FIG. 6 is a flow chart of a core encryption embodiment for the EDCE encryption/decryption by using Sylvester's equation.
  • FIG. 7 is a block diagram illustrating generating/extracting/obtaining a second shared secret key from the original shared secret multivector for a core encryption embodiment
  • FIG. 8 is a block diagram of the hardware implementation for a scalar multiplicative homomorphic encryption embodiment.
  • FIG. 9 is a flow chart of the general operation for a scalar multiplicative homomorphic encryption embodiment.
  • FIG. 10 is a flow chart of the operations for sending a multiply command for a scalar multiplicative homomorphic encryption embodiment
  • Homomorphic Encryption is a form of encryption that allows computations to be carried out on cipher text as it is encrypted without decrypting the cipher text that generates an encrypted result which, when decrypted, matches the result of operations performed on the unencrypted plaintext.
  • the essential purpose of homomorphic encryption is to allow computation on encrypted data without decrypting the data in order to perform the computation.
  • the encrypted data can remain confidential and secure while the encrypted data is processed for the desired computation.
  • useful tasks may be accomplished on encrypted (i.e., confidential and secure) data residing in untrusted environments.
  • the ability to perform computations on encrypted data may be a highly desirable capability.
  • finding a general method for computing on encrypted data is likely a highly desirable goal for cryptography.
  • the most sought after application of homomorphic encryption may be for cloud computing.
  • Data that is stored in the Cloud is typically not encrypted, and the breach of the Cloud stored, unencrypted data is ranked by the Cloud Security Alliance as the number one threat to data security. Encrypting Cloud stored data may mitigate the threat of data being compromised by a breach, but then the remote clients (owners of the data) would not then be able to perform operations (i.e., add, multiply, etc.) on the Cloud stored data while the data remains in the Cloud. In order to perform operations on encrypted data stored in the Cloud, it would be necessary to download the encrypted Cloud stored data, decrypt the data, perform all desired operations on the data locally, encrypt the resulting data and send the resulting data back to the Cloud.
  • the Cloud would require access to the user's encryption keys. It is becoming increasing undesirable to provide the Cloud access to a user's security keys as the more entities that have access to the security keys inherently increases the susceptibility of the security keys to being breached, or even stolen by an unscrupulous provider.
  • Homomorphic encryption would allow the Cloud to operate on client data without decryption, and without access to the client's security keys.
  • An embodiment may advantageously utilize Geometric Algebra to provide the encryption and decryption of numeric messages that are to be transmitted through, and possibly have operations performed by, an intermediary computing system (e.g., the broad- based computing system currently, and commonly, referred to as the Cloud, or cloud computing).
  • an intermediary computing system e.g., the broad- based computing system currently, and commonly, referred to as the Cloud, or cloud computing.
  • An embodiment of the Geometric Algebra encryption/decryption system that performs the foundational "core" encryption/decryption functions of transferring data securely using Geometric Algebra based encryption/decryption from a source system to a destination system without having arithmetic or other comparative operations performed on the transmitted encrypted data by an intermediary system may be referred to as an Enhanced Data-Centric Encryption (EDCE) system.
  • EDCE Enhanced Data-Centric Encryption
  • an EDCE system When an EDCE system is further enhanced to support and provide for arithmetic and/or other comparative operations to be performed at an intermediary computing system (e.g., the Cloud) without decrypting and re-encrypting the data at the intermediary computing system, that system may be referred to as an Enhanced Data-Centric Homomorphic Encryption (EDCHE) system.
  • EDCHE Enhanced Data-Centric Homomorphic Encryption
  • Geometric Algebra is an area of mathematics that describes the geometric interaction of vectors and other objects in a context intended to mathematically represent physical interactions of objects in the physical world.
  • the use of Geometric Algebra for cryptography represents a new, manmade use of Geometric Algebra for a purpose entirely outside of the natural basis of Geometric Algebra for representing physical interactions of objects in the real, physical, word.
  • this area of mathematics encompasses Geometric Algebra, Conformal Geometric Algebra and Clifford Algebra (referred to collectively herein as "Geometric Algebra").
  • Geometric Algebra defines the operations, such as geometric product, inverses and identities, which facilitate many features of embodiments of the core EDCE and the EDCHE systems disclosed herein. Further, Geometric Algebra allows for the organization and representation of data into the "payload" of a multivector where the data in the payload may represent, for example, plaintext, cryptotext, or identifying signatures. Consequently, Embodiments of both the core EDCE system and the E iCHE system make beneficial use of Geometric Algebra properties to provide encryption, decryption, and intermediary homomorphic operations in a relatively computationally simplistic manner while still providing robust security for both data in motion and data at rest (e.g., data stored in the Cloud).
  • methods and systems to encrypt and decrypt messages using Geometric Algebra may utilize the intrinsic algebraic homomorphic properties of Geometric Algebra to permit arithmetic and other comparative operations on encrypted messages handled by an intermediary computing system without the need for the intermediary computing system to decrypt the encrypted messages prior to performing the arithmetic and other comparative operations. Accordingly, the intermediary computing system does not need to know any information regarding any of the secret security keys of the encryption/decryption processes to properly perform the arithmetic and other comparative operations.
  • the encrypted results of the arithmetic and other comparative operations performed by the intermediary computing system when decrypted at a destination computing device, produce results equivalent to the same operations as if the operations were performed on the unencrypted plain text messages.
  • a proper data organization methodology that preserves such homomorphic properties (i.e., the mathematical relationship between the vectors utilized in the encryption process and the original plaintext messages being encrypted) should be enforced on the choice of coefficients for the vectors representing the plain text messages.
  • an embodiment of an EDCHE system provides a cryptosystem that allows unlimited multiplications and additions of cipher text (i.e., transmitted/stored encrypted messages at the intermediary/cloud computer system) due solely to the intrinsic algebraic homomorphic properties of an embodiment of the EDCHE system.
  • an embodiment of an EDCHE system may provide the homomorphic properties as a product of algebraic homomorphism without the need to use additional methods, such as "bootstrapping" (e.g., performing a recursive operation to reduce the noise associated with a cipher text) to achieve the homomorphic properties.
  • bootsstrapping e.g., performing a recursive operation to reduce the noise associated with a cipher text
  • the encrypted data values may be stored on the intermediary computing system until such time that particular arithmetic or other comparative operations are desired by a user, then the intermediary computing system may perform the requested arithmetic or other comparative operations.
  • the encrypted data values may be immediately operated on by the intermediary computing system as soon as the subject encrypted data values are received by the intermediary computing system.
  • the process of receiving the encrypted data values at the intermediary computing system inherently includes storing the encrypted data values at the intermediary computing system even if only fleetingly in an immediately used and erased Random Access Memory (RAM) location or operational register location of a computational subsystem of the intermediary computing system.
  • RAM Random Access Memory
  • Embodiments of both EDCE and EDCHE may be comprised of functional blocks, each of which may be tailored as described in more detail below according to objectives for scope, capability and security. The following sections provide a mathematical and numerical description of these functional blocks.
  • Section 1 provides a general description of embodiments of the foundational "core” EDCE system.
  • Section 2 provides additional descriptions of embodiments of the foundational "core” EDCE system, including the packing of information into multivectors, the encryption and decryption of such multivectors and the unpacking to recover the original information.
  • Section 3 provides a description of the further enhancements to embodiments of the foundational "core” EDCE system that achieve homomorphic properties for embodiments of an EDCHE system.
  • Alice and Bob are used for the sending/source and receiving/destination entities, respectively.
  • Section 1 General Core EDCE Message Encryption/Decryption
  • Section 3 Homomorphic EDCHE Enhancements to EDCE Operation
  • Section 1 General Core EDCE Message Encryption/Decryption
  • Ciphers such as RSA (Rivest-Shamir-Adleman), DES (Data Encryption Standard) and/or AES (Advanced Encryption Standard) are little more than static "machinery" that bogs down communication efficiency. The actual problem is much bigger. How can robust security be provided when: a) End-point computational resources are limited (e.g., the Internet of Things— IoT). b) Encryption/decryption must be near-real time
  • a "core" embodiment may be described as enhanced data-centric encryption, or EDCE.
  • EDCE is computationally simplistic while providing robust security over the span of the communication channel.
  • EDCE security is scalable from tiny embedded IoT (Internet of Things) devices up to server farms.
  • EDCE functionality enables many cipher schemes that show speed and bandwidth advantages over current methods.
  • encryption/decryption of data is that the EDCE encryption/decryption may be implemented using basic arithmetic operations of addition, subtraction, multiplication, and division.
  • EDCE does not require a complex operation to select a large prime number, to calculate a logarithm function, to calculate a natural logarithm function, and/or to calculate other complex and computationally intensive mathematical functions (i.e., prime numbers, logarithms, natural logarithms, and/or other complex mathematical operations are not required in the Geometric Algebra calculations disclosed herein).
  • Geometric Algebra an area of mathematics that has not been utilized before in encryption.
  • Geometric Algebra as used herein is an area of mathematics that encompasses Geometric Algebra, Conformal Geometric Algebra and Clifford Algebra (collectively herein, "Geometric Algebra").
  • Geometric Algebra allows for the organization and representation of data into the "payload" of a multivector where the data may be plaintext, cryptotext, or signatures, for example.
  • Geometric Algebra defines the operations, such as geometric product, inverses and identities, which are the enablers of encryption/decryption calculations of various embodiments.
  • Multivectors are simply the additive combination of a scalar, a vector, a bi- vector and so forth up to an n-dimension vector.
  • the unit vectors follow the algebraic structure of quaternions (Hamilton) and non-commutative algebra (Grassman). These two types of algebra allowed Clifford to conceive of the Geometric Product which is used by the various embodiments as one of the "primitive" functions of the embodiments of EDCE and EDCHE systems.
  • An example of a two-dimension (2D) multivector A that includes a scalar and a vector is: where is a unit vector along the i-axis and represents the orientation of the area created by a 12 .
  • the operations of Geometric Algebra on multivectors are discussed more fully in Appendix A: Geometric Algebra Overview" of the parent patent application Serial No.
  • each of the a t values in the multivector A above may be "packed" with information and each ai value may range from zero to very large (e.g., >256,000 bits or an entire message).
  • each ai value may range from zero to very large (e.g., >256,000 bits or an entire message).
  • the inverse of A when multiplied by A yields unity, or:
  • the destination can recover B through:
  • the "payload" may be packed in the values of the scalers and coefficients of the multivector elements.
  • the packing method may define, among many things, the Geometric Algebra operations permissible for EDCE and/or EDCHE embodiments. For example, the Rationalize operation on multivectors yields zero when all multivector coefficients are equal. Such multivectors having all equal coefficients have no inverse and the geometric product of such multivectors having all equal coefficients with another multivector has no inverse.
  • the decryption methodology for EDCE and EDCHE systems utilize the inverse of the cryptotext multivector being decrypted and of the security key(s) multivector to perform the decryption.
  • the cryptotext multivector being decrypted should not have all equal value coefficients.
  • One means to ensure that the cryptotext multivector being decrypted does not have all equal value coefficients is to have the packing/coefficient distribution method ensure that not all coefficients are equal to each other (i.e., at least one coefficient should be different than the other coefficients) when creating the shared security multivector(s) and the data message multi vectors. For an embodiment of the EDCE that simply transfers the data message, this will ensure that the cryptotext multivector to be decrypted will not have all equivalent coefficients.
  • the same packing/coefficient distribution method to ensure that the source message multivectors do not have all equivalent coefficients will minimize the potential for the cryptotext multivector being decrypted from having all equivalent coefficients, but, when various addition and subtraction operations are performed with multiple distinctly different cryptotext multivectors, there is a remote possibility that the cryptotext multivector result of the homomorphic operations will have all equivalent coefficients.
  • the destination computing device may simply assert that such a result cryptotext multivector is "undefined," or, the destination or intermediary computing system may provide a means to update the result cryptotext multivector so the result cryptotext multivector does not have all equivalent coefficients. Great care should be taken to ensure that such an update of the result cryptotext multivector does not change the ultimate value of the result plaintext value of the result cryptotext multivector after decryption.
  • the "packed" multivector that represents the original plaintext numeric message have a mathematical relationship (i.e., the homomorphic preserving mathematical relationship) to the original plaintext numeric message.
  • the term homomorphism refers to a structure-preserving map between two algebraic structures, such as groups, rings, or vector spaces.
  • An algebra homomorphism between two algebras is one that preserves the algebra structure.
  • the method by which numbers are "packed" into multivector elements must remain a representation of the original number.
  • One such relationship for packing the coefficients of the multivector that preserves homomorphic properties is to ensure that the coefficients of the multivector representation of the plaintext numeric message follow a mathematical data organization between the value of the plaintext numeric message and at least one of the values of the coefficients of the multivector representation of the plaintext numeric message where the mathematical operations incorporating the one or more values of the multivector coefficients have a result equal to the original plaintext numeric message value.
  • the mathematical relationship may include: addition of at least one coefficient of the multivector coefficients, subtraction of at least one coefficient of the multivector coefficients, addition of a constant value, subtraction of a constant value, multiplication of at least one coefficient of the multivector coefficients by a constant value, and division of at least one coefficient of the multivector coefficients by a constant value.
  • the location of the various mathematical operations relative to the particular locations of the coefficients in the multivector representation should also be consistently applied to all source numeric data messages converted to a multivector as well as for result multivectors converted to a result numeric data value in a particular encryption/decryption pathway.
  • separate multivectors may be encoded for many purposes, such as a shared secret (defined below), authentication information, and timestamps.
  • a shared secret defined below
  • authentication information such as password, password, and password.
  • timestamps such as time and time.
  • the EDCE multivector format and Geometric Algebra foundation of a core EDCE embodiment may enable a single transmission to contain far more than just cryptotext, including dummy data to increase encryption security, command instructions for additional operations, and/or configuration data for the additional operations.
  • Fig. 1 is a block diagram 100 of the hardware implementation for an embodiment.
  • a first computing device 102 is connected over an electronic network/bus connection 104 to a second computing device 106.
  • the first computing device 102 acts as the source device 102 that sends the encrypted message 108 over the network/bus connection 104.
  • the second computing device 106 acts as the destination device 106 that receives the encrypted message 108 from the network/bus connection 104.
  • communications including encrypted communications, are bidirectional such that the first 102 and second 106 computing devices may change roles as the source device 102 and destination device 106 as is necessary to accommodate the transfer of data back and forth between the first 102 and second 106 computing devices.
  • the first computing device 102 appears to be a laptop computer and the second computing device 106 appears to be a tablet device.
  • any computing device capable of communication over any form of electronic network or bus communication platform may be one, or both of the first 102 and second 106 computing devices. Further, the first 102 and second computing devices 106 may actually be the same physical computing device communicating over an internal bus connection 104 with itself, but still desiring encrypted communication to ensure that an attacker cannot monitor the internal communications bus 104 to obtain sensitive data communications in an unencrypted format.
  • Various embodiments may implement the network/bus communications channel 104 using any communications channel 104 capable of transferring electronic data between the first 102 and second 106 computing devices.
  • the network/bus may implement the network/bus communications channel 104 using any communications channel 104 capable of transferring electronic data between the first 102 and second 106 computing devices.
  • the network/bus may implement the network/bus communications channel 104 using any communications channel 104 capable of transferring electronic data between the first 102 and second 106 computing devices.
  • the network/bus communications channel 104 may implement the network/bus communications channel 104 using any communications channel 104 capable of transferring electronic data between the first 102 and second 106 computing devices.
  • the network/bus may implement the network/bus communications channel 104 using any communications channel 104 capable of transferring electronic data between the first 102 and second 106 computing devices.
  • the network/bus may implement the network/bus communications channel 104 using any communications channel 104 capable of transferring electronic data between the first 102 and second 106 computing devices.
  • the network/bus may implement the network/bus
  • connection 104 may be an Internet connection routed over one or more different communications channels during transmission from the first 102 to the second 106 computing devices.
  • the network/bus communication connection 104 may be an internal communications bus of a computing device, or even the internal bus of a processing or memory storage Integrated Circuit (IC) chip, such as a memory chip or a Central
  • the network/bus communication channel 104 may utilize any medium capable of transmitting electronic data communications, including, but not limited to: wired communications, wireless electro-magnetic communications, fiber-optic cable communications, light/laser communications, sonic/sound communications, etc., and any combination thereof of the various communication channels.
  • the various embodiments may provide the control and management functions detailed herein via an application operating on the first 102 and/or second 106 computing devices.
  • the first 102 and/or second 106 computing devices may each be a computer or computer system, or any other electronic devices device capable of performing the communications and computations of an embodiment.
  • the first 102 and second 104 computing devices may include, but are not limited to: a general purpose computer, a laptop/portable computer, a tablet device, a smart phone, an industrial control computer, a data storage system controller, a CPU, a Graphical Processing Unit (GPU), an Application Specific Integrated Circuit (ASI), and/or a Field Programmable Gate Array (FPGA).
  • GPU Graphical Processing Unit
  • ASI Application Specific Integrated Circuit
  • FPGA Field Programmable Gate Array
  • the first 102 and second 106 computing devices may be the storage controller of a data storage media (e.g., the controller for a hard disk drive) such that data delivered to/from the data storage media is always encrypted so as to limit the ability of an attacker to ever have access to unencrypted data.
  • Embodiments may be provided as a computer program product which may include a computer-readable, or machine-readable, medium having stored thereon instructions which may be used to program/operate a computer (or other electronic devices) or computer system to perform a process or processes in accordance with the various embodiments.
  • the computer-readable medium may include, but is not limited to, hard disk drives, floppy diskettes, optical disks, Compact Disc Read-Only Memories (CD-ROMs), Digital Versatile Disc ROMS (DVD-ROMs), Universal Serial Bus (USB) memory sticks, magneto-optical disks, ROMs, random access memories (RAMs), Erasable Programmable ROMs (EPROMs), Electrically Erasable Programmable ROMs (EEP OMs), magnetic optical cards, flash memory, or other types of media/machine-readable medium suitable for storing electronic instructions.
  • the computer program instructions may reside and operate on a single computer/electronic device or various portions may be spread over multiple computers/devices that comprise a computer system.
  • embodiments may also be downloaded as a computer program product, wherein the program may be transferred from a remote computer to a requesting computer by way of data signals embodied in a carrier wave or other propagation medium via a communication link (e.g., a modem or network connection, including both wired/cabled and wireless connections).
  • a communication link e.g., a modem or network connection, including both wired/cabled and wireless connections.
  • Fig. 2 is a flow chart 200 of the general operation for an embodiment.
  • a shared secret numeric data value (S s ) is shared between the source 202 and destination 204.
  • the various embodiments may share the shared secret numeric data value (S s ) between the source 202 and destination 204 via any means desired by the users.
  • S s shared secret numeric data value
  • the shared secret numeric data value (S s ) may be shared between the source 202 and destination 204 by means including, but not limited to: pre-conditioning the source 202 computing device and the destination 204 computing device with the shared secret numeric value (S s ), a standard public/private key exchange technique, RSA (Rivest-Shamir-Adleman) key exchange, and/or Diffie-Hellman key exchange
  • the original shared secret may be an alphanumeric string in ASCII (American Standard Code for Information Exchange) or another encoding protocol that is converted to a numeric value based on the associated encoding protocol, such as: ASCII, other English language/alphabetic coding systems, foreign language encoding for non-alphabetic languages (e.g., katakana for Japanese), or even pure symbol to numeric values such as for emoji's.
  • ASCII American Standard Code for Information Exchange
  • other English language/alphabetic coding systems e.g., foreign language encoding for non-alphabetic languages (e.g., katakana for Japanese), or even pure symbol to numeric values such as for emoji's.
  • both the source 202 and destination 204 need to know and use the same alphanumeric text conversion into a numeric value process to ensure that results of both the source 202 and the destination 204 are the same.
  • the source 202 converts any alphanumeric text in the message into numeric message data (M ) based on the alphanumeric encoding protocol (e.g., ASCII, other English language/alphabetic coding systems, foreign language encoding for non- alphabetic languages (e.g., katakana for Japanese), or even pure symbol to numeric values such as for emoji's) of the original text
  • the alphanumeric encoding protocol e.g., ASCII, other English language/alphabetic coding systems, foreign language encoding for non- alphabetic languages (e.g., katakana for Japanese), or even pure symbol to numeric values such as for emoji's
  • numeric message data that is, but is not limited to: positive numbers, negative numbers, zero, integer numbers, rational numbers (including fractions), and/or real numbers.
  • the source 202 distributes the numeric message data (M) into message multivector (M ) coefficients.
  • the encryption system will work with just one non-zero message multivector (M ) coefficient, but, the more non-zero message multivector (M ) coefficients there are, the stronger the encryption will become, so it is desirable to have more than one non-zero message multivector (M ) coefficient.
  • the source 202 distributes shared secret numeric value (S s ) into shared secret multivector (S s ) coefficients.
  • S s shared secret numeric value
  • the encryption system will work with just one non-zero shared secret multivector (S s ) coefficient, but, the more non-zero shared secret multivector (S s ) coefficients there are, the stronger the encryption will become, so, again, it is desirable to have more than one non-zero shared secret multivector (S s ) coefficient.
  • S s shared secret numeric value
  • the primary requirement for the distribution process from the numeric values of the message (M ) and the shared secret (S s ) to the multivector coefficient values (Af and S s ) is that the source 202 and the destination 204 both know the processes 210/222 and 212/224 such that the destination 204 can reconstruct the original message (M ).
  • the distribution of numeric data to multivector coefficients may be performed differently between the message (M ) and the shared secret (S s ).
  • the various embodiments may perform the encryption process with multivector coefficient values for both the message (M ) and shared that are, but are not limited to: positive numbers, negative numbers, zero, integer numbers, rational numbers (including fractions), and/or real numbers.
  • the distributing/packing method defines, among many things, the Geometric Algebra operations permissible for EDCE and/or DCHE embodiments.
  • the Rationalize operation on multivectors yields zero when all multivector coefficients are equal.
  • Such multivectors having all equal coefficients have no inverse and the geometric product of such multivectors having all equal coefficients with another multivector has no inverse.
  • the Rationalize operation on multivectors yields zero when all multivector coefficients are equal.
  • Such multivectors having all equal coefficients have no inverse and the geometric product of such multivectors having all equal coefficients with another multivector has no inverse.
  • the decryption methodology for EDCE and EDCHE systems utilize the inverse of the cryptotext multivector being decrypted and of the security key(s) multivector to perform the decryption. Therefore, the cryptotext multivector being decrypted should not have all equal value coefficients.
  • One means to ensure that the cryptotext multivector being decrypted does not have all equal value coefficients is to have the packing/coefficient distribution method ensure that not all coefficients are equal to each other (i.e., at least one coefficient should be different than the other coefficients) when creating the shared security multivector(s) and the data message multivectors.
  • the same packing/coefficient distribution method to ensure that the source message multivectors do not have all equivalent coefficients will minimize the potential for the cryptotext multivector being decrypted from having all equivalent coefficients, but, when various addition and subtraction operations are performed with multiple distinctly different cryptotext multivectors, there is a remote possibility that the cryptotext multivector result of the homomorphic operations will have all equivalent coefficients.
  • the destination computing device may simply assert that such a result cryptotext multivector is "undefined," or, the destination or intermediary computing system may provide a means to update the result cryptotext multivector so the result cryptotext multivector does not have all equivalent coefficients.
  • the "packed" multivector that represents the original plaintext numeric message have a mathematical relationship (i.e., the homomorphic preserving mathematical relationship) to the original plaintext numeric message.
  • the term homomorphism refers to a structure-preserving map between two algebraic structures, such as groups, rings, or vector spaces.
  • An algebra homomorphism between two algebras is one that preserves the algebra structure.
  • the method by which numbers are "packed" into multivector elements must remain a representation of the original number.
  • One such relationship for packing the coefficients of the multivector that preserves homomorphic properties is to ensure that the coefficients of the multivector representation of the plaintext numeric message follow a mathematical data organization between the value of the plaintext numeric message and at least one of the values of the coefficients of the multivector representation of the plaintext numeric message where the mathematical operations incorporating the one or more values of the multivector coefficients have a result equal to the original plaintext numeric message value.
  • the mathematical relationship may include: addition of at least one coefficient of the multivector coefficients, subtraction of at least one coefficient of the multivector coefficients, addition of a constant value, subtraction of a constant value, multiplication of at least one coefficient of the multivector coefficients by a constant value, and division of at least one coefficient of the multivector coefficients by a constant value.
  • the location of the various mathematical operations relative to the particular locations of the coefficients in the multivector representation should also be consistently applied to all source numeric data messages converted to a multivector as well as for result multivectors converted to a result numeric data value in a particular encryption/decryption pathway.
  • the number of potential coefficients is directly related to the size/dimension (N) of the multivectors such that the number of coefficients increases by a factor of 2 (i.e., 2*) for each incremental increase in the size/dimension (N) of the multivector.
  • using multivectors of at least two dimensions will provide at least four coefficients to distribute the numeric data of the message (M) and the shared secret (S s ).
  • N the number of dimensions of multivectors beyond two-dimension
  • the confusion and/or diffusion security characteristics will also be increased due to the additionally available multivector coefficients. Further, with the additionally available coefficients it is also possible to transfer more data in a single multivector message (M) payload using the additionally available multivector coefficients.
  • M multivector message
  • the source 202 encrypts a cryptotext multivector (C) as a function of at least one Geometric Algebra geometric product operation on the message multivector (M) and the shared secret multivector (S s ).
  • the source 202 converts the cryptotext multivector (C) into cryptotext numeric data (C) in accord with reverse operation of a cryptotext data coefficient distribution algorithm that is known to both the source 202 and the destination 204. While not typical of most encryption systems, an embodiment may also omit process 216 and directly send a representation of the cryptotext multivector (C) without first converting the cryptotext multivector (C) into cryptotext numeric data (C).
  • the transmission may be implemented as a series of transfers of the coefficients or as some form of records/packets that define a data structure that carries the coefficient data of the cryptotext multivector (C).
  • the various embodiments will include process 216 to convert the cryptotext multivector (C) into cryptotext numeric data (C) in order to maintain compatibility with legacy and/or third parry systems as well as to obtain the additional confusion and diffusion characteristics of encapsulating the cryptotext multivector (£) coefficients into a single cryptotext numeric data (C) value.
  • process 216 is used to convert the cryptotext multivector (C) into cryptotext numeric data (C)
  • any computing device/system that wishes to operate on the cryptotext multivector (C) it is necessary for any computing device/system that wishes to operate on the cryptotext multivector (C) to have knowledge of the particular conversion methodology so that computing device/system may properly recreate the cryptotext multivector (C).
  • Some, but not all, of the potential geometric product calculations to encrypt the message data (M ) include: a geometric product of the message multivector ( ) and the shared secret multivector geometric product "sandwich" ( and multivector based Sylvester's equation .
  • the source 202 sends the cryptotext numeric data (C) to the destination 204.
  • the destination 204 receives the cryptotext numeric data (C) sent by the source 202.
  • the destination distributes the cryptotext numeric data (C) into the cryptotext multivector (C) using the cryptotext data coefficient distribution algorithm that is known to both the source 202 and the destination 204. For the less typical
  • process 222 is also omitted as the cryptotext multivector was transmitted directly so there is not a need to convert the cryptotext numeric data (C) back into the cryptotext multivector
  • the destination 204 distributes shared secret numeric value (S s ) into shared secret multivector (S s ) coefficients in the same fashion as was done for the source 202 at process 212.
  • the destination decrypts the cryptotext multivector as a function of at least one Geometric Algebra geometric product operation on the cryptotext multivector and an inverse ( *) of the shared secret multivector (S s ) back into the message multivector ( f).
  • Geometric Algebra an inverse of the shared secret multivector (S s ) back into the message multivector ( f).
  • M include: a geometric product of the cryptotext multivector (C) and the inverse of the shared secret multivector ( geometric product "sandwich" (
  • the destination 204 converts the message multivector into the message numeric data (M ) in accord with reverse operation of the message data coefficient distribution algorithm of the source 202 at process 210.
  • the destination 202 converts the numeric message data (M ) back into the original alphanumeric text message as a reverse function of the process of the source 202 at step 208 that converted that alphanumeric text to the numeric message data (M) using standard computer character encoding characteristics.
  • Fig. 3A is a flow chart 300 of the source computing device symmetric key operation for an embodiment.
  • the encryption process 214 of the source 202 of Fig. 2 may further include processes 302-306 to use symmetric shared secret security keys to further enhance the security of an embodiment
  • the source computing device may generate/extract/obtain a second shared secret key from the original shared secret
  • the 0-Blade Reduction Operation may be found as a geometric product
  • the source computing device distributes the second shared secret key numeric value into second shared secret multivector coefficients where also
  • the distribution of numeric data to multivector coefficients may be performed differently between the message (M), the original shared secret and the second shared secret key
  • the source computing device encrypts the cryptotext multivector (C) as a function of Geometric Algebra geometric product operations on the message multivector (M), the shared secret multivector (3 ⁇ 4), and the second shared secret multivector (3 ⁇ 4).
  • C cryptotext multivector
  • M message multivector
  • shared secret multivector 3 ⁇ 4
  • second shared secret multivector 3 ⁇ 4
  • Some, but not all, of the potential geometric product calculations to encrypt the message data (M) include: geometric product "sandwich" to encrypt); and multivector based Sylvester's equation
  • Fig. 3B is a flow chart 310 of the destination computing device symmetric key operation for an embodiment.
  • the decryption process 226 of the destination 204 of Fig. 2 in conjunction with the operation of the source computing device as described in the disclosure above with respect to Fig. 3 A, may include processes 312-316 to use symmetric shared secret security keys to further enhance the security of an embodiment.
  • the destination computing device may independently generate/extract/obtain the second shared secret key (S s _,) from the original shared secret multivector by performing the 0-Blade Reduction Operation on the original shared secret multivector to obtain a scalar numerical value for the second shared secret key Again, the 0-Blade Reduction
  • Operation may be found as a geometric product ( of the geometric product of the original shared secret multivector ) and a Clifford conjugate ) of the original shared secret multivector and the geometric reverse of the geometric product of the shared secret multivector S s ) and the Clifford conjugate
  • the destination computing device also distributes the second shared secret key numeric value into the second shared secret multivector ) coefficients.
  • the primary requirement for the distribution process from the numeric values of the second shared secret key S s _,) to the second shared secret multivector coefficient values ) is that the source computing device (of Fig. 3A) and the destination computing device (of Fig. 3B) both know the process 304/314 such that the destination computing device can reconstruct the original message (M) by being able to independently recreate the second shared secret multivector from the
  • the distribution of numeric data to multivector coefficients may be performed differently between the message (M), the original shared secret (S s ), and the second shared secret key ( 3 ⁇ 4
  • the destination computing device decrypts the cryptotext multivector ) as a function of Geometric Algebra geometric product operations on the cryptotext multivector ( of the original shared secret multivector
  • Fig. 4A is a flow chart 400 of the source computing device symmetric key and cryptotext masking operation for an embodiment. Similar to the disclosure with respect to Fig. 3A above, the encryption process 214 of the source 202 of Fig. 2 may further include processes 402-406 to use symmetric shared secret security keys to further enhance the security of an embodiment. At process 402, the source computing device may
  • a second shared secret key (S 3 ⁇ 4 ) from the original shared secret multivector by performing a 0-Blade Reduction Operation on the original shared secret multivector to obtain a scalar numerical value for the second shared secret key
  • the 0-Blade Reduction Operation may be found as a geometric product
  • the source computing device distributes the second shared secret key numeric value ) into second shared secret multivector coefficients where also
  • the distribution of numeric data to multivector coefficients may be performed differently between the message (M), the original shared secret (S s ), and the second shared secret key
  • the source computing device encrypts the cryptotext multivector ( ) as a function of Geometric Algebra geometric product operations on the message multivector (M), the shared secret multivector (S s ), and the second shared secret multivector 3 ⁇ 4 .
  • Geometric Algebra there are many possible variations of the geometric product operations that will provide similar degrees of confusion and diffusion.
  • Some, but not all, of the potential geometric product calculations to encrypt the message data (M) include: geometric product "sandwich" to encrypt); and multivector based Sylvester's equation
  • the cryptotext multivector ( ) is first converted into a pre-cipher cryptotext (C) in accord with reverse operation of a cryptotext data coefficient distribution algorithm that is known to both the source computing device (Fig. 4A) and the destination computing device (Fig. 4B).
  • Fig. 4B is a flow chart 410 of the destination computing device symmetric key and cryptotext masking operation for an embodiment.
  • the destination computing device then distributes the pre-cipher cryptotext numeric data (C') into the cryptotext multivector (C) using the cryptotext data coefficient distribution algorithm that is known to both the source and destination computing devices.
  • the remaining decryption process 226 of the destination 204 of Fig. 2 in conjunction with the operation of the source computing device as described in the disclosure above with respect to Fig. 4A, may include processes 414-418 to use symmetric shared secret security keys to further enhance the security of an embodiment.
  • the destination computing device may independently generate/extract/obtain the second shared secret key (3 ⁇ 4) from the original shared secret multivector (S s ) by per
  • the destination computing device also distributes the second shared secret key numeric valu ) into the second shared secret multivector
  • the destination computing device can reconstruct the original message (M ) by being able to independently recreate the second shared secret multivector (S s _) from the second shared secret key numerical value (S s2 ).
  • the distribution of numeric data to multivector coefficients may be performed differently between the message (M), the original shared secret (S s ), and the second shared secret key
  • the destination computing device decrypts the cryptotext multivector (C) as a function of Geometric Algebra geometric product operations on the cryptotext multivector of the original shared secret multivector and an inverse a of the second shared secret multivector back into the message multivector
  • C cryptotext multivector
  • Geometric Algebra geometric product operations on the cryptotext multivector of the original shared secret multivector and an inverse a of the second shared secret multivector back into the message multivector
  • FIG. 2-4 describe a methodology that may be embodied as a method or process
  • another embodiment may be recognized as a computer system, and/or as a source computer system and a destination computer system, that encrypts data, transfers the data, and decrypts the data by implementing the processes described above with respect to the flow chart and flow chart details of Figs. 2-4.
  • the computer system, and/or the source computer system and the destination computer system that encrypts data, transfers the data, and decrypts the data
  • one, or more, individual processes described above for the methodology may be broken down and represented as a subsystem of the overall encryption computer system.
  • a subsystem of the computer system, and/or the source computer system and the destination computer system, that encrypts data, transfers the data, and decrypts the data may be assigned, in whole or in part, to a particular hardware implemented system, such as a dedicated Application Specific Integrated Circuit (ASIC) or Field Programmable Gate Array (FPGA).
  • ASIC Application Specific Integrated Circuit
  • FPGA Field Programmable Gate Array
  • One or more subsystems, in whole or in part, may alternatively be implemented as software or firmware instructions defining the operation of a computer system with specific regard to the one or more subsystems implemented as software or firmware instructions.
  • the software or firmware instructions may cause the Central
  • Processing Unit memory, and/or other systems of a computer system to operate in particular accordance with the particular one or more subsystems designated features.
  • the disclosure below provides a simplified example of the operations and data relationships during the performance of a fundamental "core" EDCE embodiment
  • the amount of data, the type of data, and the particular data values shown and described in the example are not meant to represent any particular real system, but are provided only for the purpose of showing the operations and data relationships of an embodiment. Further, the embodiments described below are not meant to restrict operations to particular data types, encryption shared secret key exchange techniques, text to numeric and back conversion techniques, and/or number to multivector coefficient assignment techniques.
  • the various embodiments may be comprised of functional blocks, each of which may be tailored as described according to objectives for scope, capability and security.
  • the following sections provide a mathematical and numerical description of one or more example embodiments of these functional blocks.
  • the numerical results in the examples are generally derived from Geometric Algebra executing in the C programming language.
  • each text message needs to be converted to a number in order to become a valid operational unit for all DCE computations.
  • the numbers are typically shown in base 10, but the various embodiments may choose other number bases as desired by the system designer.
  • a hex (base 16) representation may provide particular advantages when dealing with ASCII numerical representations as standard ASCII has a representation based on the numbers 0-127 (i.e., 2 7 ), which is one power of two (i.e., hex is 2 8 ) less than the typical 8 bits represented by a hex number of xFF.
  • symbols such as the letters a, b, c and so on are represented in order formats (such as binary, decimal, octets, hexadecimal, etc.), which are described in the ASCII printable code chart, a table that presents the relationship between formats. So the letters “a,” “b” and “c” in ASCII decimal code are 97, 98 and 99, respectively.
  • ASCII_array_from_"message [109, 101. 115, 115, 97, 103, 101]
  • n n * 256 + ascii_array_from_message[i]
  • n 1835365235
  • entropy may be added at this step by performing transformations on the ASCII codes, such as addition or modulo operations, but those entropy adding operations may affect whether intermediary homomorphic operations may properly be performed on the message data as those entropy adding operations may adversely affect the mathematical relationship to the original message values. No such entropy adding transformations are used in the examples that follow.
  • a base 10 number is transmitted and received. From the above example of a message multivector, the coefficients are concatenated to form a number string.
  • the "number to text" conversion process for this number string also uses the ASCII printable code chart, but the recovery routine is different from the "text to number” conversion. The procedure is described below:
  • variable s which is an empty string that will become the final text recovered from the input number. (Note: the symbol “ "" " is from the C-language and means empty string)
  • the input number is 30792318992869221.
  • n 30792318992869221
  • any number in base 10 may be a coefficient of a multivector element.
  • a multivector may contain arbitrary data, or data that is a result of a series of operations.
  • a base 10 number may also be represented in multivector form by distributing pieces of this number string to the coefficients in the multivector.
  • Multivectors that are 2D have 4 elements/coefficients available to pack with pieces of this number string, a 3D multivector has 8 elements, and 4D has 16.
  • EDCE has been
  • this string may be a single coefficient of, say, a 2D multivector, as follows:
  • EDCE has been demonstrated where the number string distributed to an element of the multivector exceeds 4,000 digits. However, the base 10 number in our example will typically be "distributed" in an ad hoc manner across all the multivector elements, such as:
  • the above distribution is called “number to multivector.”
  • the method of distributing the number string may be according to any of a variety of algorithms as long as the method is known and used by both the sending and receiving entities.
  • the distribution algorithm may include shuffling of the assignments to elements, performing functional operations on numbers assigned to elements or changing the algorithm between messages in a conversation. More operations increase encryption entropy.
  • shuffling and other algorithms to increase cryptographic confusion may break the potential for
  • the distributing/packing method defines, among many things, the Geometric Algebra operations permissible for EDCE and/or EDC E embodiments. For example, the
  • One means to ensure that the cryptotext multivector being decrypted does not have all equal value coefficients is to have the packing/coefficient distribution method ensure that not all coefficients are equal to each other (i.e., at least one coefficient should be different than the other coefficients) when creating the shared security multivector(s) and the data message multivectors. For an embodiment of the EDCE that simply transfers the data message, this will ensure that the cryptotext multivector to be decrypted will not have all equivalent coefficients.
  • the same packing/coefficient distribution method to ensure that the source message multivectors do not have all equivalent coefficients will minimize the potential for the cryptotext multivector being decrypted from having all equivalent coefficients, but, when various addition and subtraction operations are performed with multiple distinctly different cryptotext multivectors, there is a remote possibility that the cryptotext multivector result of the homomorphic operations will have all equivalent coefficients.
  • the destination computing device may simply assert that such a result cryptotext multivector is "undefined," or, the destination or intermediary computing system may provide a means to update the result cryptotext multivector so the result cryptotext multivector does not have all equivalent coefficients. Great care should be taken to ensure that such an update of the result cryptotext multivector does not change the ultimate value of the result plaintext value of the result cryptotext multivector after decryption.
  • the "packed" multivector that represents the original plaintext numeric message have a mathematical relationship (i.e., the homomorphic preserving mathematical relationship) to the original plaintext numeric message.
  • the term homomorphism refers to a structure-preserving map between two algebraic structures, such as groups, rings, or vector spaces.
  • An algebra homomorphism between two algebras is one that preserves the algebra structure.
  • the method by which numbers are "packed" into multivector elements must remain a representation of the original number.
  • One such relationship for packing the coefficients of the multivector that preserves homomorphic properties is to ensure that the coefficients of the multivector representation of the plaintext numeric message follow a mathematical data organization between the value of the plaintext numeric message and at least one of the values of the coefficients of the multivector representation of the plaintext numeric message where the mathematical operations incorporating the one or more values of the multivector coefficients have a result equal to the original plaintext numeric message value.
  • the mathematical relationship may include: addition of at least one coefficient of the multivector coefficients, subtraction of at least one coefficient of the multivector coefficients, addition of a constant value, subtraction of a constant value, multiplication of at least one coefficient of the multivector coefficients by a constant value, and division of at least one coefficient of the multivector coefficients by a constant value.
  • the location of the various mathematical operations relative to the particular locations of the coefficients in the multivector representation should also be consistently applied to all source numeric data messages converted to a multivector as well as for result multivectors converted to a result numeric data value in a particular encryption/decryption pathway.
  • the number may be sent using a numeric variable representation such as an integer or floating point data type.
  • an embodiment may also simply skip the step of converting the multivector (C) into cryptotext numeric data (C), and directly send a representation of the cryptotext multivector (C) without first converting the cryptotext multivector (C) into cryptotext numeric data (C).
  • the transmission may be implemented as a series of transfers of the coefficients or as some form of records/packets that define a data structure that carries the coefficient data of the cryptotext multivector (C).
  • C cryptotext multivector
  • C cryptotext numeric data
  • a "Shared Secret” is a fundamental element in cryptography.
  • a Shared Secret enables secure communication between two or more parties.
  • the Shared Secret is a number string of digits that may be packed into a multivector in the he "Shared Secret Multivector" may be used to operate on other ating the geometric product of the Shared Secret Multivector and the message multivector.
  • the Diffie-Hellman protocol uses the multiplicative group of integers modulo p (see, for example,
  • Alice chooses a secret integer a (Alice's password) and creates her signature s and sends it to Bob. (Note: the superscript 0 is a placeholder for later use, if any.)
  • Diffie— Hellman protocol is not limited to negotiating a key shared by only two participants. Any number of users can take part in the agreement by performing iterations of the protocol and exchanging intermediate data.
  • the cryptotext is created using the EDCE primitive which is the geometric product of the Message multivector and one or more other multivectors.
  • the cryptotext multivector may be the geometric product of the Message multivector and the Shared Secret Multivector.
  • the cryptotext multivector can be defined as the geometric product:
  • C In order to be transmitted, as a payload, C now may be converted to a base 10 number, through the "multivector to number" conversion process described above.
  • the Geometric Product of the Message Multivector may be taken with more than one other multivector or by using the same multivector twice to form a sandwich or by the addition of left and right multivector operations on the same Shared Secret Multivector.
  • Two examples of these types are The use of these primitives and their inverse is shown in the flow charts in Figs. 3 and 4.
  • Cryptotext Multivector there are several alternative methods to construct the Cryptotext Multivector.
  • One alternative is to encrypt the plaintext message using a conventional symmetric cipher such as AES, converting the number string output of that cipher to multivector format and use this multivector in calculating the geometric product with S s , which yields C.
  • This alternative may be practiced during the transition to EDCE within the enterprise to preserve backward compatibility with legacy encryption systems.
  • the multivector M is converted to a base 10 number:
  • FIG. 5 is a flow chart 500 of an embodiment for the core Enhanced Data-Centric Encryption (EDCE) encryption/decryption performed by using a geometric product
  • EDCE Enhanced Data-Centric Encryption
  • Setup (502) The sequence is initiated by establishing the signature and shared secret multivectors.
  • the Diffie-Hellman procedure 508 is shown but other asymmetric key ciphers such as RSA may be used to generate a number string known only to the source 504 and the destination 506.
  • end-point devices may be "pre-conditioned" with a secret (number string) known to the system administrator from which the session multivectors may be constructed.
  • the Diffie-Hillman procedure 508 sets up/creates the shared secret keys 510 and then the setup 502 creates multivectors of the Diffie-Hillman keys 510 in the multivector setup 512.
  • Source (504) The Message Multivector 516 is constructed at the create message operation 514 by concatenating the message ASCII code string to a number string and then distributing that number to the coefficients of the message multivector at 514.
  • the method of distributing to coefficients uses a prescribed algorithm known and used by both the source 504 and the destination 506.
  • the Message Multivector 516 is then encrypted 518 by computing the geometric product of the message and Shared Secret multivectors.
  • Fig. 5 shows the Cryptotext Multivector 520 as the "sandwich" of geometric products The coefficients of the Cryptotext Multivector 520 are then concatenated into a base 10 number string, C (524), and transmitted through a user-defined dispatch function 526 over an electronic network/bus communication channel 528.
  • Destination (506) C (532) is received through a user-defined operation 530 and converted back to the Cryptotext Multivector 536 using the prescribed distribution method 534.
  • the destination 506 computes the multivector inverse of the Shared Secret Multivector and uses this result in the decrypt equations 538 such as to recover the
  • the Message Multivector 540 is then converted to a number string and then to plaintext at 542.
  • FIG. 6 is a flow chart 600 of an embodiment for the EDCE
  • Setup (602) The sequence is initiated by establishing the signature and shared secret multivectors.
  • the Diffie-Hellman procedure 608 is shown but other asymmetric key ciphers such as RSA may be used to generate a number string known only to the source 604 and the destination 606.
  • end-point devices may be "pre-conditioned" with a secret (number string) known to the system administrator from which the session multivectors may be constructed.
  • the Diffie-Hillman procedure 608 sets up/creates the shared secret keys 610 and then the setup 602 creates multivectors 612 of the Diffie-Hillman keys in the multivector setup 612.
  • Source (604) The Message Multivector 616 is constructed at the create message operation 614 by concatenating the message ASCII code string to a number string and then distributing that number to the coefficients of the message multivector at 614.
  • the method of distributing to coefficients uses a prescribed algorithm known and used by both the source 604 and the destination 606.
  • the Message Multivector 616 is then encrypted 618 by computing the geometric product of the message and Shared Secret multivectors.
  • Fig. 6 shows the Cryptotext Multivector as the sum of two geometric products The coefficients of the
  • Cryptotext Multivector 620 are then concatenated into a base 10 number string, C (624), and transmitted through a user-defined dispatch function 626 over an electronic network/bus communication channel 628.
  • Destination (606) C (632) is received through a user-defined operation 630 and converted back to the Cryptotext Multivector 636 using the prescribed distribution method 634.
  • the destination 606 computes the multivector inverse of the Shared Secret Multivector and uses this result in the decrypt equations 638 such as A
  • the decryption applies multivector based "Sylvester's Equation" to recover the Message Multivector 640.
  • the essage Multivector 640 is then converted to a number string and then to plaintext at 642.
  • a pair of symmetric shared secret keys may be used instead of a single shared secret key.
  • the original shared secret numeric key (S s ) may be loaded into a multivector representation, which may be denoted as
  • a 0-Blade Reduction Operation on the original shared secret multivector may be performed in order to extract/generate/obtain a scalar value that is the second shared secret numeric key
  • Fig. 7 is a block diagram 700 illustrating generating/extracting/obtaining a second shared secret multivector key (3 ⁇ 4) 712 from the original shared secret multivector (S s ) 704 for an embodiment.
  • the original shared secret multivector (S s ) 704 may be used to encrypt and decrypt data as the first shared secret multivector key of a pair of symmetric shared secret multivector keys.
  • the original shared secret multivector ( 704 is operated on by the O-Blade Reduction Operation
  • the 0-Blade Reduction Operation 706 results in the scalar value of the second shared secret numeric key (5 3 ⁇ 4 ) 708.
  • a number to multivector coefficient distribution process 710 converts the second shared secret numeric key ( 2 into a second shared secret multivector 712.
  • the second shared secret multivector may then be
  • the first encryption primitive can be created through a sequence of geometric products using the pair of keys generated via the 0-Blade Reduction Operation (described herein, above) as follows:
  • the decryption operation involves the closed-form solution of the Sylvester's equation for 3-dimensional multivector space as follows:
  • a multivector may act as a Geometric Algebra object such that components of multi-dimensions and Clifford k-vectors are present.
  • An example is: which shows the components:
  • a typical, but not the only, arithmetic function used for secret sharing is the Diffie-Hellman function, which is based on cyclic groups with element g; for example: where S s is a shared secret which can be used by both the source and destination sides and where the operation g ab mod p yields S s . This is standard in the cyber security field. . Unbreakable Primitive
  • the shared secret S s is changed to a multivector in the same or a similar manner, such as:
  • the multivector-based Sylvester's equation may be used to create a cipher.
  • the cryptotext multivector C is: because S is a result of a one-way function and S s is a shared secret, which, when operated on by the O-BIade Reduction Operation (which may sometimes be referenced herein as the function Z b O) becomes a new result embedding a new one-way function.
  • the first encryption primitive may be created through a sequence of geometric products using the pair of keys generated via the 0-Blade Reduction Operation (described above) as follows:
  • the decryption process may comprise the following steps:
  • the multivector based Sylvester's equation may be employed here to generate a second encryption primitive which also uses the pair of symmetric shared secret keys generated via the 0-Blade Reduction Operation (described above), as follows:
  • the cipher multivector C which is a result of the multivector based Sylvester's equation above, is converted into a number, denoted by C' and defined as a pre-cipher. This number is the information to be sent from the source computing device to the destination computing device.
  • the decryption operation involves the closed-form solution of the multivector based Sylvester's equation for 3-dimensional multivector space and the XOR 'unmask' previously described for the "sandwich" / triple product above.
  • the summarized processes are given below:
  • Section 3 Homomorphic EDCHE Enhancements to EDCE Operation
  • EDCHE is an extension to the EDCE cryptosystem described in more detail in Sections 1 and 2 above.
  • the extension to support homomorphic operations requires additional considerations, particularly in the organization of the data message multivector coefficients, but, for the most part, the extension to support homomorphic operations relies on the intrinsic algebraic homomorphism of the Geometric Algebra foundation that are part of the encryption/decryption functions of the core EDCE.
  • the Geometric Algebra geometric product operations that perform the actual encryption and decryption operations remain the same for both EDCE and EDCHE embodiments.
  • the handling of the security keys also remains the same for both EDCE and EDCHE embodiments, including the data organization for "number to multivector" operations and any restrictions thereon.
  • any operations to convert text to a number and/or operations to convert a number also remain the same for both EDCE and EDCHE embodiments.
  • the choice of whether or not to convert a cryptotext multivector (C) into cryptotext numeric data (C) prior to transmitting the encrypted data to a destination system (or to an intermediary system along the path to the final destination) remains the same for both EDCE and EDCHE embodiments and the processes to convert between cryptotext multivector (C) and cryptotext numeric data (C) also remain the same between EDCE and EDCHE, except there may be some restrictions on the types of permissible operations allowed for EDCHE embodiments to ensure that an intermediary computing system operating on the encrypted data does not need knowledge of any security keys involved in the encryption/decryption process.
  • the EDCHE embodiments add further restrictions that the data organization preserves homomorphic properties (i.e., have a homomorphic preserving mathematical relationship between the vectors utilized in the encryption process and the original plaintext messages being encrypted).
  • an EDCHE embodiment simply adds restrictions to an EDCE system regarding data organization operations for the multivector representation of the data being encrypted as well as to conversions between a cryptotext multivector (C) and a cryptotext numeric data (C)
  • EDCHE embodiments being a subset of EDCE embodiments may operate as EDCE embodiments, but EDCE embodiments may not all necessarily operate as EDCHE embodiments.
  • Potential homomorphic encryption operations for an EDCHE embodiment may include multiple operations, such as, but not limited to: encrypted addition/subtraction, scalar addition/subtraction, encrypted multiplication, scalar multiplication, encrypted searching, and encrypted sorting.
  • Each of the potential homomorphic encryption operations involves operations particular to the particular operation. For this reason, separate disclosures for each particular operation may be presented to so as to make the clear the specific details comprising the implementation of each particular operation.
  • An additional document that briefly presents the combination of the potential homomorphic operations may also be separately presented that provides a brief summary of each operation and provides the additional details for performing combinations of the potential encryption operations. In view of potential disclosures, this particular disclosure is intended to address the specific details that comprise the particular details of scalar multiplicative homomorphic encryption operations.
  • client may be used to describe the owner of the operand, the issuer of the instructions/command, and/or receiver of the result data (i.e., the source, command, and/or destination computing device/system), while the generic term “cloud” may be used for data that is at rest in an untrusted environment (i.e., the intermediary computing system/device).
  • the terms “client” and “cloud” may more closely reflect a real world application where the source, command, and destination are the same entity, sending data to the intermediary "cloud” for storage, then requesting an operation from the cloud when needed by the client (e.g., a scalar multiplication product of a stored transaction dollar amount stored in encrypted format on the cloud against some unencrypted scalar value such as an interest rate).
  • client e.g., a scalar multiplication product of a stored transaction dollar amount stored in encrypted format on the cloud against some unencrypted scalar value such as an interest rate.
  • the terms source, command, destination, and intermediary reflect the relative operations being performed by a computing system/device, and do not necessarily define the computing system/device as a whole.
  • the source, command, destination, and intermediary operations/systems may reside as a particular application on one or more computing systems/devices.
  • source, command, destination, and intermediary computing systems/devices may be general purpose computing systems capable of running multiple applications at the same time, it is inherently possible that the source, command, destination, and intermediary operations are encapsulated as separate applications/functions that may permit, one, two, or all of the separate
  • a single interconnected computer system of a single owner/client may have untrusted environments that include data that is at rest (i.e., stored) in the owner/client's own end-point devices outside of the owner/client's digital secure perimeter that would have a security benefit by acting as the intermediary system without access to the security key(s).
  • homomorphism refers to a structure-preserving map between two algebraic structures, such as groups, rings, or vector spaces.
  • An algebra homomorphism between two algebras is one that preserves the algebra structure.
  • the method by which numbers are "packed" (i.e., distributed) into multivector coefficient elements should necessarily maintain some mathematical representation of the original number. Consequently, the packing/distribution method may define, among many things, the Geometric Algebra operations permissible for an EDCHE embodiment. For example, the Rationalize operation on multivectors yields zero when all multivector coefficients are equal.
  • Such multivectors having all equal coefficients have no inverse and the geometric product of such multivectors having all equal coefficients with another multivector has no inverse.
  • the decryption methodology for EDCE and EDCHE systems utilize the inverse of the cryptotext multivector being decrypted and of the security key(s) multivector to perform the decryption. Therefore, the cryptotext multivector being decrypted should not have all equal value coefficients.
  • One means to ensure that the cryptotext multivector being decrypted does not have all equal value coefficients is to have the packing/coefficient distribution method ensure that not all coefficients are equal to each other (i.e., at least one coefficient should be different than the other coefficients) when creating the shared security multivector(s) and the data message multivectors. For an embodiment of the EDCE that simply transfers the data message, this will ensure that the cryptotext multivector to be decrypted will not have all equivalent coefficients. For an EDC E system that may perform operations involving multiple encrypted data values, the same packing/coefficient distribution method to ensure that the source message multivectors do not have all equivalent coefficients will minimize the potential for the cryptotext multivector being decrypted from having all equivalent coefficients.
  • the methods and systems that encrypt and decrypt messages using Geometric Algebra may utilize the intrinsic algebraic homomorphic properties of Geometric Algebra to permit arithmetic and other comparative operations on encrypted messages handled by an intermediary computing system without the need for the intermediary computing system to decrypt the encrypted messages prior to performing the arithmetic and other comparative operations. Accordingly, the intermediary computing system does not need to know any information regarding any of the secret security keys of the encryption/decryption processes to properly perform the arithmetic and other comparative operations.
  • the encrypted results of the arithmetic and other comparative operations performed by the intermediary computing system when decrypted at a destination computing device, produce results equivalent to the same operations as if the operations were performed on the unencrypted plain text messages.
  • a proper data organization methodology i.e., packing/distributing coefficients into a multivector
  • preserves such homomorphic properties i.e., the mathematical relationship between the vectors utilized in the encryption process and the original plaintext messages being encrypted
  • the distribution/packing data arrangement should also preserve a commutative mathematical relationship to the original numeric value being encrypted.
  • the method by which numbers are "packed" into multivector elements must remain a representation of the original number.
  • One such relationship for packing the coefficients of the multivector that preserves homomorphic properties is to ensure that the coefficients of the multivector representation of the plaintext numeric message follow a mathematical data organization between the value of the plaintext numeric message and at least one of the values of the coefficients of the multivector representation of the plaintext numeric message where the mathematical operations incorporating the one or more values of the multivector coefficients have a result equal to the original plaintext numeric message value (i.e., the homomorphic preserving mathematical relationship).
  • the mathematical relationship may include: addition of at least one coefficient of the multivector coefficients, subtraction of at least one coefficient of the multivector coefficients, addition of a constant value, subtraction of a constant value, multiplication of at least one coefficient of the multivector coefficients by a constant value, and division of at least one coefficient of the multivector coefficients by a constant value.
  • the location of the various mathematical operations relative to the particular locations of the coefficients in the multivector representation should also be consistently applied to all source numeric data messages converted to a multivector as well as for result multivectors converted to a result numeric data value in a particular encryption/decryption pathway.
  • an embodiment of an EDCHE system provides a cryptosystem mat allows unlimited multiplications and additions of cipher text (i.e., transmitted/stored encrypted messages at the intermediary/cloud computer system) due solely to the intrinsic algebraic homomorphic properties of an embodiment of the EDC E system.
  • an embodiment of an EDCHE system may provide the homomorphic properties as a product of algebraic homomorphism without the need to use additional methods, such as "bootstrapping" (e.g., performing a recursive operation to reduce the noise associated with a cipher text) to achieve the homomorphic properties.
  • bootsstrapping e.g., performing a recursive operation to reduce the noise associated with a cipher text
  • homomorphism refers to a structure-preserving map between two algebraic structures, such as groups, rings, or vector spaces.
  • An algebra homomorphism between two algebras is one that preserves the algebra structure.
  • the method by which numbers are "packed" into multivector elements must remain a representation of the original number.
  • One such relationship for packing the coefficients of the multivector that preserves homomorphic properties is to ensure that the coefficients of the multivector representation of the plaintext numeric message follow a mathematical data organization between the value of the plaintext numeric message and at least one of the values of the coefficients of the multivector representation of the plaintext numeric message where the mathematical operations incorporating the one or more values of the multivector coefficients have a result equal to the original plaintext numeric message value.
  • the mathematical relationship may include: addition of at least one coefficient of the multivector coefficients, subtraction of at least one coefficient of the multivector coefficients, addition of a constant value, subtraction of a constant value, multiplication of at least one coefficient of the multivector coefficients by a constant value, and division of at least one coefficient of the multivector coefficients by a constant value.
  • the location of the various mathematical operations relative to the particular locations of the coefficients in the multivector representation should also be consistently applied to all source numeric data messages converted to a multivector as well as for result multivectors converted to a result numeric data value in a particular encryption/decryption pathway. For example, in a homomorphic preserving mathematical relationship that includes both addition and subtraction operations might be, for a three dimensional multivector there are eight possible coefficients in the multivector representation
  • the actual values of the coefficients may be selected as desired by a user so long as the homomorphic preserving mathematical relationship equals the original numeric value being encrypted.
  • the only other restriction for multivectors representing different numeric values within a homomorphic operation is that the multivectors all share the same dimensionality (i.e., all multivectors are 2D, all multivectors are 3D, all multivectors 4D, etc.).
  • the first step in representing a numeric message value in a multivector is to define the number of coefficients that are present in the multivector.
  • the total number of coefficient elements to be determined is eight
  • the coefficient selections may encompass any number that may be represented in the data type chosen by a system designer to hold the coefficient values, including, but not limited to: positive numbers, negative numbers, zero, integer numbers, rational numbers (including fractions), and/or real numbers.
  • positive numbers negative numbers
  • integer numbers integer numbers
  • rational numbers including fractions
  • real numbers real numbers.
  • the format of a multivector will always assume that the coefficients add together. For instance, the form for a 3D multivector representing the numeric value N would be:
  • the EDCHE embodiment performs the homomorphic preserving mathematical relationship process such that all coefficients are added together and there are not any coefficients that are subtracted.
  • a first step might be to factorize the numeric message N and write the value N as a sum of other numbers (ci) that are to represent the numeric message value in a message multivector such that:
  • One skilled in the art will recognize that at this point there are many ways to assign the coefficient values (ci) that will satisfy the homomorphic preserving mathematical relationship equation to enable proper operation of an EDCHE embodiment.
  • N mod n i.e., 5487 mod 8
  • cm last coefficient value
  • N C0 + Cl + C2 + C3 + CI2 + CJ3 + C23 + CJ23
  • the assignment of which coefficient value receives the addition of the N mod n operation may be random, or the assignment may be of a predetermined form designed to increase entropy to enhance the encryption security.
  • N C0 + CJ + C2 + C3 + C12 + C ⁇ 3 + C23 + CJ23
  • the EDCHE embodiment performs the homomorphic preserving mathematical relationship process such that all coefficients are added together and there are not any coefficients that are subtracted.
  • the homomorphic preserving mathematical relationship equation to represent the numeric value N would, again, be:
  • N C0 + C] + C2 + C3 + C/2 + CJ3 + C23 + C123
  • N C0 + CI + C2 + C3 + C12 + CI3 + C23 + C123
  • the homomorphic preserving mathematical relationship equation that is set equal to the numeric value N may be defined to include some subtraction of some coefficients, addition of a constant value, and multiplication of coefficient values by a constant, as well as the omission of one of the coefficients (i.e., cni) from the homomorphic preserving mathematical relationship.
  • the homomorphic preserving mathematical relationship equation to represent the numeric value N might now be:
  • N CO + C1 - C2 + C3 - C12 + 3 * C/3 + C23 + 23
  • a modified version of the methodology of either the first example (use a mathematical equation to calculate the coefficient values) or the second example (randomly generated coefficient values) described above may be used to determine the coefficient values given that the homomorphic preserving mathematical relationship now subtracts some coefficients, adds a constant, multiplies a coefficient by a constant, and omits one coefficient from the homomorphic preserving mathematical relationship altogether.
  • the homomorphic preserving mathematical relationship may be written as:
  • N C0 + Cr - C2 + C3 - C12 + 3 * C13 + C23 + 23
  • multivector N co + C1 e 1 + C2 e 2 + C3 e 3 + c12 e 1 2 + en e 1 3 + C23 e_a + C123 e 1 23
  • multivector N (901) + (985)e1 + (185)e 2 + (-584)e 3 + (286)e1 2 + (882)e1 3 + (1987)e 2 3 +
  • c123 value of 333 is a dummy value not included in the homomorphic preserving mathematical relationship, but may potentially be used to provide other features such as signature capability and/or passing of command or other information.
  • a homomorphic preserving mathematical relationship that includes some subtractive elements has the advantage of being able to represent negative numbers and zero without the coefficient values being negative for a user that prefers to not have negative coefficient values.
  • numeric value from the coefficients of a numeric data message multivector is relatively simple and straight forward.
  • To obtain the numeric data message value simply perform the homomorphic preserving mathematical relationship equation for the numeric data message multivector using the values of the multivector coefficients plugged into the homomorphic preserving mathematical relationship equation.
  • the examples given below provide the "multivector to number” process appropriate for the same example number as described above for the "number to multivector" process.
  • homomorphic preserving mathematical relationship process such that all coefficients are added together and there are not any coefficients that are subtracted.
  • the homomorphic preserving mathematical relationship equation to represent the result numeric value N would be:
  • N ' CO + CI + C2 + C3 + C12 + C13 + C23 + CJ23
  • the multivector has the form of: multivector N
  • N C0 + CI + C2 + C3 + C12 + CJ3 + C23 + CJ23
  • N (725) + (21) + (685) + (286) + (721) + (85) + (601) + (192)
  • N 725 + 21 + 685 + 286 + 721 + 85 + 601 + 192 such that result numeric value N would be:
  • the EDCHE embodiment performed the homomorphic preserving mathematical relationship process such that all coefficients are added together and there are not any coefficients that are subtracted, which is the same homomorphic preserving mathematical relationship equation as for the first example above. Consequently, the "multivector to number" process is identical to that as described for the "multivector to number” process of the first example given above.
  • the EDCHE embodiment performed the homomorphic preserving mathematical relationship process such that all coefficients are added together and there are not any coefficients that are subtracted.
  • the third example from above changed the homomorphic preserving mathematical relationship equation to include some subtraction of some coefficients, addition of a constant value, and multiplication of coefficient values by a constant, as well as the omission of one of the coefficients (i.e., cm) from the homomorphic preserving mathematical relationship.
  • N ' CO + C1 - C2 + C3 - C12 + 3 * C13 + C23 + 23
  • multivector N 725 + 21e 1 + 685 e2 + 286e3 - 721e 1 2 + 85e 13 + 601e23 + 192e 1 23 and knowing the multivector is of the form: multivector N
  • multivector N (725) + (21)e1 + (685)e 2 + (286)e 3 + (-721)e1 2 + (85)e1 3 + (601)e 2 3 +
  • N C0 + Cl - C2 + C3 - C12 + 3 * CJ3 + C23 + 23
  • N 1947 where the C123 value of 192 is ignored as a dummy value not included in the homomorphic preserving mathematical relationship. Note that the third example result of 1947 does not equal the first example result of 3316 for the same result multivector. Thus, demonstrating the necessity of using the same homomorphic preserving mathematical relationship equation for all encrypted multivectors of any homomorphic operations that may involve multiple encrypted multivectors.
  • multivectorN 333 - 201e 1 + 248e 2 + 506e 3 - 71e 1 2 + 80e 13 + 211e23 - 743en3 and knowing the result multivector with the given homomorphic preserving mathematical relationship equation is of the form:
  • multivector N (333) + (-201)e1 + (248)e2 + (506)e3 + (-71)e12 + (80)e1 3 + (21 l)e 2 3 + (-
  • N CO + Cl - C2 + C3 - C12 + 3 * CI3 + C23 + 23
  • N (333) + (-201) - (248) + (506) - (-71) + 3 * (80) + (211) + 23
  • N 333 - 201 - 248 + 506 + 71 + 240 + 211 + 23 such that result numeric value N would be:
  • ⁇ is the encryption operation
  • k is a scalar (i.e, not a vector) value multiplied (or divided) equivalently against all portions of an encrypted vector representation of the value xi.
  • the unencrypted "product" of a plaintext message and a scalar value is equal to the comparable encrypted "product" of the encrypted message multiplied by the same scalar value the operation is scalar multiplicative homomorphic.
  • the term "multiplied" for scalar multiplicative homomorphic operations includes both multiplication and division operations.
  • the plaintext message and the corresponding encrypted messages should be multiplied and
  • the interest rate (ir) represents the interest rate for a particular period of time, which is usually one year for most financial transactions.
  • the investment may be represented as multivector as follows:
  • the investment may then be encrypted by applying the triple product technique as follows:
  • the intermediary i.e., cloud
  • Alice acting as the source computing device may send the encrypted value (C) of the investment Ij to the intermediary (i.e., cloud) computing system.
  • the intermediary computing system may perform additional operations to achieve a desired result based on the given information.
  • the intermediary/cloud computing system may then multiply the encrypted multivector value by the unencrypted scalar value FVir in order to obtain the future value (FV) of the investment (//) as updated encrypted value
  • computing system may then send the updated encrypted value to Alice, now acting as the destination computing system.
  • Alice receives the calculated encrypted information ) from the cloud, she may decrypt the information and see the updated future value (FV) of her investment.
  • FV future value
  • the future value (FV) of the investment may be calculated by the intermediary/cloud computing system as follows:
  • the intermediary/cloud computing system may then send the encrypted multivector to Alice and Alice may then recover the updated investment ) as follows:
  • the intermediary/cloud system may calculate the scalar data value to be (1 + 0.008) 5 which equals 1.02525, then multiply the encrypted investment value by 1.02525 instead of 1.008 as was detailed in the example above to obtain a FV value of $10,765.14.
  • Alice may define the interest rate, number of compounding periods, and or other information in the instructions sent to the intermediary/cloud computing system.
  • Alice may instruct the intermediary/cloud computing system to look up values, such as looking up a public interest rate or other information to combine to achieve the same result.
  • the unencrypted scalar data value that is to be multiplied by the encrypted data value may be a simple predefined data value or the unencrypted scalar data value may be the result of several operations on data provided in the multiply command instructions and/or data looked up and obtained from privately or publicly available data storage.
  • the EDCHE system may support decimal (i.e., floating point, fixed point, or other real number representation) arithmetic of the encrypted data value and/or the unencrypted scalar data value to be multiplied with the encrypted data value.
  • Fig. 8 is a block diagram 800 of the hardware implementation for a scalar multiplicative homomorphic encryption embodiment.
  • a source computing device 802 is connected over an electronic network/bus connection 804 to an intermediary (e.g., cloud) computing device 806.
  • the source computing device 802 sends the cryptotext multivector 810 that will be scalar multiplied with an unencrypted scalar data value through the scalar multiplicative homomorphism of an EDCHE embodiment at the intermediary computing system 806 over the network/bus connection 804 to the intermediary computing system 806.
  • the intermediary computing system 806 receives the cryptotext multivector 810 from the network/bus connection 804.
  • the unencrypted scalar data value that is to be scalar-vector multiplied with the cryptotext multivector 810 may be predefined at the intermediary computing system, such as for the case of operations regarding a fixed interest rate with regard to encrypted investment values. Further, the predefined interest rate may further be used to perform additional calculations such as compounded interest calculations over multiple compounding periods. One skilled in the art will further recognize that other types of calculations outside of interest rate calculations may involve multiple arithmetic operations in order to obtain the final unencrypted scalar data value that will be scalar multiplied with the cryptotext multivector 810.
  • While one embodiment may have predefined scalar data values stored or otherwise accessible to the intermediary computing system 806, other embodiments may receive some, or all, data necessary to determine the scalar data value for multiplication in scalar multiplication instructions 816 sent over the communications network/bus 804 from a command computing system 814.
  • the scalar multiplication instructions may contain scalar data value information that defines the scalar data value multiplied with the cryptotext multivector 810.
  • the scalar data value information may define the unencrypted scalar data value that is to be multiplied by the encrypted data value as a simple predefined data value, or the scalar data value may be defined to be the result of several operations on data provided in the scalar data value information of the scalar multiplication instructions 816, and/or the scalar data value information may provide instructions to look up some or all of the necessary data in privately or publicly available data storage accessible by the intermediary computing system 806. Many, maybe even most, of the time, the command computing device 814 may be the same device as the source computing device 802 as it is often the case that the provider of the encrypted data is also the entity that wishes to have operations performed on the encrypted data.
  • the ultimate destination computing device 808 is the same as the source 802 and command 814 computing devices. While the source 802, command 814, and/or destination 808 computing devices may commonly be one and the same device, the operation of an EDCHE embodiment does not require that the functions be performed on a single computing device. Likewise, while less common in actual practice, the intermediary computing system 806 may also be the same computing device as one, two or all of the source 802, command 814, and/or destination 808 computing devices. Again, while devices may share the implementation of the functions of source 802, command 814, intermediary 806, and/or destination 808 computing systems, there is not a requirement for any such sharing for an EDCHE embodiment.
  • the intermediary computing system 806 may immediately perform a "scalar multiplication" of the cryptotext multivector 810 and a scalar data value using scalar-vector multiplication (as instructed by a user 816 or simply in response to receiving the cryptotext multivector 810) or the intermediary computing system 806 may store the cryptotext multivector 810 until such time that the intermediary computing system 806 is instructed 816 to perform the homomorphic scalar multiplication operation. Once the homomorphic scalar multiplication operation is completed by the intermediary computing system 806, the intermediary computing system sends the encrypted homomorphic scalar multiplicative result multivector 812 to the destination computing system 808 over the network/bus
  • the destination computing system 808 receives the encrypted homomorphic scalar multiplicative result multivector 812 from the network/bus communication connection 804 and decrypts the encrypted homomorphic scalar
  • multiplicative result multivector 812 to obtain the desired plaintext scalar multiplication result.
  • the cryptotext multivector 810 may be converted to a non-multivector cryptotext when being sent over the network/bus communication connection 804, then converted back into the cryptotext multivector at the intermediary computing system 806 for scalar multiplicative homomorphic operations.
  • the encrypted homomorphic scalar multiplication result multivector 812 may be converted to a non-multivector scalar multiplication result cryptotext when being sent over the network/bus communication connection 804, then converted back into the encrypted homomorphic scalar multiplication result multivector 812 at the destination computing device 808 for decryption by the destination computing device 808 into the plaintext scalar multiplicative result.
  • communications are bidirectional such that the source computing device 802, the command computing device 814, the intermediary computing system 806, and/or the destination computing device 808 may change roles so as to operate as a source computing device 802, the command computing device 814, the intermediary computing system 806, and/or the destination computing device 808 as is necessary to accommodate the transfer of data back and forth between the source 802, command 814, and destination 808 computing devices as well as for computation of homomorphic scalar multiplication products at the intermediary computing system 806.
  • the source computing device 802 and command computing device 814 appear to be laptop computers and the destination computing device 808 appears to be a tablet device.
  • communication over any form of electronic network or bus communication platform 804 may be one, multiple or all of the source computing device 802, the command computing device 814, the intermediary computing system 806, and/or the destination computing device.
  • source 802, command 814, intermediary 806, and destination computing devices/systems 808 may actually be the same physical computing device communicating over an internal bus connection 804 with itself, but still desiring encrypted communication to ensure that an attacker cannot monitor the internal communications bus 804 or hack an unprotected area of the computing system (i.e., the intermediary section 806) in order to obtain sensitive data communications in an unencrypted format
  • Various embodiments may implement the network/bus communications channel 804 using any communications channel 804 capable of transferring electronic data between the source 802, command 814, intermediary 806, and/or destination 808 computing devices/systems.
  • the network/bus communication connection 804 may be an Internet connection routed over one or more different communications channels during transmission from the source 802 and/or command 814 computing devices to the intermediary 806 computing system, and men onto the destination computing device 808.
  • the network/bus communication connection 804 may be an internal
  • the network/bus communication channel 804 may utilize any medium capable of transmitting electronic data communications, including, hut not limited to: wired communications, wireless electro-magnetic communications, fiber-optic cable
  • the various embodiments may provide the control and management functions detailed herein via an application operating on the source 802, command 814, intermediary 806, and/or destination 808 computing devices/systems.
  • T ' he source 802, command 814, intermediary 806, and/or destination 808 computing devices/systems may each be a computer or computer system, or any other electronic devices device capable of performing the communications and computations of an embodiment.
  • the source 802, command 814, intermediary 806, and/or destination 808 computing devices/systems may include, but are not limited to: a general purpose computer, a laptop/portable computer, a tablet device, a smart phone, an industrial control computer, a data storage system controller, a CPU, a Graphical Processing Unit (GPU), an Application Specific integrated Circuit (ASI), and/or a Field Programmable Gate Array (PPGA).
  • the first 102 and second 106 computing devices may he the storage controller of a data storage media (e.g., the controller for a hard disk drive) such that data delivered to/from the data storage media is always encrypted so as to limit the ability of an attacker to ever have access to unencrypted data.
  • a data storage media e.g., the controller for a hard disk drive
  • Embodiments may be provided as a computer program product which may include a computer-readable, or machine-readable, medium, having stored thereon instructions which may be used to program/operate a computer (or other electronic devices) or computer system to perform a process or processes in accordance with the various embodiments.
  • the computer-readable medium may include, but is not limited to, hard disk drives, floppy diskettes, optical disks, Compact Disc Read-Only Memories (CD-ROMs), Digital Versatile Disc ROMS (DVD- ROMs), Universal Serial Bus (USB) memory sticks, magneto-optical disks, ROMs, random access memories (RAMs), Erasable Programmable ROMs (EPROMs), Electrically Erasable Programmable ROMs (EEPROMs), magnetic optical cards, flash memory, or other types of media/machine-readable medium suitable for storing electronic instructions.
  • the computer program instructions may reside and operate on a single computer/electronic device or various portions may be spread over multiple computers/devices that comprise a computer system.
  • embodiments may also be downloaded as a computer program product, wherein the program may be transferred from a remote computer to a requesting computer by way of data signals embodied in a carrier wave or other propagation medium via a communication link (e.g., a modem or network connection, including both wired/cabled and wireless connections).
  • a communication link e.g., a modem or network connection, including both wired/cabled and wireless connections.
  • Fig. 9 is a flow chart 900 of the general operation for a scalar multiplicative homomorphic encryption embodiment
  • a shared secret numeric data value (S s ) is shared between the source 902 and destination 906.
  • the various embodiments may share the shared secret numeric data value (S s ) between the source 902 and destination 906 via any means desired by the users.
  • S s shared secret numeric data value
  • the shared secret numeric data value (S s ) may be shared between the source 906 and destination 906 by means including, but not limited to: pre-conditioning the source 902 computing device and the destination 906 computing device with the shared secret numeric value (S s ), a standard public/private key exchange technique, RSA (Rivest-Shamir-Adleman) key exchange, and/or Diffie-Hellman key exchange (disclosed in more detail herein, below).
  • the original shared secret may be an alphanumeric string in ASCII (American Standard Code for Information Exchange) or another encoding protocol that is converted to a numeric value based on the associated encoding protocol, such as ASCII.
  • ASCII American Standard Code for Information Exchange
  • ASCII American Standard Code for Information Exchange
  • the source 902 distributes the numeric message data (M) into message multivector (M) coefficients in accord with a homomorphic mathematical relationship equation between a plaintext data value and coefficients of a multivector that represents the plaintext data value.
  • the source 902 and the destination 906 should use the same homomorphic preserving mathematical relationship to preserve the homomorphism of the encrypted data.
  • the encryption system will work with just one non-zero message multivector (M) coefficient, but, the more non-zero message multivector (M) coefficients there are, the stronger the encryption will become, so it is desirable to have more than one non-zero message multivector (M) coefficient.
  • the source 202 distributes the shared secret numeric value (S s ) into shared secret multivector (S s ) coefficients.
  • S s shared secret multivector
  • the encryption system will work with just one non-zero shared secret multivector (S s ) coefficient, but, the more non-zero shared secret multivector (S s ) coefficients there are, the stronger the encryption will become, so, again, it is desirable to have more than one non-zero shared secret multivector coefficient.
  • One skilled in the art will recognize that there are many approaches for distributing numeric data into several coefficients of a multivector (see herein, above for disclosure of some example packing/distribution methods).
  • the primary requirement for the distribution process from the numeric values of the message (M) and the shared secret (S s ) to the multivector coefficient values (A? and S s ) is that the source 902 and the destination 906 both know the processes 910/930 and 912/926 such that the destination 904 can get the proper value for the homomorphic scalar multiplicative result (SMR). As long as it is known to both the source 902 and the destination 904, the distribution of numeric data to multivector coefficients may be performed differently between the message (M )/scalar multiplicative result (SMR) pair and the shared secret (S s ).
  • the distributing/packing method defines, among many things, the Geometric Algebra operations permissible for EDCE and/or EDCHE embodiments.
  • the Rationalize operation on multivectors yields zero when all multivector coefficients are equal.
  • Such multivectors having all equal coefficients have no inverse and the geometric product of such multivectors having all equal coefficients with another multivector has no inverse.
  • the Rationalize operation on multivectors yields zero when all multivector coefficients are equal.
  • Such multivectors having all equal coefficients have no inverse and the geometric product of such multivectors having all equal coefficients with another multivector has no inverse.
  • the decryption methodology for EDCE and EDCHE systems utilize the inverse of the cryptotext multivector being decrypted and of the security key(s) multivector to perform the decryption. Therefore, the cryptotext multivector being decrypted should not have all equal value coefficients.
  • One means to ensure that the cryptotext multivector being decrypted does not have all equal value coefficients is to have the packing/coefficient distribution method ensure that not all coefficients are equal to each other (i.e., at least one coefficient should be different than the other coefficients) when creating the shared security multivector(s) and the data message multivectors.
  • the cryptotext multivector to be decrypted will not have all equivalent coefficients.
  • the "packed" multivector that represents the original plaintext numeric message have a mathematical relationship (i.e., the homomorphic preserving mathematical relationship) to the original plaintext numeric message.
  • the term homomorphism refers to a structure-preserving map between two algebraic structures, such as groups, rings, or vector spaces. An algebra homomorphism between two algebras is one that preserves the algebra structure.
  • One such relationship for packing the coefficients of the multivector that preserves homomorphic properties is to ensure that the coefficients of the multivector representation of the plaintext numeric message follow a mathematical data organization between the value of the plaintext numeric message and at least one of the values of the coefficients of the multivector representation of the plaintext numeric message where the mathematical operations incorporating the one or more values of the multivector coefficients have a result equal to the original plaintext numeric message value.
  • the mathematical relationship may include: addition of at least one coefficient of the multivector coefficients, subtraction of at least one coefficient of the multivector coefficients, addition of a constant value, subtraction of a constant value, multiplication of at least one coefficient of the multivector coefficients by a constant value, and division of at least one coefficient of the multivector coefficients by a constant value.
  • the location of the various mathematical operations relative to the particular locations of the coefficients in the multivector representation should also be consistently applied to all source numeric data messages converted to a multivector as well as for result multivectors converted to a result numeric data value in a particular encryption decryption pathway.
  • the distribution (i.e., "packing") of the shared secret multivector (3 ⁇ 4) may be performed in any fashion so long as the distribution (i.e., "packing") method of the shared secret multivector (S s ) is known and used consistently by the source 902 and destination 906 computing devices as, ultimately, the shared secret multivector (S s ) used by the source 902 and destination 906 should be equal to each other to ensure that the decryption operations 928 work properly in relation to the encryption 914 operations.
  • the number of potential coefficients is directly related to the size/dimension (N) of the multivectors such that the number of coefficients increases by a factor of 2 (i.e., 2") for each incremental increase in the size/dimension (N) of the multivector.
  • using multivectors of at least two dimensions will provide at least four coefficients to distribute the numeric data of the message (M) and the shared secret (S S ).
  • the confusion and/or diffusion security characteristics will also be increased due to the additionally available multivector coefficients. Further, with the additionally available coefficients it is also possible to transfer more data in a single multivector message (A?) payload using the additionally available multivector coefficients.
  • the source 902 encrypts a cryptotext multivector (C) as a function of at least one Geometric Algebra geometric product operation on the message multivector (M) and the shared secret multivector Due to the nature of the geometric product operation of Geometric Algebra, there are many possible variations of the geometric product application that will provide similar degrees of confusion and diffusion.
  • Some, but not all, of the potential geometric product calculations to encrypt the message data (M) include: a geometric product of the message multivector and the shared secret multivector ; geometric product "sandwich” and multivector based Sylvester's equation
  • the source 902 sends the cryptotext multivector (C) to the intermediary computing system 904.
  • Various embodiments may optionally convert the cryptotext multivector (C) into cryptotext numeric data (C) in accord with reverse operation of a cryptotext data coefficient distribution algorithm that is known to the source 902, intermediary 904, and the destination 906 computing systems/devices.
  • An embodiment may also skip conversion to cryptotext numeric data (C) and directly send a representation of the cryptotext multivector ) without first converting the cryptotext multivector into cryptotext numeric data (C).
  • the transmission may be implemented as a series of transfers of the coefficients or as some form of records/packets that define a data structure that carries the coefficient data of the cryptotext multivector .
  • Not converting the cryptotext multivector (C) into cryptotext numeric data (C) has the advantage of avoiding the processing time for the conversion as well as having the advantage that, for homomorphic operations performed at an intermediary computing system, the intermediary computing system need not have any knowledge of the methodology used to create the cryptotext multivector (C). If process 916 is used to convert the cryptotext multivector (C) into cryptotext numeric data (C), it is necessary for any computing device/system that wishes to operate on the cryptotext multivector (C) to have knowledge of the particular conversion methodology so that the computing device/system may properly recreate the cryptotext multivector (C).
  • cry totext multivector (C) into cryptotext numeric data (C) is that it may be possible to include additional confusion/diffusion features in conversion to cryptotext numeric data.
  • the intermediary computing system 904 receives the cryptotext multivector (C) sent by the source 902.
  • the intermediary performs the user desired scalar multiplication of the cryptotext multivector (C) sent by the source 902 and an unencrypted scalar data value (V).
  • the intermediary 904 sends the scalar multiplicative result cryptotext multivector (SMRC) to the destination 906.
  • the destination 906 receives the scalar multiplicative result cryptotext multivector (SMRC) sent by the intermediary 904.
  • the destination 906 distributes shared secret numeric value (S s ) into shared secret multivector (S s ) coefficients in the same fashion as was done for the source 902 at process 912.
  • the destination decrypts the scalar multiplicative result cryptotext multivector (SMRC) as a function of at least one Geometric Algebra geometric product operation on the scalar multiplicative result cryptotext multivector (SMRC) and an inverse of the shared secret multivector (S s ) into a scalar result message multivector (SMR).
  • SMRC scalar multiplicative result cryptotext multivector
  • the destination 906 converts the scalar multiplicative result cryptotext multivector (SMRC) into the scalar multiplicative result numeric value (SMR) in accord with reverse operation of homomorphic preserving mathematical relationship of the source 902 at process 910.
  • SMRC scalar multiplicative result cryptotext multivector
  • SMR scalar multiplicative result numeric value
  • Fig. 10 is a flow chart 1000 of the operations for sending a multiply command for a scalar multiplicative homomorphic encryption embodiment.
  • the information used to determine the scalar data value (V) may be derived from scalar data value information contained in scalar multiplication instructions received from a command computing device 1002.
  • the command computing device 1002 sends scalar multiplication instructions, including scalar data value information that defines the scalar data value (V) to be scalar multiplied with the cryptotext multivector (C).
  • the intermediary computing system 1004 receives the scalar multiplication instructions from the command computing device 1002.
  • the intermediary computing system determines the scalar data value (7) based on the scalar value information delivered as part of the scalar multiplication instructions.
  • the scalar data value information may define the unencrypted scalar data value (V) that is to be multiplied by the encrypted cryptotext multivector (C) as a simple predefined data value, or the scalar data value (V) may be defined to be the result of several operations on data provided in the scalar data value information of the scalar multiplication instructions, and/or the scalar data value information may provide instructions to the intermediary computing system 1004 to look up some or all of the necessary data in privately or publicly available data storage accessible by the intermediary computing system 1004.
  • [REFERENCE2] is given in [REFERENCE 1 ] as: and is used by the destination to unpack, the cryptotext prior to the decryption process.

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Abstract

Disclosed are methods and systems for encrypting a numeric message using Geometric Algebra on a source computing device, performing scalar-vector multiplication on the encrypted numeric message and an unencrypted scalar data value to get an encrypted scalar multiplicative result without decrypting the encrypted numeric message on an intermediary computing system that does not have knowledge of the encryption security keys, and decrypting using Geometric Algebra the encrypted scalar multiplicative result on a destination computing device such that the decrypted result is equal to multiplication of the unencrypted numeric message and the scalar data value. Encrypt operations use the geometric product (Clifford Product) of multivectors created from plain text/data of the numeric data message with one or more other multivectors that carry encryption keys. Decrypt operation decrypts the scalar multiplicative result by employing geometric algebra operations such as multivector inverse, Clifford conjugate and others along with the geometric product.

Description

METHODS AND SYSTEMS FOR ENHANCED DATA-CENTRIC SCALAR MULTIPLICATIVE HOMOMORPHIC ENCRYPTION SYSTEMS USING
GEOMETRIC ALGEBRA
Cross Reference to Related Applications
[0001] This application is a continuation-in-part of U.S. patent applications Serial No. 15/667,325, filed August 2, 2017, entitled "Methods and Systems for Enhanced Data-Centric Encryption Systems Using Geometric Algebra;" and Serial No. 15/884,047, filed January 30, 2018, entitled "Methods and Systems for Enhanced Data-Centric Encryption Additive Homomorphic Systems Using Geometric Algebra," all of which are specifically incorporated herein by reference for all that they disclose and teach. This application is further based upon and claims the benefit of U.S. provisional applications Serial No. 62/483,227, filed April 7, 2017, entitled "Methods and Systems for Enhanced Data-Centric Scalar Multiplicative Homomorphic Encryption Systems Using Geometric Algebra;" Serial No. 62/572,955, filed October 16, 2017, entitled "Methods and System for Enhanced Data-Centric Homomorphic Encryption Searching Using Geometric Algebra;" and Serial No. 62/572,970, filed October 16, 2017, entitled "Methods and System for Enhanced Data-Centric Homomorphic
Encryption Sorting Using Geometric Algebra;" all of which are also specifically incorporated herein by reference for all that they disclose and teach.
Background of the Invention
[0002] In the last several decades, personal computers and other consumer computing devices, such has hand-held devices and smart phones, have become ubiquitous among the general public. As the proliferation of personal computers and other computing devices became prevalent, the usefulness of the computers and other computing devices was increased by interconnected communications between different computers/computing devices via various electronic networking communications systems. With the advent of the publicly accessible Internet and the establishment of the World Wide Web (WWW) for common communications between computers and/or other computing devices on the Internet, it became common for private identification and financial information to be transferred over the publicly accessible Internet. To ensure that the private information is not accessed by parties that are not intended to be privy to the private information, various encryption techniques have been applied to the private data being transferred over the Internet. As data storage has become accessible over networking technologies, including over the publicly accessible Internet, it has also become prudent to store sensitive data in an encrypted format
[0003] Modern encryption employs mathematical techniques that manipulate positive integers or binary bits. Asymmetric encryption, such as RSA (Rivest-Shamir-Adleman), relies on number theoretic one-way functions that are predictably difficult to factor and can be made more difficult with an ever increasing size of the encryption keys. Symmetric encry tion, such as DES (Data Encryption Standard) and AES (Advanced Encryption Standard), uses bit manipulations within registers to shuffle the cryptotext to increase "diffusion" as well as register based operations with a shared key to increase "confusion." Diffusion and confusion are measures for the increase in statistical entropy on the data payload being transmitted. The concepts of diffusion and confusion in encryption are normally attributed as first being identified by Claude Shannon in the 1940s. Diffusion is generally thought of as complicating the mathematical process of generating unencrypted (plain text) data from the encrypted (cryptotext) data, thus, making it difficult to discover the encryption key of the encryption process by spreading the influence of each piece of the unencrypted (plain) data across several pieces of the encrypted (cryptotext) data.
Consequently, an encryption system that has a high degree of diffusion will typically change several characters of the encrypted (cryptotext) data for the change of a single character in the unencrypted (plain) data making it difficult for an attacker to identify changes in the unencrypted (plain) data. Confusion is generally thought of as obscuring the relationship between the unencrypted (plain) data and the encrypted (cryptotext) data. Accordingly, an encryption system that has a high degree of confusion would entail a process that drastically changes the unencrypted (plain) data into the encrypted (cryptotext) data in a way that, even when an attacker knows the operation of the encryption method (such as the public standards of RSA, DES, and/or AES), it is still difficult to deduce the encryption key.
[0004] Homomorphic Encryption is a form of encryption that allows computations to be carried out on cipher text as it is encrypted without decrypting the cipher text that generates an encrypted result which, when decrypted, matches the result of operations performed on the unencrypted plaintext
[0005] The word homomorphism comes from the ancient Greek language: όμός (homos) meaning "same" and μορφή (morphe) meaning "form" or "shape." Homomorphism may have different definitions depending on the field of use. In mathematics, for example, homomorphism may be considered a transformation of a first set into a second set where the relationship between the elements of the first set are preserved in the relationship of the elements of the second set.
[0006] For instance, a map / between sets A and B is a homomorphism of A into B if
Figure imgf000005_0001
where "op" is the respective group operation defining the relationship between A and B.
[0007] More specifically, for abstract algebra, the term homomorphism may be a structure-preserving map between two algebraic structures such as groups, rings, or vector spaces. Isomorphisms, automorphisms, and endomorphisms are typically considered special types of homomorphisms. Among other more specific definitions of homomorphism, algebra homomorphism may be considered a homomorphism that preserves the algebra structure between two sets.
Summary of the Invention
[0008] An embodiment of the present invention may comprise a method for performing homomorphic scalar multiplication on a cryptotext encrypted data representation of a corresponding plain text data value and an unencrypted scalar data value, the method comprising: distributing by a source computing device a numeric message data value (M ) into coefficients of a message multivector
Figure imgf000005_0002
) representing the numeric message data value ( M ) in accord with a homomorphic preserving mathematical relationship between an unencrypted numeric data value and multivector coefficients representing the unencrypted numeric data value that is known to the source computing device and a destination computing device; distributing by the source computing device a shared secret numeric value (Ss) into coefficients of a shared secret multivector in accord with a shared secret coefficient
Figure imgf000005_0003
distribution algorithm that is known to the source computing device and the destination computing device, such that the shared secret numeric value (Ss) is known or knowable to the source computing device and the destination computing device, but is kept secret from other devices not intended to have access to the numeric message data including an intermediary computing system; encrypting by the source computing device a cryptotext multivector ) as
Figure imgf000005_0007
an encryption function of at least one Geometric Algebra geometric product operation on the message multivector ) and the shared secret multivector ; sending by the source
Figure imgf000005_0004
Figure imgf000005_0006
computing device the cryptotext multivector
Figure imgf000005_0005
) to the intermediary computing system; receiving by the intermediary computing system the cryptotext multivector ( sent by the
Figure imgf000006_0007
source computing device; multiplying by the intermediary computing system an unencrypted scalar data value (7) and the cryptotext multivector
Figure imgf000006_0006
using scalar-vector multiplication in order to obtain a scalar multiplicative result cryptotext multivector (SMRC); sending by the intermediary computing system the scalar multiplicative result cryptotext multivector (SMRC) to the destination computing device; receiving by the destination computing device the scalar multiplicative result cryptotext multivector (SMRC) sent by the intermediary computing system; distributing by the destination computing device the shared secret numeric value (Ss) into the shared secret multivector (Ss) in accord with the shared secret coefficient distribution algorithm; decrypting by the destination computing device the scalar multiplicative result cryptotext multivector (SMRC) as a decryption function of at least one Geometric Algebra geometric product operation on the scalar multiplicative result cryptotext multivector (SMRC) and an inverse
Figure imgf000006_0001
of the shared secret multivector
Figure imgf000006_0005
) into a scalar multiplicative result multivector (SMR) such that the decryption function provides a corresponding decryption operation for the encryption process of the cryptotext multivector ; and converting by the destination computing device the scalar multiplicative result multivector (SMR) into a scalar multiplicative result value (SMR) in accord with the homomorphic preserving mathematical relationship such that the scalar multiplicative result value (SMR) is equal to a multiplication product of the unencrypted numeric message data value (M) and the unencrypted scalar data value (V).
[0009] An embodiment of the present invention may further comprise a method for encrypting a numeric message data value (M) on a source computing device in order to transfer a cryptotext multivector
Figure imgf000006_0003
( ) encrypted representation of the numeric message data value (M) to an intermediary computing system that will perform homomorphic scalar multiplication of the cryptotext multivector ) and an unencrypted scalar data value (V) and
Figure imgf000006_0002
deliver a result of the homomorphic scalar multiplication to a destination computing device, the method comprising: distributing by the source computing device the numeric message data value (M) into coefficients of a message multivector
Figure imgf000006_0004
( ) in accord with a homomorphic preserving mathematical relationship between an unencrypted numeric data value and multivector coefficients representing the unencrypted numeric data value that is known to the source computing device and the destination computing device; distributing by the source computing device a shared secret numeric value (Ss) into coefficients of a shared secret multivector ) in accord with a shared secret coefficient distribution algorithm that is known to the source computing device and the destination computing device, the shared secret numeric value (Ss) being known or knowable to the source computing device and the destination computing device, but is kept secret from other devices not intended to have access to the numeric message data including the intermediary computing system; encrypting by the source computing device the cryptotext multivector (C) as an encryption function of at least one Geometric Algebra geometric product operation on the message multivector (M) and the shared secret multivector (Ss); and sending by the source computing device the cryptotext multivector (C) to the intermediary computing system.
[0010] An embodiment of the present invention may further comprise a method for performing homomorphic scalar multiplication on an intermediary computer system of a cryptotext multivector (C) encrypted data representation of a corresponding plain text numeric data value received from a source computing device and an unencrypted scalar data value (V) and delivering a homomorphic scalar multiplicative result cryptotext multivector (SMRC) to a destination computing device, the method comprising: receiving by the intermediary computing system the cryptotext multivector (C) sent by the source computing device; multiplying by the intermediary computing system the unencrypted scalar data value (V) and the cryptotext multivector (C) using scalar-vector multiplication in order to obtain a scalar multiplicative result cryptotext multivector (SMRC); and sending by the intermediary computing system the scalar multiplicative result cryptotext multivector (SMRC) to the destination computing device.
[0011] An embodiment of the present invention may further comprise a method for decrypting a scalar multiplicative result cryptotext multivector (SMRC) on a destination computing device received from an intermediary computing system that performed homomorphic scalar multiplication of a cryptotext multivector (C) originated from a source computing device and an unencrypted scalar data value (V), the method comprising:
receiving by the destination computing device the scalar multiplicative result cryptotext multivector (SMRC) sent by the intermediary computing system; distributing by the source computing device a shared secret numeric value (Ss) into a shared secret multivector
Figure imgf000007_0001
in accord with a shared secret coefficient distribution algorithm that is known to the source computing device and the destination computing device, the shared secret numeric value (Ss) being known or knowable to the source computing device and the destination computing device, but is kept secret from other devices not intended to have access to the numeric message data including the intermediary computing system; decrypting by the destination computing device the scalar multiplicative result cryptotext multivector (SMRC) as a decryption function of at least one Geometric Algebra geometric product operation on the scalar multiplicative result cryptotext multivector (SMRC) and an inverse of the shared secret multivector (Ss) into a scalar multiplicative result multivector (SMR) such that the decryption function provides a corresponding decryption operation for an encryption process of the cryptotext multivector (C); and converting by the destination computing device the scalar multiplicative result multivector (SMR) into a scalar multiplicative result data value (SMR) in accord with the homomorphic preserving mathematical relationship between an unencrypted numeric data value and multivector coefficients representing the unencrypted numeric data value that is known to the source computing device and the destination computing device such that the scalar multiplicative result value (SMR) is equal to a multiplication product of an unencrypted numeric message data value (M) represented by the cryptotext multivector (C) and the unencrypted scalar data value (V).
[0012] An embodiment of the present invention may further comprise a scalar multiplicative homomorphic Enhanced Data-Centric Encryption (EDCE) system for scalar multiplicative homomorphic multiplication of a cryptotext encrypted data representation of a corresponding plain text data value and an unencrypted scalar data value, the scalar multiplicative homomorphic EDCE system comprising: a source computing device, wherein the source computing device further comprises: a source numeric message distribution subsystem that distributes a numeric message data value (M) into coefficients of a message multivector (M ) representing the numeric message data value (M) in accord with a homomorphic preserving mathematical relationship between an unencrypted numeric data value and multivector coefficients representing the unencrypted numeric data value that is known to the source computing device and a destination computing device; a source numeric shared secret distribution subsystem that distributes a shared secret numeric value (Ss) into coefficients of a shared secret multivector (Ss) in accord with a shared secret coefficient distribution algorithm that is known to the source computing device and the destination computing device, such that the shared secret numeric value (Ss) is known or knowable to the source computing device and the destination computing device, but is kept secret from other devices not intended to have access to the numeric message data including an intermediary computing system; a source encryption subsystem that encrypts a cryptotext multivector (C) as an encryption function of at least one Geometric Algebra geometric product operation on the message multivector (M) and the shared secret multivector (Ss); and a source send subsystem that sends the cryptotext multivector (C) to the intermediary computing system; the intermediary computing system, wherein the intermediary computing system further comprises: an intermediary receive subsystem that receives the cryptotext multivector (C) sent by the source computing device; an intermediary homomorphic scalar multiplication subsystem that multiplies an unencrypted scalar data value (V) and the cryptotext multivector (C) using scalar-vector multiplication in order to obtain a scalar multiplicative result cryptotext multivector (SMRC); and an intermeidary send subsystem that sends the scalar multiplicative result cryptotext multivector (SMRC) to the destination computing device; and the destination computing device, wherein the destination computing device further comprises: a destination receive subsystem that receives the scalar multiplicative result cryptotext multivector (SMRC) sent by the intermediary computing system; a destination numeric shared secret distribution subsystem that distributes the shared secret numeric value (Ss) into the shared secret multivector (Ss) in accord with the shared secret coefficient distribution algorithm; a destination decryption subsystem that decrypts the scalar multiplicative result cryptotext multivector (SMRC) as a decryption function of at least one Geometric Algebra geometric product operation on the scalar multiplicative result cryptotext multivector (SMRC) and an inverse (Ss 1) of the shared secret multivector (Ss) into a scalar multiplicative result multivector (SMR) such that the decryption function provides a corresponding decryption operation for the encryption process of the cryptotext multivector (C); and a destination convert multivector subsystem that converts the scalar multiplicative result multivector (SMR) into a scalar multiplicative result value (SMR) in accord with the homomorphic preserving mathematical relationship such that the scalar multiplicative result value (SMR) is equal to a multiplication product of the unencrypted numeric message data value (M) and the unencrypted scalar data value (V).
[0013] An embodiment of the present invention may further comprise a scalar multiplicative homomorphic Enhanced Data-Centric Encryption (EDCE) system source computing device for encrypting a numeric message data value (M) in order to transfer a cryptotext multivector (C) encrypted representation of the numeric message data value (M) to an intermediary computing system that will perform homomorphic scalar multiplication of the cryptotext multivector (C) and an unencrypted scalar data value (V) and deliver a result of the homomorphic scalar multiplication to a destination computing device, the scalar multiplicative homomorphic EDCE system source computing device comprising: a source numeric message distribution subsystem that distributes the numeric message data value (M) into coefficients of a message multivector (M) in accord with a homomorphic preserving mathematical relationship between an unencrypted numeric data value and multivector coefficients representing the unencrypted numeric data value that is known to the source computing device and the destination computing device; a source numeric shared secret distribution subsystem that distributes a shared secret numeric value (Ss) into coefficients of a shared secret multivector (S5) in accord with a shared secret coefficient distribution algorithm that is known to the source computing device and the destination computing device, the shared secret numeric value (Ss) being known or knowable to the source computing device and the destination computing device, but is kept secret from other devices not intended to have access to the numeric message data including the intermediary computing system; a source encryption subsystem that encrypts the cryptotext multivector (C) as an encryption function of at least one Geometric Algebra geometric product operation on the message multivector (M) and the shared secret multivector (Ss); and a source send subsystem that sends the cryptotext multivector (C) to the intermediary computing system.
[0014] An embodiment of the present invention may further comprise a scalar multiplicative homomorphic Enhanced Data-Centric Encryption (EDCE) system
intermediary computing system for performing homomorphic scalar multiplication of a cryptotext multivector (C) encrypted data representation of a corresponding plain text numeric data value received from a source computing device and an unencrypted scalar data value (V) and delivering a homomorphic scalar multiplicative result cryptotext multivector (SMRC) to a destination computing device, the scalar multiplicative homomorphic EDCE system intermediary computing system comprising: an intermediary receive subsystem that receives the cryptotext multivector (C) sent by the source computing device; an intermediary homomorphic scalar multiplication subsystem that multiplies the unencrypted scalar data value (V) and the cryptotext multivector (C) using scalar-vector multiplication in order to obtain a scalar multiplicative result cryptotext multivector (SMRC); and an intermeidary send subsystem that sends the scalar multiplicative result cryptotext multivector (SMRC) to the destination computing device.
[0015] An embodiment of the present invention may further comprise a scalar multiplicative homomorphic Enhanced Data-Centric Encryption (EDCE) system destination computing device for decrypting a scalar multiplicative result cryptotext multivector (SMRC) received from an intermediary computing system that performed homomorphic scalar multiplication of a cryptotext multivector (C) originated from a source computing device and an unencrypted scalar data value (V), the scalar multiplicative homomorphic EDCE system destination computing device comprising: a destination receive subsystem that receives the scalar multiplicative result cryptotext multivector (SMRC) sent by the intermediary computing system; a destination numeric shared secret distribution subsystem that distributes a shared secret numeric value (Ss) into a shared secret multivector (Ss) in accord with a shared secret coefficient distribution algorithm that is known to the source computing device and the destination computing device, the shared secret numeric value (Ss) being known or knowable to the source computing device and the destination computing device, but is kept secret from other devices not intended to have access to the numeric message data including the intermediary computing system; a destination decryption subsystem that decrypts the scalar multiplicative result cryptotext multivector (SMRC) as a decryption function of at least one Geometric Algebra geometric product operation on the scalar multiplicative result cryptotext multivector (SMRC) and an inverse (Ss 1) of the shared secret multivector (Ss) into a scalar multiplicative result multivector (SMR) such that the decryption function provides a corresponding decryption operation for an encryption process of the cryptotext multivector (C); and a destination convert multivector subsystem that converts the scalar multiplicative result multivector (SMR) into a scalar multiplicative result data value (SMR) in accord with the homomorphic preserving mathematical relationship between an unencrypted numeric data value and multivector coefficients representing the unencrypted numeric data value that is known to the source computing device and the destination computing device such that the scalar multiplicative result value (SMR) is equal to a multiplication product of an unencrypted numeric message data value (M) represented by the cryptotext multivector (C) and the unencrypted scalar data value (V).
Brief Description of the Drawings
[0016] In the drawings,
[0017] FIG. 1 is a block diagram of the hardware implementation for a core encryption embodiment (i.e., a core Enhanced Data-Centric Encryption— EDCE— embodiment).
[0018] FIG. 2 is a flow chart of the general operation for a core encryption
embodiment.
[0019] FIG. 3 A is a flow chart of the source computing device symmetric key operation for a core encryption embodiment. [0020] FIG. 3B is a flow chart of the destination computing device symmetric key operation for a core encryption embodiment.
[0021] FIG. 4A is a flow chart of the source computing device symmetric key and cryptotext masking operation for a core encryption embodiment.
[0022] FIG. 4B is a flow chart of the destination computing device symmetric key and cryptotext masking operation for an encryption embodiment.
[0023] FIG. 5 is a flow chart of a core encryption embodiment for the EDCE encryption/decryption by using a geometric product "sandwich."
[0024] FIG. 6 is a flow chart of a core encryption embodiment for the EDCE encryption/decryption by using Sylvester's equation.
[0025] FIG. 7 is a block diagram illustrating generating/extracting/obtaining a second shared secret key from the original shared secret multivector for a core encryption embodiment
[0026] FIG. 8 is a block diagram of the hardware implementation for a scalar multiplicative homomorphic encryption embodiment.
[0027] FIG. 9 is a flow chart of the general operation for a scalar multiplicative homomorphic encryption embodiment.
[0028] FIG. 10 is a flow chart of the operations for sending a multiply command for a scalar multiplicative homomorphic encryption embodiment
Detailed Description of the Embodiments
[0029] Homomorphic Encryption is a form of encryption that allows computations to be carried out on cipher text as it is encrypted without decrypting the cipher text that generates an encrypted result which, when decrypted, matches the result of operations performed on the unencrypted plaintext.
[0030] The essential purpose of homomorphic encryption is to allow computation on encrypted data without decrypting the data in order to perform the computation. In this way, the encrypted data can remain confidential and secure while the encrypted data is processed for the desired computation. Accordingly, useful tasks may be accomplished on encrypted (i.e., confidential and secure) data residing in untrusted environments. In a world of distributed computation and heterogeneous networking, the ability to perform computations on encrypted data may be a highly desirable capability. Hence, finding a general method for computing on encrypted data is likely a highly desirable goal for cryptography. [0031] The most sought after application of homomorphic encryption may be for cloud computing. Data that is stored in the Cloud is typically not encrypted, and the breach of the Cloud stored, unencrypted data is ranked by the Cloud Security Alliance as the number one threat to data security. Encrypting Cloud stored data may mitigate the threat of data being compromised by a breach, but then the remote clients (owners of the data) would not then be able to perform operations (i.e., add, multiply, etc.) on the Cloud stored data while the data remains in the Cloud. In order to perform operations on encrypted data stored in the Cloud, it would be necessary to download the encrypted Cloud stored data, decrypt the data, perform all desired operations on the data locally, encrypt the resulting data and send the resulting data back to the Cloud. Alternatively, if a user wants the Cloud services provider to perform the computations, the Cloud would require access to the user's encryption keys. It is becoming increasing undesirable to provide the Cloud access to a user's security keys as the more entities that have access to the security keys inherently increases the susceptibility of the security keys to being breached, or even stolen by an unscrupulous provider.
Homomorphic encryption would allow the Cloud to operate on client data without decryption, and without access to the client's security keys.
[0032] An embodiment may advantageously utilize Geometric Algebra to provide the encryption and decryption of numeric messages that are to be transmitted through, and possibly have operations performed by, an intermediary computing system (e.g., the broad- based computing system currently, and commonly, referred to as the Cloud, or cloud computing). An embodiment of the Geometric Algebra encryption/decryption system that performs the foundational "core" encryption/decryption functions of transferring data securely using Geometric Algebra based encryption/decryption from a source system to a destination system without having arithmetic or other comparative operations performed on the transmitted encrypted data by an intermediary system may be referred to as an Enhanced Data-Centric Encryption (EDCE) system. When an EDCE system is further enhanced to support and provide for arithmetic and/or other comparative operations to be performed at an intermediary computing system (e.g., the Cloud) without decrypting and re-encrypting the data at the intermediary computing system, that system may be referred to as an Enhanced Data-Centric Homomorphic Encryption (EDCHE) system.
[0033] Geometric Algebra is an area of mathematics that describes the geometric interaction of vectors and other objects in a context intended to mathematically represent physical interactions of objects in the physical world. The use of Geometric Algebra for cryptography represents a new, manmade use of Geometric Algebra for a purpose entirely outside of the natural basis of Geometric Algebra for representing physical interactions of objects in the real, physical, word. As used herein, this area of mathematics encompasses Geometric Algebra, Conformal Geometric Algebra and Clifford Algebra (referred to collectively herein as "Geometric Algebra"). Generally, Geometric Algebra defines the operations, such as geometric product, inverses and identities, which facilitate many features of embodiments of the core EDCE and the EDCHE systems disclosed herein. Further, Geometric Algebra allows for the organization and representation of data into the "payload" of a multivector where the data in the payload may represent, for example, plaintext, cryptotext, or identifying signatures. Consequently, Embodiments of both the core EDCE system and the E iCHE system make beneficial use of Geometric Algebra properties to provide encryption, decryption, and intermediary homomorphic operations in a relatively computationally simplistic manner while still providing robust security for both data in motion and data at rest (e.g., data stored in the Cloud).
[0034] For an embodiment of an EDCHE system, methods and systems to encrypt and decrypt messages using Geometric Algebra may utilize the intrinsic algebraic homomorphic properties of Geometric Algebra to permit arithmetic and other comparative operations on encrypted messages handled by an intermediary computing system without the need for the intermediary computing system to decrypt the encrypted messages prior to performing the arithmetic and other comparative operations. Accordingly, the intermediary computing system does not need to know any information regarding any of the secret security keys of the encryption/decryption processes to properly perform the arithmetic and other comparative operations. The encrypted results of the arithmetic and other comparative operations performed by the intermediary computing system, when decrypted at a destination computing device, produce results equivalent to the same operations as if the operations were performed on the unencrypted plain text messages. To properly achieve the homomorphic effect for intermediary arithmetic and other comparative operations, a proper data organization methodology that preserves such homomorphic properties (i.e., the mathematical relationship between the vectors utilized in the encryption process and the original plaintext messages being encrypted) should be enforced on the choice of coefficients for the vectors representing the plain text messages. As discussed in more detail below, ensuring that the coefficients of the multivector representation of the plaintext numeric message follow a mathematical data organization methodology (i.e., a homomorphic preserving mathematical relationship) between the value of the plaintext numeric message and at least one of the values of the coefficients of the multivector representation of the plaintext numeric message where the mathematical operations incorporating the one or more values of the multivector coefficients have a result equal to the original plaintext numeric message value will provide the proper data organization to preserve the homomorphic properties of the Geometric Algebra operations of the core
Figure imgf000015_0001
use of the proper data organization, an embodiment of an EDCHE system provides a cryptosystem that allows unlimited multiplications and additions of cipher text (i.e., transmitted/stored encrypted messages at the intermediary/cloud computer system) due solely to the intrinsic algebraic homomorphic properties of an embodiment of the EDCHE system. Thus, an embodiment of an EDCHE system may provide the homomorphic properties as a product of algebraic homomorphism without the need to use additional methods, such as "bootstrapping" (e.g., performing a recursive operation to reduce the noise associated with a cipher text) to achieve the homomorphic properties.
[0035] The encrypted data values may be stored on the intermediary computing system until such time that particular arithmetic or other comparative operations are desired by a user, then the intermediary computing system may perform the requested arithmetic or other comparative operations. Likewise, the encrypted data values may be immediately operated on by the intermediary computing system as soon as the subject encrypted data values are received by the intermediary computing system. However, as one skilled in the art will recognize, the process of receiving the encrypted data values at the intermediary computing system inherently includes storing the encrypted data values at the intermediary computing system even if only fleetingly in an immediately used and erased Random Access Memory (RAM) location or operational register location of a computational subsystem of the intermediary computing system.
[0036] Embodiments of both EDCE and EDCHE may be comprised of functional blocks, each of which may be tailored as described in more detail below according to objectives for scope, capability and security. The following sections provide a mathematical and numerical description of these functional blocks.
[0037] In order to help minimize the potential confusion of the complex subject matter herein, the descriptions below have been split up to separately cover foundational "core" EDCE concepts and the additional enhancements concepts that permit homomorphic operations for EDCHE. In view of that, Section 1 provides a general description of embodiments of the foundational "core" EDCE system. Section 2 provides additional descriptions of embodiments of the foundational "core" EDCE system, including the packing of information into multivectors, the encryption and decryption of such multivectors and the unpacking to recover the original information. Section 3 provides a description of the further enhancements to embodiments of the foundational "core" EDCE system that achieve homomorphic properties for embodiments of an EDCHE system. Generally, in this description, as is the typical convention, for particular examples of operations, Alice and Bob are used for the sending/source and receiving/destination entities, respectively. Thus, the arrangement of the disclosure may be summarized as follows:
Section 1: General Core EDCE Message Encryption/Decryption
A. Hardware Implementation for Core EDCE Embodiments (Fig. 1)
B. General Core EDCE Operational Flow Charts (Figs. 2-4)
Section 2: Additional Descriptions of Core EDCE Message
Encryption/Decryption
A. Packing and unpacking multivectors
1 ) Text to number
2) Number to text
3) Multivector Data Structure
4) Number to multivector
5) Multivector to number
B. Shared Secret
C. Cryptotext creation
D. Decryption
E. EDCE Flow Chart (Figs. 5-6)
F. Symmetric Key Pair Encryption/Decryption from O-Blade Reduction
Operation (Fig. 7)
* O-Blade Reduction Operation
* Geometric Algebra Encryption Primitives
* Numerical Examples for encryption and decryption with double shared secret in 3 dimensions
G. An Unbreakable Primitive Using Geometric Algebra and Arithmetic
Functions
'"Example with secret sharing and 3D multivectors
Section 3: Homomorphic EDCHE Enhancements to EDCE Operation
A. Homomorphic Data Organization (i.e., Coefficient Distribution/"Packing") B. Number to multivector that supports Ηοmomorphism
* First example
* Second example
* Third example
C. Multivector to number that supports Homomorphism
* First example
* Second example
* Third example
D. Scalar Multiplicative Homomorphism in EDCHE
* General Scalar Multiplicative EDCHE example
E. Hardware Implementation for Scalar Multiplicative Homomorphic
EDCHE Embodiments (Fig. 8)
F. Scalar Multiplicative Homomorphic EDCHE Operational Flow Charts (Figs. 9 & 10)
Section 1: General Core EDCE Message Encryption/Decryption
[0038] With the arrival of the internet and many forms of mobile devices, the volume of encrypted data is growing exponentially. Portable devices like "thumb drives," "smart cards" and Solid State Disks (SSDs) contain both plain text and or encrypted "passive" data storage. Passive data storage is found on the tiny devices for the Internet of Things (IoT) as well as the large memories in server farms.
[0039] When data leaves storage, when it is in motion, it is even more vulnerable to attack. Current encryption techniques have not evolved alongside network security infrastructure and they are not well suited for the sheer volume of data in motion. As we move towards "cloud computing," as mobile devices move us towards "perimeter-less" network security, the industry is moving away from trusting just the security of networks, servers or applications and focusing toward data-centric encryption. With data-centric encryption and authentication there are controls that are traveling with the data rather than just happening at the application layer or the final destination in a network.
[0040] However, the fluidity of this data in motion stalls with the computationally intensive mathematics that remain at the heart of current encryption infrastructures. Ciphers such as RSA (Rivest-Shamir-Adleman), DES (Data Encryption Standard) and/or AES (Advanced Encryption Standard) are little more than static "machinery" that bogs down communication efficiency. The actual problem is much bigger. How can robust security be provided when: a) End-point computational resources are limited (e.g., the Internet of Things— IoT). b) Encryption/decryption must be near-real time
c) Authentication of the source and destination must be continuously reasserted
[0041] A "core" embodiment may be described as enhanced data-centric encryption, or EDCE. Compared to incumbent encryption schemes, EDCE is computationally simplistic while providing robust security over the span of the communication channel. EDCE security is scalable from tiny embedded IoT (Internet of Things) devices up to server farms. EDCE functionality enables many cipher schemes that show speed and bandwidth advantages over current methods. One aspect of EDCE that provides speed enhancement in the
encryption/decryption of data is that the EDCE encryption/decryption may be implemented using basic arithmetic operations of addition, subtraction, multiplication, and division.
Notably, EDCE does not require a complex operation to select a large prime number, to calculate a logarithm function, to calculate a natural logarithm function, and/or to calculate other complex and computationally intensive mathematical functions (i.e., prime numbers, logarithms, natural logarithms, and/or other complex mathematical operations are not required in the Geometric Algebra calculations disclosed herein).
[0042] A central feature of the various embodiments is the use of Geometric Algebra, an area of mathematics that has not been utilized before in encryption. Geometric Algebra as used herein is an area of mathematics that encompasses Geometric Algebra, Conformal Geometric Algebra and Clifford Algebra (collectively herein, "Geometric Algebra").
Geometric Algebra allows for the organization and representation of data into the "payload" of a multivector where the data may be plaintext, cryptotext, or signatures, for example. Geometric Algebra defines the operations, such as geometric product, inverses and identities, which are the enablers of encryption/decryption calculations of various embodiments.
[0043] Multivectors are simply the additive combination of a scalar, a vector, a bi- vector and so forth up to an n-dimension vector. However, the unit vectors follow the algebraic structure of quaternions (Hamilton) and non-commutative algebra (Grassman). These two types of algebra allowed Clifford to conceive of the Geometric Product which is used by the various embodiments as one of the "primitive" functions of the embodiments of EDCE and EDCHE systems. [0044] An example of a two-dimension (2D) multivector A that includes a scalar and a vector is:
Figure imgf000019_0001
where is a unit vector along the i-axis and
Figure imgf000019_0002
represents the orientation of the area created by a12. The operations of Geometric Algebra on multivectors are discussed more fully in Appendix A: Geometric Algebra Overview" of the parent patent application Serial No.
15/667,325, entitled "Methods and Systems for Enhanced Data-Centric Encryption Systems Using Geometric Algebra," but some general observations may be helpful to the description of the various embodiments disclosed below. First, each of the at values in the multivector A above may be "packed" with information and each ai value may range from zero to very large (e.g., >256,000 bits or an entire message). Secondly, the inverse of A when multiplied by A yields unity, or:
Figure imgf000019_0003
Thus, if a second multivector B is created and the geometric product is transmitted then
Figure imgf000019_0005
the destination can recover B through:
Figure imgf000019_0004
[0045] For the various embodiments, the "payload" may be packed in the values of the scalers and coefficients of the multivector elements. The packing method may define, among many things, the Geometric Algebra operations permissible for EDCE and/or EDCHE embodiments. For example, the Rationalize operation on multivectors yields zero when all multivector coefficients are equal. Such multivectors having all equal coefficients have no inverse and the geometric product of such multivectors having all equal coefficients with another multivector has no inverse. As discussed in more detail below, the decryption methodology for EDCE and EDCHE systems utilize the inverse of the cryptotext multivector being decrypted and of the security key(s) multivector to perform the decryption. Therefore, the cryptotext multivector being decrypted should not have all equal value coefficients. One means to ensure that the cryptotext multivector being decrypted does not have all equal value coefficients is to have the packing/coefficient distribution method ensure that not all coefficients are equal to each other (i.e., at least one coefficient should be different than the other coefficients) when creating the shared security multivector(s) and the data message multi vectors. For an embodiment of the EDCE that simply transfers the data message, this will ensure that the cryptotext multivector to be decrypted will not have all equivalent coefficients. For an EDCHE system that may perform operations involving multiple encrypted data values, the same packing/coefficient distribution method to ensure that the source message multivectors do not have all equivalent coefficients will minimize the potential for the cryptotext multivector being decrypted from having all equivalent coefficients, but, when various addition and subtraction operations are performed with multiple distinctly different cryptotext multivectors, there is a remote possibility that the cryptotext multivector result of the homomorphic operations will have all equivalent coefficients. The destination computing device may simply assert that such a result cryptotext multivector is "undefined," or, the destination or intermediary computing system may provide a means to update the result cryptotext multivector so the result cryptotext multivector does not have all equivalent coefficients. Great care should be taken to ensure that such an update of the result cryptotext multivector does not change the ultimate value of the result plaintext value of the result cryptotext multivector after decryption.
[0046] For embodiments that intend to retain homomorphic properties for encrypted data messages such as an embodiment of an EDCHE system, there is an additional restriction that the "packed" multivector that represents the original plaintext numeric message have a mathematical relationship (i.e., the homomorphic preserving mathematical relationship) to the original plaintext numeric message. In abstract algebra, the term homomorphism refers to a structure-preserving map between two algebraic structures, such as groups, rings, or vector spaces. An algebra homomorphism between two algebras is one that preserves the algebra structure. In order to preserve the algebra structure between arithmetic and Geometric Algebra operations, the method by which numbers are "packed" into multivector elements must remain a representation of the original number. One such relationship for packing the coefficients of the multivector that preserves homomorphic properties is to ensure that the coefficients of the multivector representation of the plaintext numeric message follow a mathematical data organization between the value of the plaintext numeric message and at least one of the values of the coefficients of the multivector representation of the plaintext numeric message where the mathematical operations incorporating the one or more values of the multivector coefficients have a result equal to the original plaintext numeric message value. The mathematical relationship may include: addition of at least one coefficient of the multivector coefficients, subtraction of at least one coefficient of the multivector coefficients, addition of a constant value, subtraction of a constant value, multiplication of at least one coefficient of the multivector coefficients by a constant value, and division of at least one coefficient of the multivector coefficients by a constant value. The location of the various mathematical operations relative to the particular locations of the coefficients in the multivector representation should also be consistently applied to all source numeric data messages converted to a multivector as well as for result multivectors converted to a result numeric data value in a particular encryption/decryption pathway.
[0047] Additionally, separate multivectors may be encoded for many purposes, such as a shared secret (defined below), authentication information, and timestamps. In addition to the encryption and decryption of a message, the EDCE multivector format and Geometric Algebra foundation of a core EDCE embodiment may enable a single transmission to contain far more than just cryptotext, including dummy data to increase encryption security, command instructions for additional operations, and/or configuration data for the additional operations.
A. Hardware Implementation for Core EDCE Embodiments (Fig. 1)
[0048] Fig. 1 is a block diagram 100 of the hardware implementation for an embodiment. A first computing device 102 is connected over an electronic network/bus connection 104 to a second computing device 106. In the embodiment shown in Fig. 1, the first computing device 102 acts as the source device 102 that sends the encrypted message 108 over the network/bus connection 104. The second computing device 106 acts as the destination device 106 that receives the encrypted message 108 from the network/bus connection 104. Generally, communications, including encrypted communications, are bidirectional such that the first 102 and second 106 computing devices may change roles as the source device 102 and destination device 106 as is necessary to accommodate the transfer of data back and forth between the first 102 and second 106 computing devices.
[0049] Further, as shown in Fig. 1 , the first computing device 102 appears to be a laptop computer and the second computing device 106 appears to be a tablet device.
Generally, any computing device capable of communication over any form of electronic network or bus communication platform may be one, or both of the first 102 and second 106 computing devices. Further, the first 102 and second computing devices 106 may actually be the same physical computing device communicating over an internal bus connection 104 with itself, but still desiring encrypted communication to ensure that an attacker cannot monitor the internal communications bus 104 to obtain sensitive data communications in an unencrypted format.
[0050] Various embodiments may implement the network/bus communications channel 104 using any communications channel 104 capable of transferring electronic data between the first 102 and second 106 computing devices. For instance, the network/bus
communication connection 104 may be an Internet connection routed over one or more different communications channels during transmission from the first 102 to the second 106 computing devices. Likewise, the network/bus communication connection 104 may be an internal communications bus of a computing device, or even the internal bus of a processing or memory storage Integrated Circuit (IC) chip, such as a memory chip or a Central
Processing Unit (CPU) chip. The network/bus communication channel 104 may utilize any medium capable of transmitting electronic data communications, including, but not limited to: wired communications, wireless electro-magnetic communications, fiber-optic cable communications, light/laser communications, sonic/sound communications, etc., and any combination thereof of the various communication channels.
[0051] The various embodiments may provide the control and management functions detailed herein via an application operating on the first 102 and/or second 106 computing devices. The first 102 and/or second 106 computing devices may each be a computer or computer system, or any other electronic devices device capable of performing the communications and computations of an embodiment. The first 102 and second 104 computing devices may include, but are not limited to: a general purpose computer, a laptop/portable computer, a tablet device, a smart phone, an industrial control computer, a data storage system controller, a CPU, a Graphical Processing Unit (GPU), an Application Specific Integrated Circuit (ASI), and/or a Field Programmable Gate Array (FPGA).
Notably, the first 102 and second 106 computing devices may be the storage controller of a data storage media (e.g., the controller for a hard disk drive) such that data delivered to/from the data storage media is always encrypted so as to limit the ability of an attacker to ever have access to unencrypted data. Embodiments may be provided as a computer program product which may include a computer-readable, or machine-readable, medium having stored thereon instructions which may be used to program/operate a computer (or other electronic devices) or computer system to perform a process or processes in accordance with the various embodiments. The computer-readable medium may include, but is not limited to, hard disk drives, floppy diskettes, optical disks, Compact Disc Read-Only Memories (CD-ROMs), Digital Versatile Disc ROMS (DVD-ROMs), Universal Serial Bus (USB) memory sticks, magneto-optical disks, ROMs, random access memories (RAMs), Erasable Programmable ROMs (EPROMs), Electrically Erasable Programmable ROMs (EEP OMs), magnetic optical cards, flash memory, or other types of media/machine-readable medium suitable for storing electronic instructions. The computer program instructions may reside and operate on a single computer/electronic device or various portions may be spread over multiple computers/devices that comprise a computer system. Moreover, embodiments may also be downloaded as a computer program product, wherein the program may be transferred from a remote computer to a requesting computer by way of data signals embodied in a carrier wave or other propagation medium via a communication link (e.g., a modem or network connection, including both wired/cabled and wireless connections).
B. General Core EDCE Operational Flow Charts (Figs. 2-4)
[0052] Fig. 2 is a flow chart 200 of the general operation for an embodiment. At process 206 a shared secret numeric data value (Ss) is shared between the source 202 and destination 204. The various embodiments may share the shared secret numeric data value (Ss) between the source 202 and destination 204 via any means desired by the users. To ensure the shared secret numeric data value (Ss) is kept secret, it is likely that some type of handshaking/setup encrypted key transfer mechanism will be desired to share the shared secret numeric data value (Ss). For example, the shared secret numeric data value (Ss) may be shared between the source 202 and destination 204 by means including, but not limited to: pre-conditioning the source 202 computing device and the destination 204 computing device with the shared secret numeric value (Ss), a standard public/private key exchange technique, RSA (Rivest-Shamir-Adleman) key exchange, and/or Diffie-Hellman key exchange
(disclosed in more detail herein, below). Further, the original shared secret may be an alphanumeric string in ASCII (American Standard Code for Information Exchange) or another encoding protocol that is converted to a numeric value based on the associated encoding protocol, such as: ASCII, other English language/alphabetic coding systems, foreign language encoding for non-alphabetic languages (e.g., katakana for Japanese), or even pure symbol to numeric values such as for emoji's. However, both the source 202 and destination 204 need to know and use the same alphanumeric text conversion into a numeric value process to ensure that results of both the source 202 and the destination 204 are the same.
[0053] At process 208, the source 202 converts any alphanumeric text in the message into numeric message data (M ) based on the alphanumeric encoding protocol (e.g., ASCII, other English language/alphabetic coding systems, foreign language encoding for non- alphabetic languages (e.g., katakana for Japanese), or even pure symbol to numeric values such as for emoji's) of the original text Again, both the source 202 and destination 204 need to know and use the same alphanumeric text conversion into a numeric value process to ensure that results of both the source 202 and the destination 204 are the same. If the message data is already in numeric form, it is not necessary to perform process 208 as the original numeric message data (M ) may be used as is. The various embodiments may perform the encryption process with numeric message data (M) that is, but is not limited to: positive numbers, negative numbers, zero, integer numbers, rational numbers (including fractions), and/or real numbers. At process 210, the source 202 distributes the numeric message data (M) into message multivector (M ) coefficients. The encryption system will work with just one non-zero message multivector (M ) coefficient, but, the more non-zero message multivector (M ) coefficients there are, the stronger the encryption will become, so it is desirable to have more than one non-zero message multivector (M ) coefficient. At process 212, the source 202 distributes shared secret numeric value (Ss) into shared secret multivector (Ss) coefficients. Again, the encryption system will work with just one non-zero shared secret multivector (Ss) coefficient, but, the more non-zero shared secret multivector (Ss) coefficients there are, the stronger the encryption will become, so, again, it is desirable to have more than one non-zero shared secret multivector (Ss) coefficient. One skilled in the art will recognize that there are many approaches for distributing numeric data into several coefficients of a multivector (see herein, below for disclosure of some example
packing/distribution methods). The primary requirement for the distribution process from the numeric values of the message (M ) and the shared secret (Ss) to the multivector coefficient values (Af and Ss) is that the source 202 and the destination 204 both know the processes 210/222 and 212/224 such that the destination 204 can reconstruct the original message (M ). As long as it is known to both the source 202 and the destination 204, the distribution of numeric data to multivector coefficients may be performed differently between the message (M ) and the shared secret (Ss). Further, the various embodiments may perform the encryption process with multivector coefficient values for both the message (M ) and shared that are, but are not limited to: positive numbers, negative numbers, zero, integer numbers, rational numbers (including fractions), and/or real numbers.
[0054] The distributing/packing method defines, among many things, the Geometric Algebra operations permissible for EDCE and/or DCHE embodiments. For example, the Rationalize operation on multivectors yields zero when all multivector coefficients are equal. Such multivectors having all equal coefficients have no inverse and the geometric product of such multivectors having all equal coefficients with another multivector has no inverse. For example, the Rationalize operation on multivectors yields zero when all multivector coefficients are equal. Such multivectors having all equal coefficients have no inverse and the geometric product of such multivectors having all equal coefficients with another multivector has no inverse. As discussed in more detail below, the decryption methodology for EDCE and EDCHE systems utilize the inverse of the cryptotext multivector being decrypted and of the security key(s) multivector to perform the decryption. Therefore, the cryptotext multivector being decrypted should not have all equal value coefficients. One means to ensure that the cryptotext multivector being decrypted does not have all equal value coefficients is to have the packing/coefficient distribution method ensure that not all coefficients are equal to each other (i.e., at least one coefficient should be different than the other coefficients) when creating the shared security multivector(s) and the data message multivectors. For an embodiment of the EDCE that simply transfers the data message, this will ensure that the cryptotext multivector to be decrypted will not have all equivalent coefficients. For an EDCHE system that may perform operations involving multiple encrypted data values, the same packing/coefficient distribution method to ensure that the source message multivectors do not have all equivalent coefficients will minimize the potential for the cryptotext multivector being decrypted from having all equivalent coefficients, but, when various addition and subtraction operations are performed with multiple distinctly different cryptotext multivectors, there is a remote possibility that the cryptotext multivector result of the homomorphic operations will have all equivalent coefficients. The destination computing device may simply assert that such a result cryptotext multivector is "undefined," or, the destination or intermediary computing system may provide a means to update the result cryptotext multivector so the result cryptotext multivector does not have all equivalent coefficients. Great care should be taken to ensure that such an update of the result cryptotext multivector does not change the ultimate value of the result plaintext value of the result cryptotext multivector after decryption. Therefore, it may be desirable for the distribution/packing method to also ensure that not all coefficients are equal to each other (i.e., at least one coefficient should be different than the other coefficients).
[0055] For embodiments that intend to retain homomorphic properties for encrypted data messages such as an embodiment of an EDCHE system, there is an additional restriction that the "packed" multivector that represents the original plaintext numeric message have a mathematical relationship (i.e., the homomorphic preserving mathematical relationship) to the original plaintext numeric message. In abstract algebra, the term homomorphism refers to a structure-preserving map between two algebraic structures, such as groups, rings, or vector spaces. An algebra homomorphism between two algebras is one that preserves the algebra structure. In order to preserve the algebra structure between arithmetic and Geometric Algebra operations, the method by which numbers are "packed" into multivector elements must remain a representation of the original number. One such relationship for packing the coefficients of the multivector that preserves homomorphic properties is to ensure that the coefficients of the multivector representation of the plaintext numeric message follow a mathematical data organization between the value of the plaintext numeric message and at least one of the values of the coefficients of the multivector representation of the plaintext numeric message where the mathematical operations incorporating the one or more values of the multivector coefficients have a result equal to the original plaintext numeric message value. The mathematical relationship may include: addition of at least one coefficient of the multivector coefficients, subtraction of at least one coefficient of the multivector coefficients, addition of a constant value, subtraction of a constant value, multiplication of at least one coefficient of the multivector coefficients by a constant value, and division of at least one coefficient of the multivector coefficients by a constant value. The location of the various mathematical operations relative to the particular locations of the coefficients in the multivector representation should also be consistently applied to all source numeric data messages converted to a multivector as well as for result multivectors converted to a result numeric data value in a particular encryption/decryption pathway.
[0056] The restriction to retain homomorphic properties is only applicable to the distribution (i.e., "packing") of the message multivector (M) coefficients and not to the distribution (i.e., "packing") of the shared secret multivector . Consequently, the distribution (i.e., "packing") of the shared secret multivector may be performed in any
Figure imgf000026_0001
fashion so long as the distribution (i.e., "packing") method of the shared secret multivector (Ss) is known and used consistently by the source 202 and destination 204 computing devices as, ultimately, the shared secret multivector (Ss) used by the source 202 and destination 204 should be equal to each other to ensure that the decryption operations 226 work properly in relation to the encryption 214 operations. The number of potential coefficients is directly related to the size/dimension (N) of the multivectors such that the number of coefficients increases by a factor of 2 (i.e., 2*) for each incremental increase in the size/dimension (N) of the multivector. To increase the confusion and/or diffusion of the encryption process disclosed herein, using multivectors of at least two dimensions will provide at least four coefficients to distribute the numeric data of the message (M) and the shared secret (Ss). By increasing the number of dimensions (N) of multivectors beyond two-dimension
multivectors, the confusion and/or diffusion security characteristics will also be increased due to the additionally available multivector coefficients. Further, with the additionally available coefficients it is also possible to transfer more data in a single multivector message (M) payload using the additionally available multivector coefficients.
[0057] At process 214, the source 202 encrypts a cryptotext multivector (C) as a function of at least one Geometric Algebra geometric product operation on the message multivector (M) and the shared secret multivector (Ss). At process 216, the source 202 converts the cryptotext multivector (C) into cryptotext numeric data (C) in accord with reverse operation of a cryptotext data coefficient distribution algorithm that is known to both the source 202 and the destination 204. While not typical of most encryption systems, an embodiment may also omit process 216 and directly send a representation of the cryptotext multivector (C) without first converting the cryptotext multivector (C) into cryptotext numeric data (C). The transmission may be implemented as a series of transfers of the coefficients or as some form of records/packets that define a data structure that carries the coefficient data of the cryptotext multivector (C). Typically, the various embodiments will include process 216 to convert the cryptotext multivector (C) into cryptotext numeric data (C) in order to maintain compatibility with legacy and/or third parry systems as well as to obtain the additional confusion and diffusion characteristics of encapsulating the cryptotext multivector (£) coefficients into a single cryptotext numeric data (C) value. If process 216 is used to convert the cryptotext multivector (C) into cryptotext numeric data (C), it is necessary for any computing device/system that wishes to operate on the cryptotext multivector (C) to have knowledge of the particular conversion methodology so that computing device/system may properly recreate the cryptotext multivector (C). [0058] Due to the nature of the geometric product operation of Geometric Algebra, there are many possible variations of the geometric product application that will provide similar degrees of confusion and diffusion. Some, but not all, of the potential geometric product calculations to encrypt the message data (M ) include: a geometric product
Figure imgf000028_0002
of the message multivector
Figure imgf000028_0005
( ) and the shared secret multivector
Figure imgf000028_0001
geometric product "sandwich" (
Figure imgf000028_0004
and multivector based Sylvester's equation
Figure imgf000028_0003
. At process 218, the source 202 sends the cryptotext numeric data (C) to the destination 204.
[0059] At process 220, the destination 204 receives the cryptotext numeric data (C) sent by the source 202. At process 222, the destination distributes the cryptotext numeric data (C) into the cryptotext multivector (C) using the cryptotext data coefficient distribution algorithm that is known to both the source 202 and the destination 204. For the less typical
embodiment, if process 216 to convert the cryptotext multivector
Figure imgf000028_0017
( ) into cryptotext numeric data (C) is omitted, then process 222 is also omitted as the cryptotext multivector
Figure imgf000028_0018
was transmitted directly so there is not a need to convert the cryptotext numeric data (C) back into the cryptotext multivector
Figure imgf000028_0016
At process 224, the destination 204 distributes shared secret numeric value (Ss) into shared secret multivector (Ss) coefficients in the same fashion as was done for the source 202 at process 212. At process 226, the destination decrypts the cryptotext multivector
Figure imgf000028_0015
as a function of at least one Geometric Algebra geometric product operation on the cryptotext multivector
Figure imgf000028_0014
and an inverse
Figure imgf000028_0006
( *) of the shared secret multivector (Ss) back into the message multivector
Figure imgf000028_0013
( f). Again, due to the nature of the geometric product operation of Geometric Algebra there are many possible variations of the geometric product application that will provide similar degrees of confusion and diffusion. Some, but not all, of the potential geometric product calculations to decrypt the message data
(M) include: a geometric product
Figure imgf000028_0008
of the cryptotext multivector (C) and the inverse of the shared secret multivector ( geometric product "sandwich" (
Figure imgf000028_0009
Figure imgf000028_0010
Figure imgf000028_0007
to decrypt); and multivector based Sylvester's equation +
Figure imgf000028_0019
Figure imgf000028_0011
to decrypt). At process 228, the destination 204 converts the message multivector
Figure imgf000028_0012
into the message numeric data (M ) in accord with reverse operation of the message data coefficient distribution algorithm of the source 202 at process 210. At process 230, if necessary, the destination 202 converts the numeric message data (M ) back into the original alphanumeric text message as a reverse function of the process of the source 202 at step 208 that converted that alphanumeric text to the numeric message data (M) using standard computer character encoding characteristics.
[0060] Fig. 3A is a flow chart 300 of the source computing device symmetric key operation for an embodiment. The encryption process 214 of the source 202 of Fig. 2 may further include processes 302-306 to use symmetric shared secret security keys to further enhance the security of an embodiment For instance, at process 302, the source computing device may generate/extract/obtain a second shared secret key from the original shared
Figure imgf000029_0003
secret multivector by performing a 0-Blade Reduction Operation on the original shared secret multivector to obtain a scalar numerical value for the second shared secret key
Figure imgf000029_0002
The 0-Blade Reduction Operation may be found as a geometric product
Figure imgf000029_0004
of the geometric product of the original shared secret multivector
Figure imgf000029_0001
Figure imgf000029_0008
Figure imgf000029_0005
and a Clifford conjugate
Figure imgf000029_0010
of the original shared secret multivector and the geometric
Figure imgf000029_0007
reverse
Figure imgf000029_0011
( ) of the geometric product
Figure imgf000029_0009
¾) of the shared secret multivector and
Figure imgf000029_0006
the Clifford conjugate ¾ of the original shared secret multivector A further
Figure imgf000029_0012
Figure imgf000029_0013
discussion of the 0-Blade Reduction Operation to calculate the second shared secret key
Figure imgf000029_0014
) may be found herein, below.
[0061] At process 304, the source computing device distributes the second shared secret key numeric value
Figure imgf000029_0022
into second shared secret multivector coefficients where also
Figure imgf000029_0015
not all coefficients are equal to each other (i.e., at least one coefficient should be different than the other coefficients). Once again, one skilled in the art will recognize that there are many approaches for distributing numeric data into several coefficients of a multivector (see herein, below, for disclosure of one such distribution method). The primary requirement for the distribution process from the numeric values of the second shared secret key to the
Figure imgf000029_0017
second shared secret multivector coefficient values
Figure imgf000029_0016
2) is that the source computing device (of Fig. 3A) and the destination computing device (of Fig. 3B) both know the process 304/314 such that the destination computing device can reconstruct the original message (M) by being able to independently recreate the second shared secret multivector from the
Figure imgf000029_0018
second shared secret key numerical value As long as it is known to both the source
Figure imgf000029_0019
computing device and the destination computing device, the distribution of numeric data to multivector coefficients may be performed differently between the message (M), the original shared secret
Figure imgf000029_0021
and the second shared secret key
Figure imgf000029_0020
[0062] At process 306, the source computing device encrypts the cryptotext multivector (C) as a function of Geometric Algebra geometric product operations on the message multivector (M), the shared secret multivector (¾), and the second shared secret multivector (¾). Again, due to the nature of the geometric product operation of Geometric Algebra there are many possible variations of the geometric product operations that will provide similar degrees of confusion and diffusion. Some, but not all, of the potential geometric product calculations to encrypt the message data (M) include: geometric product "sandwich" to encrypt); and multivector based Sylvester's equation
Figure imgf000030_0001
Figure imgf000030_0002
encrypt).
[0063] Fig. 3B is a flow chart 310 of the destination computing device symmetric key operation for an embodiment. The decryption process 226 of the destination 204 of Fig. 2, in conjunction with the operation of the source computing device as described in the disclosure above with respect to Fig. 3 A, may include processes 312-316 to use symmetric shared secret security keys to further enhance the security of an embodiment. At process 312, the destination computing device may independently generate/extract/obtain the second shared secret key (Ss_,) from the original shared secret multivector
Figure imgf000030_0017
by performing the 0-Blade Reduction Operation on the original shared secret multivector
Figure imgf000030_0018
to obtain a scalar numerical value for the second shared secret key
Figure imgf000030_0019
Again, the 0-Blade Reduction
Operation may be found as a geometric product (
Figure imgf000030_0003
of the geometric product
Figure imgf000030_0008
of the original shared secret multivector
Figure imgf000030_0007
) and a Clifford conjugate ) of
Figure imgf000030_0004
the original shared secret multivector
Figure imgf000030_0006
and the geometric reverse
Figure imgf000030_0005
of the geometric product
Figure imgf000030_0009
of the shared secret multivector Ss) and the Clifford conjugate
Figure imgf000030_0010
(Ss) of the orginal shared secret multivector
Figure imgf000030_0011
(Ss). See also the further discussion of the 0- Blade Reduction Operation to calculate the second shared secret key which may be
Figure imgf000030_0012
found herein, below.
[0064] At process 314, the destination computing device also distributes the second shared secret key numeric value
Figure imgf000030_0013
into the second shared secret multivector
Figure imgf000030_0014
) coefficients. Yet again, one skilled in the art will recognize that there are many approaches for distributing numeric data into several coefficients of a multivector (see herein, below, for disclosure of one such distribution method). Again, the primary requirement for the distribution process from the numeric values of the second shared secret key
Figure imgf000030_0015
Ss_,) to the second shared secret multivector coefficient values ) is that the source computing device
Figure imgf000030_0016
(of Fig. 3A) and the destination computing device (of Fig. 3B) both know the process 304/314 such that the destination computing device can reconstruct the original message (M) by being able to independently recreate the second shared secret multivector from the
Figure imgf000031_0005
second shared secret key numerical value (
Figure imgf000031_0006
As long as it is known to both the source computing device and the destination computing device, the distribution of numeric data to multivector coefficients may be performed differently between the message (M), the original shared secret (Ss), and the second shared secret key (
Figure imgf000031_0004
¾
[0065] At process 316, the destination computing device decrypts the cryptotext multivector ) as a function of Geometric Algebra geometric product operations on the cryptotext multivector ( of the original shared secret multivector
Figure imgf000031_0003
Figure imgf000031_0007
and an inverse of the second shared secret multivector
Figure imgf000031_0008
back into the message multivector
Figure imgf000031_0009
( ) Once again, due to the nature of the geometric product operation of Geometric Algebra there are many ossible variations of the geometric product operations that will provide similar degrees of confusion and diffusion. Some, but not all, of the potential geometric product calculations to decrypt the message data (M ) include: geometric product "sandwich"
Figure imgf000031_0002
to decrypt); and multivector based Sylvester's equation (
Figure imgf000031_0001
to decrypt), the selection of which is based on the geometric product calculation used to encrypt the message data (M ).
[0066] Fig. 4A is a flow chart 400 of the source computing device symmetric key and cryptotext masking operation for an embodiment. Similar to the disclosure with respect to Fig. 3A above, the encryption process 214 of the source 202 of Fig. 2 may further include processes 402-406 to use symmetric shared secret security keys to further enhance the security of an embodiment. At process 402, the source computing device may
generate/extract/obtain a second shared secret key (S¾) from the original shared secret multivector by performing a 0-Blade Reduction Operation on the original shared secret multivector
Figure imgf000031_0011
to obtain a scalar numerical value for the second shared secret key
Figure imgf000031_0015
The 0-Blade Reduction Operation may be found as a geometric product
Figure imgf000031_0013
Figure imgf000031_0010
f me geometric product
Figure imgf000031_0012
of the original shared secret multivector
Figure imgf000031_0014
and a Clifford conjugate
Figure imgf000031_0016
of the original shared secret multivector
Figure imgf000031_0022
) and the geometric reverse
Figure imgf000031_0019
of the geometric product
Figure imgf000031_0017
) of the shared secret multivecto
Figure imgf000031_0021
r
Figure imgf000031_0023
) and the Clifford conjugate of the original shared secret multivector ). A further
Figure imgf000031_0020
Figure imgf000031_0018
discussion of the O-Blade Reduction Operation to calculate the second shared secret key
Figure imgf000032_0002
may be found herein, below.
[0067] At process 404, the source computing device distributes the second shared secret key numeric value ) into second shared secret multivector coefficients where also
Figure imgf000032_0003
Figure imgf000032_0001
not all coefficients are equal to each other (i.e., at least one coefficient should be different than the other coefficients). Once again, one skilled in the art will recognize that there are many approaches for distributing numeric data into several coefficients of a multivector (see herein, below, for disclosure some such distribution methods). The primary requirement for the distribution process from the numeric values of the second shared secret key to the
Figure imgf000032_0008
second shared secret multivector coefficient values
Figure imgf000032_0007
is that the source computing device (of Fig. 4A) and the destination computing device (of Fig. 4B) both know the process 404/416 such that the destination computing device can reconstruct the original message (M) by being able to independently recreate the second shared secret multivector from the
Figure imgf000032_0006
second shared secret key numerical value
Figure imgf000032_0004
( sz) As long as it is known to both the source computing device and the destination computing device, the distribution of numeric data to multivector coefficients may be performed differently between the message (M), the original shared secret (Ss), and the second shared secret key
Figure imgf000032_0005
[0068] At process 406, the source computing device encrypts the cryptotext multivector ( ) as a function of Geometric Algebra geometric product operations on the message multivector (M), the shared secret multivector (Ss), and the second shared secret multivector ¾ . Again, due to the nature of the geometric product operation of Geometric Algebra there are many possible variations of the geometric product operations that will provide similar degrees of confusion and diffusion. Some, but not all, of the potential geometric product calculations to encrypt the message data (M) include: geometric product "sandwich" to encrypt); and multivector based Sylvester's equation
Figure imgf000032_0009
Figure imgf000032_0010
encrypt).
[0069] At process 408, in the process of the source computing device for converting the cryptotext multivector
Figure imgf000032_0011
) into cryptotext numeric data (C) (see, for example, the disclosure above with respect to process 216 of Fig. 2), the cryptotext multivector
Figure imgf000032_0012
( ) is first converted into a pre-cipher cryptotext (C) in accord with reverse operation of a cryptotext data coefficient distribution algorithm that is known to both the source computing device (Fig. 4A) and the destination computing device (Fig. 4B). The pre-cipher cryptotext (C') is then masked as an exclusive or (C = C' XOR Ss) of the pre-cipher cryptotext (C) and the original shared secret numeric value (Ss) to obtain/create the cryptotext numeric data (C) to send to the destination computing device.
[0070] Fig. 4B is a flow chart 410 of the destination computing device symmetric key and cryptotext masking operation for an embodiment. At process 412, in the process of the destination computing device for distributing the cryptotext numeric data (C) into the coeffecients for the cryptotext multivector (C) (see, for example, the disclosure above with respect to process 222 of Fig. 2), is first unmasked through an exclusive or (C = C XOR Ss) of the sent cryptotext numeric data (C) and the original shared secret numeric value (Ss) in order to obtain/create the pre-cipher cryptotext numeric data (C) on the destination computing device. The destination computing device then distributes the pre-cipher cryptotext numeric data (C') into the cryptotext multivector (C) using the cryptotext data coefficient distribution algorithm that is known to both the source and destination computing devices.
[0071] The remaining decryption process 226 of the destination 204 of Fig. 2, in conjunction with the operation of the source computing device as described in the disclosure above with respect to Fig. 4A, may include processes 414-418 to use symmetric shared secret security keys to further enhance the security of an embodiment. At process 414, the destination computing device may independently generate/extract/obtain the second shared secret key (¾) from the original shared secret multivector (Ss) by per
Reduction Operation on the original shared secret multivector (Ss) to
numerical value for the second shared secret key
Figure imgf000033_0010
. Again, the 0-B
Operation may be found as a geometric product
Figure imgf000033_0001
product original shared secret multivector and a Cli if
Figure imgf000033_0005
Figure imgf000033_0002
the original shared secret multivector
Figure imgf000033_0003
and the geometric reverse (
geometric product
Figure imgf000033_0006
) of the shared secret multivector
Figure imgf000033_0004
) and the
) of the orginal shared secret multivector
Figure imgf000033_0007
See also the further
Blade Reduction Operation to calculate the second shared secret key (
found herein, below.
[0072] At process 416, the destination computing device also distributes the second shared secret key numeric valu ) into the second shared secret multivector
Figure imgf000033_0008
Figure imgf000033_0009
coefficients. Yet again, one skilled in the art will recognize that there are many approaches for distributing numeric data into several coefficients of a multivector (see herein, below, for disclosure of one such distribution method). Again, the primary requirement for the distribution/packing process from the numeric values of the second shared secret key ( to
Figure imgf000034_0008
the second shared secret multivector coefficient values is that the source computing
Figure imgf000034_0009
device (of Fig. 4A) and the destination computing device (of Fig. 4B) both know the process 404/416 such that the destination computing device can reconstruct the original message (M ) by being able to independently recreate the second shared secret multivector (Ss_) from the second shared secret key numerical value (Ss2). As long as it is known to both the source computing device and the destination computing device, the distribution of numeric data to multivector coefficients may be performed differently between the message (M), the original shared secret (Ss), and the second shared secret key
Figure imgf000034_0004
[0073] At process 416, the destination computing device decrypts the cryptotext multivector (C) as a function of Geometric Algebra geometric product operations on the cryptotext multivector
Figure imgf000034_0001
of the original shared secret multivector
Figure imgf000034_0003
and an inverse
Figure imgf000034_0005
a of the second shared secret multivector
Figure imgf000034_0002
back into the message multivector Once again, due to the nature of the geometric product operation of Geometric Algebra there are many possible variations of the geometric product operations that will provide similar degrees of confusion and diffusion. Some, but not all, of the potential geometric product calculations to decrypt the message data ( M) include: geometric product "sandwich" to decrypt); and multivector based Sylvester's
Figure imgf000034_0006
equation to decrypt), the selection of
Figure imgf000034_0007
which is based on the geometric product calculation used to encrypt the message data (M).
[0074] Additionally, while the flow charts and flow chart details described above with respect to Figs. 2-4 describe a methodology that may be embodied as a method or process, another embodiment may be recognized as a computer system, and/or as a source computer system and a destination computer system, that encrypts data, transfers the data, and decrypts the data by implementing the processes described above with respect to the flow chart and flow chart details of Figs. 2-4. Further, in describing the computer system, and/or the source computer system and the destination computer system, that encrypts data, transfers the data, and decrypts the data, one, or more, individual processes described above for the methodology may be broken down and represented as a subsystem of the overall encryption computer system. A subsystem of the computer system, and/or the source computer system and the destination computer system, that encrypts data, transfers the data, and decrypts the data may be assigned, in whole or in part, to a particular hardware implemented system, such as a dedicated Application Specific Integrated Circuit (ASIC) or Field Programmable Gate Array (FPGA). One or more subsystems, in whole or in part, may alternatively be implemented as software or firmware instructions defining the operation of a computer system with specific regard to the one or more subsystems implemented as software or firmware instructions. The software or firmware instructions may cause the Central
Processing Unit, memory, and/or other systems of a computer system to operate in particular accordance with the particular one or more subsystems designated features.
Section 2: Additional Descriptions of Core EDCE Message Encryption/Decryption
[0075] The disclosure below provides a simplified example of the operations and data relationships during the performance of a fundamental "core" EDCE embodiment The amount of data, the type of data, and the particular data values shown and described in the example are not meant to represent any particular real system, but are provided only for the purpose of showing the operations and data relationships of an embodiment. Further, the embodiments described below are not meant to restrict operations to particular data types, encryption shared secret key exchange techniques, text to numeric and back conversion techniques, and/or number to multivector coefficient assignment techniques.
[0076] In addition to the utilization of the Geometric Algebra geometric product as a novel encryption primitive, the various embodiments may be comprised of functional blocks, each of which may be tailored as described according to objectives for scope, capability and security. The following sections provide a mathematical and numerical description of one or more example embodiments of these functional blocks. The numerical results in the examples are generally derived from Geometric Algebra executing in the C programming language.
A. Packing and Unpacking Multivectors
Contents
1) Text to Number
2) Number to text
3) Multivector data structure
4) Number to multivector 5) Multivector to number
1) Text to number
[0077] For the example EDCE embodiment described herein, each text message needs to be converted to a number in order to become a valid operational unit for all DCE computations. For the embodiments shown herein, the numbers are typically shown in base 10, but the various embodiments may choose other number bases as desired by the system designer. For instance, a hex (base 16) representation may provide particular advantages when dealing with ASCII numerical representations as standard ASCII has a representation based on the numbers 0-127 (i.e., 27), which is one power of two (i.e., hex is 28) less than the typical 8 bits represented by a hex number of xFF. According to the ASCII character- encoding scheme, symbols such as the letters a, b, c and so on, are represented in order formats (such as binary, decimal, octets, hexadecimal, etc.), which are described in the ASCII printable code chart, a table that presents the relationship between formats. So the letters "a," "b" and "c" in ASCII decimal code are 97, 98 and 99, respectively.
[0078] As an example, assume that the plaintext text message is "message." In ASCII decimal code, this is represented as follows: m e s s a g e
109 101 115 115 97 103 101
[0079] With this relationship between symbols and decimal numbers, the conversion from text to number in base 10, using the text "message", is executed as follows:
[0080] The variable n represents the final number of the conversion from text to number. We start defining this variable to zero. So, n = 0.
[0081] Then we create an array with the ASCII decimal codes for each letter of the message: text = "message"
ASCII_array_from_"message" = [109, 101. 115, 115, 97, 103, 101]
[0082] This array has a size of 7 elements, thus array size = 7
[0083] Then, for each value of the array of ASCII characters, in a loop, we will (i) multiply n by 256 (we chose 256 because it is a power of two greater than the largest number in the ASCII printable code chart, so we are reserving a space of 8 bits, since 28 = 256)
(ii) sum with the equivalent ASCII decimal code, assigning the result to the same initial n variable, as follows:
For i=0; i < array_size; i++
n = n * 256 + ascii_array_from_message[i]
Note the details of each iteration below: n = 0 For i = 3
array = [109, 101, 115, 115, 97, 103, n = 7169395 * 256 + 115
101] n = 1835365235
array_size = 7
For i = 4
For i = 0 n = 1835365235 * 256 + 97 n = 0 * 256 + 109 n = 469853500257
n = 109
For i = 5
For i = 1 n = 469853500257 * 256 + 103 n = 109 * 256 + 101 n = 120282496065895
n = 28005
For i = 6
For i = 2 n = 120282496065895 * 256 + 101 n = 28005 * 256 + 115 n = 30792318992869221 n = 7169395
Figure imgf000037_0001
[0084] By performing the above calculation, the final value of n is:
30792318992869221
[0085] Thus, the plain text "message" as a number in base 10 is equal to
30792318992869221. Once we have a base 10 number it is possible to perform the calculations described herein for message encryption. If desired, entropy may be added at this step by performing transformations on the ASCII codes, such as addition or modulo operations, but those entropy adding operations may affect whether intermediary homomorphic operations may properly be performed on the message data as those entropy adding operations may adversely affect the mathematical relationship to the original message values. No such entropy adding transformations are used in the examples that follow.
2) Number to text
[0086] After performing various calculations, a base 10 number is transmitted and received. From the above example of a message multivector, the coefficients are concatenated to form a number string. The "number to text" conversion process for this number string also uses the ASCII printable code chart, but the recovery routine is different from the "text to number" conversion. The procedure is described below:
[0087] We start with the variable s, which is an empty string that will become the final text recovered from the input number. (Note: the symbol " "" " is from the C-language and means empty string)
The input number is 30792318992869221. n = 30792318992869221
[0088] Now, we perform a loop until n is "emptied", since this number refers to an actual text message. This means the loop will stop when n is equal to zero. In each loop iteration, we will recover, from the last to the first, each ASCII decimal code correspondent to the text that we are retrieving. To do that, we will perform a bitwise AND operation using the value OxFF (which is 256-1 in hexadecimal format or in base 16). We will convert the code to character symbols and concatenate with the current string, always putting the most recent recovered character in the front of the string. Lastly, we will update the value of n by performing a right shift of 8 bits.
[0089] Let's say that the function "get_char" converts the ASCII decimal code to a character symbol. [0090] The procedure is as follows: while n > 0
s = get_char(n AND OxFF) + s
[0091] Note the details of each iteration below: s = ""
n = 30792318992869221
while n > 0
s = get_char(n AND OxFF) + s
Figure imgf000040_0001
[0092] Thus, the number 30792318992869221 is converted to the text string 'message," which agrees with the original plaintext 3) Multivector Data Structure
[0093] For the example embodiment discussed herein, any number in base 10 may be a coefficient of a multivector element. A multivector may contain arbitrary data, or data that is a result of a series of operations. A base 10 number may also be represented in multivector form by distributing pieces of this number string to the coefficients in the multivector.
Multivectors that are 2D have 4 elements/coefficients available to pack with pieces of this number string, a 3D multivector has 8 elements, and 4D has 16. EDCE has been
demonstrated up to at 7D. A 4D multivector with 16 elements is written as:
Figure imgf000041_0001
4) Number to multivector
[0094] Given the base 10 number string 30792318992869221, this string may be a single coefficient of, say, a 2D multivector, as follows:
0 + 30792318992869221e1 + e2 + e12
[0095] EDCE has been demonstrated where the number string distributed to an element of the multivector exceeds 4,000 digits. However, the base 10 number in our example will typically be "distributed" in an ad hoc manner across all the multivector elements, such as:
30792 + 31899e1 + 28692e2 + 21e12
[0096] The above distribution is called "number to multivector." For a core EDCE embodiment, the method of distributing the number string may be according to any of a variety of algorithms as long as the method is known and used by both the sending and receiving entities. To increase cryptographic "confusion," the distribution algorithm may include shuffling of the assignments to elements, performing functional operations on numbers assigned to elements or changing the algorithm between messages in a conversation. More operations increase encryption entropy. However, it should be noted that shuffling and other algorithms to increase cryptographic confusion may break the potential for
homomorphic operations by and EDCHE embodiment. [0097] In order to ensure that the Geometric Algebra operations properly impart encryption security and also provide decryption capability, there are also some other rules that should be followed when performing the "number to multivector" process. For instance, the distributing/packing method defines, among many things, the Geometric Algebra operations permissible for EDCE and/or EDC E embodiments. For example, the
Rationalize operation on multivectors yields zero when all multivector coefficients are equal. Such multivectors having all equal coefficients have no inverse and the geometric product of such multivectors having all equal coefficients with another multivector has no inverse. As discussed in more detail below, the decryption methodology for EDCE and EDCHE systems utilize the inverse of the cryptotext multivector being decrypted and of the security key(s) multivector to perform the decryption. Therefore, the cryptotext multivector being decrypted should not have all equal value coefficients. One means to ensure that the cryptotext multivector being decrypted does not have all equal value coefficients is to have the packing/coefficient distribution method ensure that not all coefficients are equal to each other (i.e., at least one coefficient should be different than the other coefficients) when creating the shared security multivector(s) and the data message multivectors. For an embodiment of the EDCE that simply transfers the data message, this will ensure that the cryptotext multivector to be decrypted will not have all equivalent coefficients. For an EDCHE system that may perform operations involving multiple encrypted data values, the same packing/coefficient distribution method to ensure that the source message multivectors do not have all equivalent coefficients will minimize the potential for the cryptotext multivector being decrypted from having all equivalent coefficients, but, when various addition and subtraction operations are performed with multiple distinctly different cryptotext multivectors, there is a remote possibility that the cryptotext multivector result of the homomorphic operations will have all equivalent coefficients. The destination computing device may simply assert that such a result cryptotext multivector is "undefined," or, the destination or intermediary computing system may provide a means to update the result cryptotext multivector so the result cryptotext multivector does not have all equivalent coefficients. Great care should be taken to ensure that such an update of the result cryptotext multivector does not change the ultimate value of the result plaintext value of the result cryptotext multivector after decryption.
[0098] The simple distribution method used in some of the core EDCE embodiment examples below is described as follows: Let the input base 10 number string =
30792318992869221. We count the number of digits and determine that the number size is 17 digits. We then determine how to distribute these digits to the elements of a multivector. Considering a multivector of 2D, which has 4 elements we apply the following equation:
Figure imgf000043_0001
[0099] Where ep is "each portion" length.
[0100] Now we have the original base 10 number and its size (17), the multivector structure (2D, 8 elements) and the length of each element (5). Now we need to "slice" the base 10 number in order to distribute each part as a coefficient of the new multivector.
[0101] Computationally, the procedure is as follows:
Figure imgf000043_0002
[0102] This creates the following multivector:
30792 + 31899e1 + 28692e2 + 21e: 12
Handling special cases:
[0103] Regardless of the method of distribution, the leading digit in any coefficient must be non-zero. For example, let the number to be converted to multivector
be 30792318990869221. Applying the distribution method shown above would result in: 30792 + 31899e1 + 08692e2 + 21e
[0104] Note the third element = 08692e2. The computer will treat this number as 8692. When converting back from multivector to number, instead of
30,792,318,990,869,221 we would have 3,079,231,899,869,221, which is not the same number (commas added only for comparability).
[0105] To avoid this outcome, it is necessary to include verification in the algorithm that the first number of a coefficient is non-zero. If it is zero, this number should be placed as the last number in the coefficient of the previous element of the multivector. So, the correct result of the conversion of the number 30792318990869221 to a 2D multivector is:
30792 + 318990e1 + 8692e2 + 21e12
Homomorphic "number to multivector" distribution method restrictions:
[0106] For embodiments that intend to retain homomorphic properties for encrypted data messages such as an embodiment of an EDCHE system, there is an additional restriction that the "packed" multivector that represents the original plaintext numeric message have a mathematical relationship (i.e., the homomorphic preserving mathematical relationship) to the original plaintext numeric message. In abstract algebra, the term homomorphism refers to a structure-preserving map between two algebraic structures, such as groups, rings, or vector spaces. An algebra homomorphism between two algebras is one that preserves the algebra structure. In order to preserve the algebra structure between arithmetic and Geometric Algebra operations, the method by which numbers are "packed" into multivector elements must remain a representation of the original number. One such relationship for packing the coefficients of the multivector that preserves homomorphic properties is to ensure that the coefficients of the multivector representation of the plaintext numeric message follow a mathematical data organization between the value of the plaintext numeric message and at least one of the values of the coefficients of the multivector representation of the plaintext numeric message where the mathematical operations incorporating the one or more values of the multivector coefficients have a result equal to the original plaintext numeric message value. The mathematical relationship may include: addition of at least one coefficient of the multivector coefficients, subtraction of at least one coefficient of the multivector coefficients, addition of a constant value, subtraction of a constant value, multiplication of at least one coefficient of the multivector coefficients by a constant value, and division of at least one coefficient of the multivector coefficients by a constant value. The location of the various mathematical operations relative to the particular locations of the coefficients in the multivector representation should also be consistently applied to all source numeric data messages converted to a multivector as well as for result multivectors converted to a result numeric data value in a particular encryption/decryption pathway.
[0107] Additional and more detailed examples of homomorphism conserving "number to multivector" processes are described in more detail below in Section 3 that covers scalar multiplicative homomorphic EDCHE embodiments in more detail
5) Multivector to number
[0108] The simple distribution method used in some of the core EDCE embodiment examples below is described as follows:
[0109] For the simple distribution (i.e., "packing") method disclosed above for parsing the string representation of a base 10 number to obtain the coefficient values, converting a multivector to a base 10 number is simply the reverse process of concatenating the coefficients of the multivector in order to form a base 10 number.
[0110] As an example: The multivector: 30792 + 31899ex + 28692e2 + 21e12 becomes: 30792318992869221.
[0111] Note that in the core EDCE protocol of some of the example embodiments herein, only base 10 number strings are transmitted, not multivectors, but sending only base 10 number strings is not a requirement for an embodiment. In some embodiments, the number may be sent using a numeric variable representation such as an integer or floating point data type. Further, while not typical of most encryption systems, instead of sending a single cryptotext number (C), an embodiment may also simply skip the step of converting the multivector (C) into cryptotext numeric data (C), and directly send a representation of the cryptotext multivector (C) without first converting the cryptotext multivector (C) into cryptotext numeric data (C). The transmission may be implemented as a series of transfers of the coefficients or as some form of records/packets that define a data structure that carries the coefficient data of the cryptotext multivector (C). As would be the case for a more typical encryption system, if the process to convert the cryptotext multivector (C) into cryptotext numeric data (C) is used to convert the cryptotext multivector (C) into cryptotext numeric data (C), it is necessary for any computing device/system that wishes to operate on the cryptotext multivector (
Figure imgf000046_0003
to have knowledge of the particular conversion methodology so that the computing device/system may properly recreate the cryptotext multivector (C).
B. Shared Secret
[0112] A "Shared Secret" is a fundamental element in cryptography. A Shared Secret enables secure communication between two or more parties. For the various embodiments the Shared Secret is a number string of digits that may be packed into a multivector in the he "Shared Secret Multivector" may be used to operate on other ating the geometric product of the Shared Secret Multivector and the message multivector.
[0113] A variety of methods are already in practice to establish the Shared Secret between sources and destinations. As disclosed herein, the conversion of a "Shared Secret" number to a "Shared Secret Multivector" is completely novel. Communication end-point devices may be "pre-conditioned" with a unique identifier (number string) known only to the system administrators. In a public/private key environment such as RS A, the Shared Secret may be encrypted by the source using only the destination's public key. The method used in the examples below is the Diffie-Hellman key exchange protocol. This is a convenient, widely adopted method to establish a number string Shared Secret. However, any method that securely produces a shared number string is suitable for use with the various embodiments.
[0114] The Diffie-Hellman protocol uses the multiplicative group of integers modulo p (see, for example,
https://en.wikipedia.org/wiki/Multiplicative groupe of integers modulo n), where p is prime
{see, for example, https://en.wikipedia.org/wiki/Prime_number), and g is a primitive root modulo p (see, for example, https://en.wildpedia.org/wiki/Primitive_root_modulo_n and https://en.wikipedia.org/wiki/Modular_arithmetic). These two values are chosen in this way to ensure that the resulting shared secret can take on any value from 1 to p-\ . A simple example of Diffie-Hellman follows:
• Alice and Bob first agree on using the same root modulo p and base g.
• Alice chooses a secret integer a (Alice's password) and creates her signature s
Figure imgf000046_0002
Figure imgf000046_0001
and sends it to Bob. (Note: the superscript 0 is a placeholder for later use, if any.)
• y, Bob chooses a secret integer b (Bob's password) and creates his signature
Figure imgf000047_0004
Figure imgf000047_0003
and sends it to Alice.
• Alice and Bob are able to compute the shared secret key Ss as follows:
Figure imgf000047_0002
• The keys computed by Alice and Bob are the same. This is the Shared Secret.
[0115] Note that Diffie— Hellman protocol is not limited to negotiating a key shared by only two participants. Any number of users can take part in the agreement by performing iterations of the protocol and exchanging intermediate data.
Numeric example
[0116] Assume the following:
Figure imgf000047_0001
= 286639204586849997936652161962
Figure imgf000048_0001
[0118] To compute the shared secret, both Alice and Bob will perform the following equation, which will generate the same value for both, thus the shared secret is reference as
Figure imgf000048_0002
Ss = 374101092446920532886590141005
[0119] The shared secret number string above may be distributed as before to create a Shared Secret Multivector:
Figure imgf000048_0003
[0120] In a similar manner the S
Figure imgf000048_0004
number string for Alice and Bob can be distributed in a multivector format to create
Figure imgf000048_0005
¾ These multivectors for Alice and Bob are essential to the additional EDCE security features discussed in Section 2.
C. Cryptotext creation
[0121] The cryptotext is created using the EDCE primitive which is the geometric product of the Message multivector and one or more other multivectors. In the most basic form, the cryptotext multivector may be the geometric product of the Message multivector and the Shared Secret Multivector.
[0122] The procedure is defined as follows. Let the plaintext message be "this is a test" By applying the "text to number" conversion, we will get the plaintext message as the number:
2361031878030638688519054699098996 [0123] By applying the "number to multivector" conversion using a 2D multivector structure the plaintext multivector is:
Figure imgf000049_0001
[0124] Using the Shared Secret multivector that was determined above:
Figure imgf000049_0002
[0125] The cryptotext multivector can be defined as the geometric product:
Figure imgf000049_0003
[0126] Using methods for calculating the geometric product of M and Ss implemented in C programming code, the cryptotext multivector C described above is calculated as:
Figure imgf000049_0004
[0127] In order to be transmitted, as a payload, C now may be converted to a base 10 number, through the "multivector to number" conversion process described above.
C10 = 5649796324893205335999076139905242395250959838376115938268771181474
[0128] To increase the entropy of the Cryptotext Multivector, the Geometric Product of the Message Multivector may be taken with more than one other multivector or by using the same multivector twice to form a sandwich or by the addition of left and right multivector operations on the same Shared Secret Multivector. Two examples of these types are
Figure imgf000049_0006
Figure imgf000049_0005
The use of these primitives and their inverse is shown in the flow charts in Figs. 3 and 4.
[0129] Note that there are several alternative methods to construct the Cryptotext Multivector. One alternative is to encrypt the plaintext message using a conventional symmetric cipher such as AES, converting the number string output of that cipher to multivector format and use this multivector in calculating the geometric product with Ss, which yields C. This alternative may be practiced during the transition to EDCE within the enterprise to preserve backward compatibility with legacy encryption systems.
D. Decryption
[0130] Since Bob has the same shared secret of the source, he can open the cryptotext by performing a geometric product of the cryptotext multivector and the inverse of the shared secret multivector. When Bob receives C10, he will apply the appropriate number to
multivector conversion to get:
C = 564979632489320S3 + 35999076139905242e1 + 39525095983837611e2
+ 5938268771181474e12
[0131] To recover the plaintext multivector M Bob determines the geometric product of the cryptotext multivector and the inverse of the Shared Secret multivector.
M = CSs'1
[0132] The method to determine Ss 1 given Ss is described in more detail in "Appendix A: Geometric Algebra Overview" of the parent patent application Serial No. 15/667,325, entitled "Methods and Systems for Enhanced Data-Centric Encryption Systems Using
Geometric Algebra." It has been computed as:
= 0.000026073076962327080434360644193616198433147472820027689698131 + 0.0000008254576382902133111810564738782973039080139959893127032226! + (-0.000003022013576864948697543940186562731020525315836555058082087)e2 + (-0.000001476164784851810000367564890720274369477622677012150966249)e3 + (-0.000001979165546727098436715224982579657444314140144210619540629)e12 + (-0.000004182897588855026711933985071316069976075801566911515708555)e13 + (-0.000002840357355486048162850225401518433913770465475305727734285)eZ3 + (-0.000000583992297267872657596889780817224702593392912954493277677)e12r Thus,
M = 236103187 + 803063868e1 + 851905469e2 + 9098996e:
[0133] The multivector M is converted to a base 10 number:
M10 = 2361031878030638688519054699098996
[0134] Finally, this number is converted to text using the "number to text" procedure described above, resulting in:
M plain text = "this is a test"
E. EDCE Flow chart (Figs. 5-6)
[0135] Fig. 5 is a flow chart 500 of an embodiment for the core Enhanced Data-Centric Encryption (EDCE) encryption/decryption performed by using a geometric product
"sandwich."
[0136] Setup (502): The sequence is initiated by establishing the signature and shared secret multivectors. Here the Diffie-Hellman procedure 508 is shown but other asymmetric key ciphers such as RSA may be used to generate a number string known only to the source 504 and the destination 506. Alternatively, end-point devices may be "pre-conditioned" with a secret (number string) known to the system administrator from which the session multivectors may be constructed. The Diffie-Hillman procedure 508 sets up/creates the shared secret keys 510 and then the setup 502 creates multivectors of the Diffie-Hillman keys 510 in the multivector setup 512.
[0137] Source (504): The Message Multivector 516 is constructed at the create message operation 514 by concatenating the message ASCII code string to a number string and then distributing that number to the coefficients of the message multivector at 514. The method of distributing to coefficients uses a prescribed algorithm known and used by both the source 504 and the destination 506.
[0138] The Message Multivector 516 is then encrypted 518 by computing the geometric product of the message and Shared Secret multivectors. Fig. 5 shows the Cryptotext Multivector 520 as the "sandwich" of geometric products The coefficients of
Figure imgf000051_0001
the Cryptotext Multivector 520 are then concatenated into a base 10 number string, C (524), and transmitted through a user-defined dispatch function 526 over an electronic network/bus communication channel 528.
[0139] Destination (506): C (532) is received through a user-defined operation 530 and converted back to the Cryptotext Multivector 536 using the prescribed distribution method 534. The destination 506 computes the multivector inverse of the Shared Secret Multivector and uses this result in the decrypt equations 538 such as to recover the
Figure imgf000052_0001
Message Multivector 540. The Message Multivector 540 is then converted to a number string and then to plaintext at 542.
[0140] FIG. 6 is a flow chart 600 of an embodiment for the EDCE
encryption/decryption by using Sylvester's equation.
[0141] Setup (602): The sequence is initiated by establishing the signature and shared secret multivectors. Here the Diffie-Hellman procedure 608 is shown but other asymmetric key ciphers such as RSA may be used to generate a number string known only to the source 604 and the destination 606. Alternatively, end-point devices may be "pre-conditioned" with a secret (number string) known to the system administrator from which the session multivectors may be constructed. The Diffie-Hillman procedure 608 sets up/creates the shared secret keys 610 and then the setup 602 creates multivectors 612 of the Diffie-Hillman keys in the multivector setup 612.
[0142] Source (604): The Message Multivector 616 is constructed at the create message operation 614 by concatenating the message ASCII code string to a number string and then distributing that number to the coefficients of the message multivector at 614. The method of distributing to coefficients uses a prescribed algorithm known and used by both the source 604 and the destination 606.
[0143] The Message Multivector 616 is then encrypted 618 by computing the geometric product of the message and Shared Secret multivectors. Fig. 6 shows the Cryptotext Multivector as the sum of two geometric products The coefficients of the
Figure imgf000052_0002
Cryptotext Multivector 620 are then concatenated into a base 10 number string, C (624), and transmitted through a user-defined dispatch function 626 over an electronic network/bus communication channel 628.
[0144] Destination (606): C (632) is received through a user-defined operation 630 and converted back to the Cryptotext Multivector 636 using the prescribed distribution method 634. The destination 606 computes the multivector inverse of the Shared Secret Multivector and uses this result in the decrypt equations 638 such as
Figure imgf000053_0001
A
Figure imgf000053_0002
Figure imgf000053_0003
to decrypt). Where the encryption uses the sum of two geometric products
Figure imgf000053_0004
the decryption applies multivector based "Sylvester's Equation" to recover the Message Multivector 640. The essage Multivector 640 is then converted to a number string and then to plaintext at 642.
F. Symmetric Key Pair Encryption/Decryption from 0-Blade Reduction Operation (Fig. 7)
0-Blade Reduction Operation
[0145] In order to increase security to the Geometric Algebra encryption primitives, a pair of symmetric shared secret keys may be used instead of a single shared secret key. The following lists the processes that may be used to generate/extract/obtain the second shared secret multivector (¾) from the original shared secret numeric value key (Ss).
1. The original shared secret numeric value (Ss), which may be obtained from a key exchange procedure, such as, but not limited to, the Diffie-Hellman key exchange process, may be used as the first shared secret numeric key of the pair of symmetric shared secret keys;
2. Using a multivector distribution operation, the original shared secret numeric key (Ss) may be loaded into a multivector representation, which may be denoted as
Figure imgf000053_0007
3. A 0-Blade Reduction Operation on the original shared secret multivector
Figure imgf000053_0008
may be performed in order to extract/generate/obtain a scalar value that is the second shared secret numeric key
Figure imgf000053_0005
4. The scalar that results from 0-Blade Reduction Operation, which has been defined as may be loaded into a multivector through another multivector coefficient distribution operation with the resulting multivector being the second shared secret multivector
Figure imgf000053_0006
[0146] Fig. 7 is a block diagram 700 illustrating generating/extracting/obtaining a second shared secret multivector key (¾) 712 from the original shared secret multivector (Ss) 704 for an embodiment. The original shared secret multivector (Ss) 704 may be used to encrypt and decrypt data as the first shared secret multivector key of a pair of symmetric shared secret multivector keys. For the Geometric Algebra encryption primitive 702, the original shared secret multivector (
Figure imgf000054_0002
704 is operated on by the O-Blade Reduction Operation
Figure imgf000054_0001
The 0-Blade Reduction Operation 706 results in the scalar value of the second shared secret numeric key (5¾) 708. A number to multivector coefficient distribution process 710 converts the second shared secret numeric key (
Figure imgf000054_0007
2 into a second shared secret multivector
Figure imgf000054_0003
712. The second shared secret multivector may then be
Figure imgf000054_0004
used to encrypt and decrypt data as the second shared secret multivector key of a pair of symmetric shared secret multivector keys.
Geometric Algebra Encryption Primitives
Primitive 1 - "Sandwich" / Triple Product
Encryption
[0147] The first encryption primitive can be created through a sequence of geometric products using the pair of keys generated via the 0-Blade Reduction Operation (described herein, above) as follows:
Figure imgf000054_0005
Decryption
[0148] The decryption process uses the previously defined inverse multivector as follows:
Figure imgf000054_0006
Primitive 2— Multivector based Sylvester's Equation
Encryption
[0149] The well-known Sylvester's equation is employed here to generate a second encryption primitive which also uses the pair of symmetric encryption keys generated via the 0-Blade Reduction Operation (described herein, above) as follows:
Figure imgf000055_0002
Decryption
[0150] The decryption operation involves the closed-form solution of the Sylvester's equation for 3-dimensional multivector space as follows:
Figure imgf000055_0001
[0151] Note that a solution for higher dimensions requires a different formula. Further note that if the original shared secret (Ss) is generated using an encrypted/secure key exchange, such as, but not limited to the Diffie-Hellman process discussed in more detail herein, there is no transmission of the original shared secret multivector ( and,
Figure imgf000055_0003
consequently, no transmission of the second shared secret multivector (
Figure imgf000055_0004
Still further note that since the second shared secret multivector ) is a result of a one-way function, the use
Figure imgf000055_0005
of the second shared secret multivector
Figure imgf000055_0006
in either of the Geometric Algebra encryption primitive operations above, further guarantees security.
Numerical examples for encryption and decryption with doubled shared-secret in 3 dimensions
[0152] Let the message multivector M be: = 23 + 24e1 + 19e2 + 31e3 + 23e12 + 30e13 + 21e23 + 268e123 and the original secret multivector Ss be: s = 29 + 22e1 + 31e2 + 28e3 + 23e12 + 17e13 + 20e23 + 215e123
[0153] From the original secret multivector
Figure imgf000055_0007
, create a scalar by applying a O-Blade Reduction Operation as follows:
Figure imgf000056_0001
scalar = 2281454761
[0154] Then create the second secret multivector by converting the scalar to a
Figure imgf000056_0003
multivector, as follows:
Figure imgf000056_0002
Geometric product "sandwich" or Geometric triple product
[0155] In order to encrypt the multivector compute the geometric product
Figure imgf000056_0005
"sandwich" as follows:
Figure imgf000056_0004
and recover the message multivector
Figure imgf000056_0006
M as follows:
Figure imgf000056_0007
Multivector based Sylvester's Equation
[0156] Another way to encrypt the message multivector
Figure imgf000056_0009
is by applying the multivector based Sylvester's equation:
Figure imgf000056_0008
and recover the message multivector
Figure imgf000057_0002
as follows:
Figure imgf000057_0001
G. An Unbreakable Primitive Using Geometric Algebra and Arithmetic Functions
Example with secret sharing and 3D multivectors
Set Up
[0157] A multivector may act as a Geometric Algebra object such that components of multi-dimensions and Clifford k-vectors are present. An example is:
Figure imgf000057_0003
which shows the components:
Figure imgf000057_0004
[0158] A typical, but not the only, arithmetic function used for secret sharing is the Diffie-Hellman function, which is based on cyclic groups with element g; for example:
Figure imgf000057_0005
where Ss is a shared secret which can be used by both the source and destination sides and where the operation gabmod p yields Ss. This is standard in the cyber security field. . Unbreakable Primitive
[01S9] Given a message M, distribute the numerical content of M over a multivector
Figure imgf000058_0008
For example, let M be represented as below:
M = m1,m2,m3 ··. m„ such that mi is a number that constitutes a placed integer value for a coefficient. Then:
Figure imgf000058_0007
[0160] Note that other multivector variations are also possible.
[0161] The shared secret Ss is changed to a multivector in the same or a similar manner, such as:
Figure imgf000058_0001
[0162] An operation known as "0-Blade Reduction" creates a new scalar from
Figure imgf000058_0002
as a second shared secret where:
Figure imgf000058_0006
Figure imgf000058_0003
Then is converted to a multivector, for example:
Figure imgf000058_0004
[0163] Finally, the multivector-based Sylvester's equation may be used to create a cipher. Thus, the cryptotext multivector C is:
Figure imgf000058_0005
because S is a result of a one-way function and Ss is a shared secret, which, when operated on by the O-BIade Reduction Operation (which may sometimes be referenced herein as the function Zb O) becomes a new result embedding a new one-way function.
[0164] Up to this point the encryption may have susceptibility to a pair of known cryptotext attacks. However, as shown in part below, the final unbreakability has been achieved.
Encryption primitives with unbreakable cipher:
Primitive 1 - "Sandwich" / Triple Product
Encryption
[0165] The first encryption primitive may be created through a sequence of geometric products using the pair of keys generated via the 0-Blade Reduction Operation (described above) as follows:
Figure imgf000059_0001
[0166] In order to add another layer of security to the cipher text C and guarantee unbreakability against standard cipher analysis, such as the known plain-text attack, it may be helpful to do the following:
• The cipher text, originally in multivector form, is converted to a number C'
Figure imgf000059_0002
yielding a 'pre-cipher' which is denoted here as C'.
• Then 'mask' this pre-cipher C' by performing an XOR operation with the shared secret Ss
C = C' XOR Ss to obtain the final cipher/crypto text C, which is sent to a destination computing device. Decryption
[0167] The decryption process may comprise the following steps:
• Receive the cryptotext C at the destination computing device.
• Recover the pre-cipher cryptotext C' from the cryptotext C using the shared secret Ss, as follows: C' = C XOR Ss
• Convert C' into a multivector yielding the cryptotext multivector C:
Figure imgf000060_0001
• Recover the message multivector M making use of the inverse multivectors with respect to
Figure imgf000060_0004
Figure imgf000060_0002
Primitive 2— Multivector-based Sylvester's Equation Encryption
[0168] The multivector based Sylvester's equation may be employed here to generate a second encryption primitive which also uses the pair of symmetric shared secret keys generated via the 0-Blade Reduction Operation (described above), as follows:
Figure imgf000060_0003
[0169] As was done above for the encryption primitive with "sandwich" / triple product, it may be beneficial to add another layer of security by using the same process as described above for XOR masking. [0170] The cipher multivector C, which is a result of the multivector based Sylvester's equation above, is converted into a number, denoted by C' and defined as a pre-cipher. This number is the information to be sent from the source computing device to the destination computing device.
Decryption
[0171] The decryption operation involves the closed-form solution of the multivector based Sylvester's equation for 3-dimensional multivector space and the XOR 'unmask' previously described for the "sandwich" / triple product above. The summarized processes are given below:
• Receive the cryptotext C as a number.
• Recover the pre-cipher cryptotext C' from cryptotext C using the shared secret Ss: C' = C XOR Ss
• Distribute C' into the coefficients of a cryptotext multivector
Figure imgf000061_0002
Figure imgf000061_0001
• The message multivector is recovered using the following closed-form solution for the multivector based Sylvester's equation:
Figure imgf000061_0003
[0172] Note that a solution for higher dimensions requires a different formula. Further note that if the original shared secret (Ss) is generated using an encrypted/secure key exchange, such as, but not limited to the Diffie-Hellman process discussed in more detail herein, there is no transmission of the original shared secret multivector ) and,
Figure imgf000061_0004
consequently, no transmission of the second shared secret multivector . Still further note
Figure imgf000061_0005
that since the second shared secret multivector (
Figure imgf000062_0001
is a result of a one-way function, the use of the second shared secret multivector
Figure imgf000062_0002
in either of the Geometric Algebra encryption primitive operations above, further guarantees security. Ultimately, as best as can be ascertained, there is no feasible way or high computer power to find or sequences of
Figure imgf000062_0003
such, which can be also dynamically updated in a continuous fashion using other Geometric Algebra operations to further increase security.
Section 3: Homomorphic EDCHE Enhancements to EDCE Operation
[0173] EDCHE is an extension to the EDCE cryptosystem described in more detail in Sections 1 and 2 above. The extension to support homomorphic operations requires additional considerations, particularly in the organization of the data message multivector coefficients, but, for the most part, the extension to support homomorphic operations relies on the intrinsic algebraic homomorphism of the Geometric Algebra foundation that are part of the encryption/decryption functions of the core EDCE. For instance, the Geometric Algebra geometric product operations that perform the actual encryption and decryption operations remain the same for both EDCE and EDCHE embodiments. The handling of the security keys also remains the same for both EDCE and EDCHE embodiments, including the data organization for "number to multivector" operations and any restrictions thereon. Likewise, any operations to convert text to a number and/or operations to convert a number also remain the same for both EDCE and EDCHE embodiments. The choice of whether or not to convert a cryptotext multivector (C) into cryptotext numeric data (C) prior to transmitting the encrypted data to a destination system (or to an intermediary system along the path to the final destination) remains the same for both EDCE and EDCHE embodiments and the processes to convert between cryptotext multivector (C) and cryptotext numeric data (C) also remain the same between EDCE and EDCHE, except there may be some restrictions on the types of permissible operations allowed for EDCHE embodiments to ensure that an intermediary computing system operating on the encrypted data does not need knowledge of any security keys involved in the encryption/decryption process. Even the data organization for "number to multivector" and "multivector to number" operations share the core EDCE restrictions between EDCE and EDCHE embodiments, the EDCHE embodiments add further restrictions that the data organization preserves homomorphic properties (i.e., have a homomorphic preserving mathematical relationship between the vectors utilized in the encryption process and the original plaintext messages being encrypted). Accordingly, since an EDCHE embodiment simply adds restrictions to an EDCE system regarding data organization operations for the multivector representation of the data being encrypted as well as to conversions between a cryptotext multivector (C) and a cryptotext numeric data (C), EDCHE embodiments being a subset of EDCE embodiments may operate as EDCE embodiments, but EDCE embodiments may not all necessarily operate as EDCHE embodiments.
[0174] Potential homomorphic encryption operations for an EDCHE embodiment may include multiple operations, such as, but not limited to: encrypted addition/subtraction, scalar addition/subtraction, encrypted multiplication, scalar multiplication, encrypted searching, and encrypted sorting. Each of the potential homomorphic encryption operations involves operations particular to the particular operation. For this reason, separate disclosures for each particular operation may be presented to so as to make the clear the specific details comprising the implementation of each particular operation. An additional document that briefly presents the combination of the potential homomorphic operations may also be separately presented that provides a brief summary of each operation and provides the additional details for performing combinations of the potential encryption operations. In view of potential disclosures, this particular disclosure is intended to address the specific details that comprise the particular details of scalar multiplicative homomorphic encryption operations. While the concept may be more typically described as "scalar multiplicative" homomorphic encryption, both scalar multiplication and scalar division operations are included under the umbrella of "scalar multiplicative" homomorphic encryption. As division is simply a modified form of multiplication in that the division operation has the same result as multiplying by the fraction of one over the operand, at the core, the division operation may still be thought of as a multiplication operation using the fractional (i.e., 1/jc) value as the scalar value to be multiplied. The scalar multiplication operation may also properly function using positive and negative numbers as well as whole numbers and tractions (i.e., decimals).
[017S] The description of EDCHE below will typically use the term "source" for the entity (e.g., computing device/system) where the numeric value that is the encrypted operand of the scalar multiplicative homomorphic operation originates and "destination" for the entity (e.g., computing device/system) that receives the result of the scalar multiplicative homomorphic operation. The term "intermediary" will typically define the "intermediate" entity acting in between the source(s) and destination on the encrypted numeric value operands originated at the source entity. In the case where an entity outside of the intermediate entity provides instructions to the intermediate entity to perform the scalar multiplicative homomorphic operation, that entity has been defined herein as the "command" entity. Alternatively, the term "client" may be used to describe the owner of the operand, the issuer of the instructions/command, and/or receiver of the result data (i.e., the source, command, and/or destination computing device/system), while the generic term "cloud" may be used for data that is at rest in an untrusted environment (i.e., the intermediary computing system/device). In some respects, the terms "client" and "cloud" may more closely reflect a real world application where the source, command, and destination are the same entity, sending data to the intermediary "cloud" for storage, then requesting an operation from the cloud when needed by the client (e.g., a scalar multiplication product of a stored transaction dollar amount stored in encrypted format on the cloud against some unencrypted scalar value such as an interest rate). Ultimately, the terms source, command, destination, and intermediary reflect the relative operations being performed by a computing system/device, and do not necessarily define the computing system/device as a whole. Thus, the source, command, destination, and intermediary operations/systems may reside as a particular application on one or more computing systems/devices. As the source, command, destination, and intermediary computing systems/devices may be general purpose computing systems capable of running multiple applications at the same time, it is inherently possible that the source, command, destination, and intermediary operations are encapsulated as separate applications/functions that may permit, one, two, or all of the separate
applications/functions to run on a single computing device/system. Also, a single interconnected computer system of a single owner/client may have untrusted environments that include data that is at rest (i.e., stored) in the owner/client's own end-point devices outside of the owner/client's digital secure perimeter that would have a security benefit by acting as the intermediary system without access to the security key(s).
[0176] To assist the reader in locating information, the arrangement of the remainder of the Section 2 disclosure below may be summarized as follows:
A. Homomorphic Data Organization (i.e., Coefficient Distribution/"Packing")
B. Number to multivector that supports Homomorphism
* First example
* Second example
* Third example C. Multivector to number that supports Homomorphism
* First example
* Second example
* Third example
D. Scalar Multiplicative Homomorphism in EDCHE
* General Scalar Multiplicative EDCHE example
E. Hardware Implementation for Scalar Multiplicative Homomorphic
EDCHE Embodiments (Fig. 1)
F. Scalar Multiplicative Homomorphic EDCHE Operational Flow Charts (Figs. 9 & 10)
A. Homomorphic Data Organization (i.e., Coefficient Distribution/Tacking")
[0177] In abstract algebra, the term homomorphism refers to a structure-preserving map between two algebraic structures, such as groups, rings, or vector spaces. An algebra homomorphism between two algebras is one that preserves the algebra structure. In order to preserve the algebra structure between arithmetic and Geometric Algebra operations, the method by which numbers are "packed" (i.e., distributed) into multivector coefficient elements should necessarily maintain some mathematical representation of the original number. Consequently, the packing/distribution method may define, among many things, the Geometric Algebra operations permissible for an EDCHE embodiment. For example, the Rationalize operation on multivectors yields zero when all multivector coefficients are equal. Such multivectors having all equal coefficients have no inverse and the geometric product of such multivectors having all equal coefficients with another multivector has no inverse. As discussed in Sections 1 and 2 above, the decryption methodology for EDCE and EDCHE systems utilize the inverse of the cryptotext multivector being decrypted and of the security key(s) multivector to perform the decryption. Therefore, the cryptotext multivector being decrypted should not have all equal value coefficients. One means to ensure that the cryptotext multivector being decrypted does not have all equal value coefficients is to have the packing/coefficient distribution method ensure that not all coefficients are equal to each other (i.e., at least one coefficient should be different than the other coefficients) when creating the shared security multivector(s) and the data message multivectors. For an embodiment of the EDCE that simply transfers the data message, this will ensure that the cryptotext multivector to be decrypted will not have all equivalent coefficients. For an EDC E system that may perform operations involving multiple encrypted data values, the same packing/coefficient distribution method to ensure that the source message multivectors do not have all equivalent coefficients will minimize the potential for the cryptotext multivector being decrypted from having all equivalent coefficients.
[0178] For an embodiment of an EDCHE system, the methods and systems that encrypt and decrypt messages using Geometric Algebra may utilize the intrinsic algebraic homomorphic properties of Geometric Algebra to permit arithmetic and other comparative operations on encrypted messages handled by an intermediary computing system without the need for the intermediary computing system to decrypt the encrypted messages prior to performing the arithmetic and other comparative operations. Accordingly, the intermediary computing system does not need to know any information regarding any of the secret security keys of the encryption/decryption processes to properly perform the arithmetic and other comparative operations. The encrypted results of the arithmetic and other comparative operations performed by the intermediary computing system, when decrypted at a destination computing device, produce results equivalent to the same operations as if the operations were performed on the unencrypted plain text messages. In order to properly achieve the homomorphic effect for intermediary arithmetic and other comparative operations, a proper data organization methodology (i.e., packing/distributing coefficients into a multivector) that preserves such homomorphic properties (i.e., the mathematical relationship between the vectors utilized in the encryption process and the original plaintext messages being encrypted) should be enforced on the choice of coefficients for the vectors representing the plain text messages. Homomorphism in the EDCHE is built upon the commutative nature of the Geometric Algebra operations of the encryption and decryption principles described above in Sections 1 and 2 for EDCE embodiments. Therefore, the distribution/packing data arrangement should also preserve a commutative mathematical relationship to the original numeric value being encrypted. In order to preserve the algebra structure between arithmetic and Geometric Algebra operations, the method by which numbers are "packed" into multivector elements must remain a representation of the original number. One such relationship for packing the coefficients of the multivector that preserves homomorphic properties is to ensure that the coefficients of the multivector representation of the plaintext numeric message follow a mathematical data organization between the value of the plaintext numeric message and at least one of the values of the coefficients of the multivector representation of the plaintext numeric message where the mathematical operations incorporating the one or more values of the multivector coefficients have a result equal to the original plaintext numeric message value (i.e., the homomorphic preserving mathematical relationship). The mathematical relationship may include: addition of at least one coefficient of the multivector coefficients, subtraction of at least one coefficient of the multivector coefficients, addition of a constant value, subtraction of a constant value, multiplication of at least one coefficient of the multivector coefficients by a constant value, and division of at least one coefficient of the multivector coefficients by a constant value. The location of the various mathematical operations relative to the particular locations of the coefficients in the multivector representation should also be consistently applied to all source numeric data messages converted to a multivector as well as for result multivectors converted to a result numeric data value in a particular encryption/decryption pathway. Ensuring that the coefficients of the vector representation of the plaintext numeric message meet the above described definition of the homomorphic preserving mathematical relationship will provide the proper commutative data organization (i.e., packing/distributing) methodology that preserves the homomorphic properties of the Geometric Algebra operations of the core EDCE encryption/decryption processes. Due to the use of the proper commutative data organization, an embodiment of an EDCHE system provides a cryptosystem mat allows unlimited multiplications and additions of cipher text (i.e., transmitted/stored encrypted messages at the intermediary/cloud computer system) due solely to the intrinsic algebraic homomorphic properties of an embodiment of the EDC E system. Thus, an embodiment of an EDCHE system may provide the homomorphic properties as a product of algebraic homomorphism without the need to use additional methods, such as "bootstrapping" (e.g., performing a recursive operation to reduce the noise associated with a cipher text) to achieve the homomorphic properties.
B. Number to multivector that supports Homomorphism
[0179] As described above, in order to preserve homomorphism, a "number to multivector" paradigm where the "packed" multivector that represents the original plaintext numeric message has a mathematical relationship (i.e., the homomorphic preserving mathematical relationship) to the original plaintext numeric message preserves the homomorphic characteristics of the Geometric Algebra operations of EDCE
encryption/decryption operations. In abstract algebra, the term homomorphism refers to a structure-preserving map between two algebraic structures, such as groups, rings, or vector spaces. An algebra homomorphism between two algebras is one that preserves the algebra structure. In order to preserve the algebra structure between arithmetic and Geometric Algebra operations, the method by which numbers are "packed" into multivector elements must remain a representation of the original number. One such relationship for packing the coefficients of the multivector that preserves homomorphic properties is to ensure that the coefficients of the multivector representation of the plaintext numeric message follow a mathematical data organization between the value of the plaintext numeric message and at least one of the values of the coefficients of the multivector representation of the plaintext numeric message where the mathematical operations incorporating the one or more values of the multivector coefficients have a result equal to the original plaintext numeric message value. The mathematical relationship may include: addition of at least one coefficient of the multivector coefficients, subtraction of at least one coefficient of the multivector coefficients, addition of a constant value, subtraction of a constant value, multiplication of at least one coefficient of the multivector coefficients by a constant value, and division of at least one coefficient of the multivector coefficients by a constant value. The location of the various mathematical operations relative to the particular locations of the coefficients in the multivector representation should also be consistently applied to all source numeric data messages converted to a multivector as well as for result multivectors converted to a result numeric data value in a particular encryption/decryption pathway. For example, in a homomorphic preserving mathematical relationship that includes both addition and subtraction operations might be, for a three dimensional multivector there are eight possible coefficients in the multivector representation
Figure imgf000068_0001
numbered so as to correspond with the unit vector associated with each coefficient), if the coefficients for the d and e12 unit vectors (i.e., C2 and C12) are subtracted in the calculation of the homomorphic preserving mathematical relationship for a first numeric data message conversion to a multivector, all other numeric messages converted to a message vector should also ensure that the ¾ and cy_> coefficients are subtracted in the conversion homomorphic preserving mathematical relationship process. Likewise, when converting a multivector into a numeric value, the homomorphic preserving mathematical relationship process should also retain the subtraction of the ¾ and c;_> coefficients. Other than the location of additions and subtractions within the homomorphic preserving mathematical relationship of the coefficients, the actual values of the coefficients may be selected as desired by a user so long as the homomorphic preserving mathematical relationship equals the original numeric value being encrypted. The only other restriction for multivectors representing different numeric values within a homomorphic operation is that the multivectors all share the same dimensionality (i.e., all multivectors are 2D, all multivectors are 3D, all multivectors 4D, etc.). One skilled in the art will recognize that there are many, perhaps even an infinite, number of ways to select coefficient values that meet the stated criteria/restrictions and that each of those ways will create a satisfactory EDCHE embodiment so long as the stated criteria/restrictions are, in fact, met
[0180] Some processes are common within each multivector coefficient selection process. For instance, the first step in representing a numeric message value in a multivector is to define the number of coefficients that are present in the multivector. The total number of coefficient elements of a multivector is directly related to the dimensionality of the multivector by the equation n = 2m where n is the total number of coefficient elements and m is equal to the dimensionality of the multivector (e.g., for a 3D multivector n = 23 = 8). Thus, in order to represent the numeric message N in a 3D multivector, the total number of coefficient elements to be determined is eight
[0181] The coefficient selections may encompass any number that may be represented in the data type chosen by a system designer to hold the coefficient values, including, but not limited to: positive numbers, negative numbers, zero, integer numbers, rational numbers (including fractions), and/or real numbers. As a base of understanding, the format of a multivector will always assume that the coefficients add together. For instance, the form for a 3D multivector representing the numeric value N would be:
Figure imgf000069_0001
[0182] Subtractions would be represented as negative coefficients. This "form" of the multivector coefficients adding together would remain consistent regardless of whether or not the corresponding coefficient is added or subtracted in the homomorphic preserving mathematical relationship between the original numeric value and the coefficients of the representative multivector.
[0183] Below are several examples of processes to automate or otherwise demonstrate methodology for the selection of coefficient values for a message multivectors that represents a numeric message value N. One skilled in the art will again recognize that many ways to select the coefficient values that meet the stated criteria/restrictions and that embodiments are not limited to the example methodologies described herein. First example:
[0184] In this first example, the EDCHE embodiment performs the homomorphic preserving mathematical relationship process such that all coefficients are added together and there are not any coefficients that are subtracted. A first step might be to factorize the numeric message N and write the value N as a sum of other numbers (ci) that are to represent the numeric message value in a message multivector such that:
Figure imgf000070_0001
where n is the number of elements in the multivector as determined by n = 2m where m is the dimension of the multivector. (e.g., n = 8 for a 3D multivector). One skilled in the art will recognize that at this point there are many ways to assign the coefficient values (ci) that will satisfy the homomorphic preserving mathematical relationship equation to enable proper operation of an EDCHE embodiment. One such method might be to perform an integer division of Nby the number of elements n. Given the numeric value iV= 5487 and a 3D multivector where n = 8, the result is 5487 integer division by 8, which is equal to 685. The remainder may be obtained by calculating N mod n (i.e., 5487 mod 8), which may be added to the last coefficient, resulting in a last coefficient value (i.e., cm) of 685 + 7, which equals 692. The sum may then be written as:
N= C0 + Cl + C2 + C3 + CI2 + CJ3 + C23 + CJ23
5487 = 685 + 685 + 685 + 685 + 685 + 685 + 685 + 692 and the resulting multivector representing the numeric value N would be:
Figure imgf000070_0002
[0185] The assignment of which coefficient value receives the addition of the N mod n operation may be random, or the assignment may be of a predetermined form designed to increase entropy to enhance the encryption security.
[0186] In the case where the numeric value is divisible by the number of coefficient elements n of the message multivector (i.e., Nmod n = 0 such that all ct are equal, which is not permitted for proper decryption operation), a number less than the value of one coefficient element (e.g., C13) may be subtracted from that coefficient element (e.g., again C13) and added to another coefficient element (e.g., C23). For example, given N= 2944 for a 3D multivector where n = 8, 2944 may be integer divided by 8 to equal 386 with no remainder. An embodiment may then simply subtract one from cn and add one to C23 to ensure that not all coefficient elements are equal such that the homomorphic preserving mathematical relationship may then be written as:
N= C0 + CJ + C2 + C3 + C12 + C}3 + C23 + CJ23
2944 = 368 + 368 + 368 + 368 + 368 + 367 + 369 + 368 and the resulting multivector representing the numeric value N would be: multivector
Figure imgf000071_0001
multivector N= 368 + 368e1 + 368e2 + 368e3 + 368e12 + 367e13 + 369e23 + 368e123
Second example:
[0187] As with the first example, the EDCHE embodiment performs the homomorphic preserving mathematical relationship process such that all coefficients are added together and there are not any coefficients that are subtracted. Thus, for a 3D multivector with coefficients co, ci, C2, C3, C12, C13, C23, and ci23, where the coefficients are numbered so as to correspond with the unit vector associated with each coefficient, the homomorphic preserving mathematical relationship equation to represent the numeric value N would, again, be:
N= C0 + C] + C2 + C3 + C/2 + CJ3 + C23 + C123
[0188] Instead of having a strict equation to follow, an algorithm that randomly finds integers that satisfy the sum of integers equation may be created. To ensure that the coefficients add up to the numeric value N, the random number for each coefficient element may be serially (i.e., successively) calculated with the random number restricted to be some amount less than the remaining sum in order to achieve the numeric value N until the last coefficient value calculated is set equal to the remaining amount in order to achieve the numeric value N. For such a random number based system, given that N is again 5487, the homomorphic preserving mathematical relationship may be written as:
N= C0 + CI + C2 + C3 + C12 + CI3 + C23 + C123
5487 = 385 + 985 + 685 + 584 + 786 + 482 + 887 + 693 and the resulting multivector representing the numeric value N would be:
Figure imgf000072_0001
multivector N= 385 + 985e1 + 685e2 + 584¾ + 786en + 482e13 + 887e23 + 693et23
Third example:
[0189] Unlike the first two examples, in this third example the homomorphic preserving mathematical relationship equation that is set equal to the numeric value Nmay be defined to include some subtraction of some coefficients, addition of a constant value, and multiplication of coefficient values by a constant, as well as the omission of one of the coefficients (i.e., cni) from the homomorphic preserving mathematical relationship. In this third example, the homomorphic preserving mathematical relationship equation to represent the numeric value N might now be:
N= CO + C1 - C2 + C3 - C12 + 3 * C/3 + C23 + 23
[0190] A modified version of the methodology of either the first example (use a mathematical equation to calculate the coefficient values) or the second example (randomly generated coefficient values) described above may be used to determine the coefficient values given that the homomorphic preserving mathematical relationship now subtracts some coefficients, adds a constant, multiplies a coefficient by a constant, and omits one coefficient from the homomorphic preserving mathematical relationship altogether. Using randomly generated numbers, and given that N is again 5487, the homomorphic preserving mathematical relationship may be written as:
N= C0 + Cr - C2 + C3 - C12 + 3 * C13 + C23 + 23
5487 = (901) + (985) - (185) + (-584) - (286) + 3 * (882) + (1987) + 23
5487 = 901 + 985 - 185 - 584 - 286 + 2646 + 1987 + 23 and the resulting multivector representing the numeric value N would be: multivector N= co + C1 e1 + C2 e2 + C3 e3 + c12 e12 + en e13 + C23 e_a + C123 e123 multivector N= (901) + (985)e1 + (185)e2 + (-584)e3 + (286)e12 + (882)e13 + (1987)e23 +
333e123
multivector N=
where the c123 value of 333 is a dummy value not included in the homomorphic preserving mathematical relationship, but may potentially be used to provide other features such as signature capability and/or passing of command or other information.
[0191] A homomorphic preserving mathematical relationship that includes some subtractive elements has the advantage of being able to represent negative numbers and zero without the coefficient values being negative for a user that prefers to not have negative coefficient values.
C. Multivector to number that supports Homomorphism
[0192] Obtaining a numeric value from the coefficients of a numeric data message multivector is relatively simple and straight forward. To obtain the numeric data message value, simply perform the homomorphic preserving mathematical relationship equation for the numeric data message multivector using the values of the multivector coefficients plugged into the homomorphic preserving mathematical relationship equation. The examples given below provide the "multivector to number" process appropriate for the same example number as described above for the "number to multivector" process. First example:
[0193] In the first example above, the EDCHE embodiment performed the
homomorphic preserving mathematical relationship process such that all coefficients are added together and there are not any coefficients that are subtracted. Thus, for a 3D multivector with coefficients C0, C1, C2, C3, C12, C13, C23, and c123, where the coefficients are numbered so as to correspond with the unit vector associated with each coefficient, the homomorphic preserving mathematical relationship equation to represent the result numeric value N would be:
N '= CO + CI + C2 + C3 + C12 + C13 + C23 + CJ23
As described above, the multivector has the form of: multivector N
Figure imgf000074_0001
[0194] Now, given the following result multivector, multivector
Figure imgf000074_0002
and knowing the multivector of the form: multivector
Figure imgf000074_0003
then the result multivector may be rewritten to highlight the appropriate positive and negative values as:
(725) + (21)e1 + (685)e2 + (286)e3 + (-721)e12 + (85)e13 + (601)e23 + (192)e123
and the calculation of the result numeric value N would proceed as follows:
N= C0 + CI + C2 + C3 + C12 + CJ3 + C23 + CJ23
N= (725) + (21) + (685) + (286) + (721) + (85) + (601) + (192) N= 725 + 21 + 685 + 286 + 721 + 85 + 601 + 192 such that result numeric value N would be:
N= 3316
[0195] Now, given a different result multivector that includes some negative coefficients, such as the following vector:
Figure imgf000075_0001
and knowing the result multivector with the given homomorphic preserving mathematical relationship equation is of the form: multivector
Figure imgf000075_0003
then the result multivector may be rewritten to highlight the appropriate positive and negative values as:
Figure imgf000075_0004
and the calculation of the result numeric value N would proceed as follows:
Figure imgf000075_0002
such that result numeric value N would be:
N= 363
Second example:
[0196] In the second example above, the EDCHE embodiment performed the homomorphic preserving mathematical relationship process such that all coefficients are added together and there are not any coefficients that are subtracted, which is the same homomorphic preserving mathematical relationship equation as for the first example above. Consequently, the "multivector to number" process is identical to that as described for the "multivector to number" process of the first example given above.
Third example:
[0197] In the first and second example above, the EDCHE embodiment performed the homomorphic preserving mathematical relationship process such that all coefficients are added together and there are not any coefficients that are subtracted. The third example from above changed the homomorphic preserving mathematical relationship equation to include some subtraction of some coefficients, addition of a constant value, and multiplication of coefficient values by a constant, as well as the omission of one of the coefficients (i.e., cm) from the homomorphic preserving mathematical relationship. Thus, for the third example, for a 3D multivector with coefficients co, c1, C2, c3, c12, c13, c23, and c123, where the coefficients are numbered so as to correspond with the unit vector associated with each coefficient, the homomorphic preserving mathematical relationship equation to represent the numeric value N would again be:
N '= CO + C1 - C2 + C3 - C12 + 3 * C13 + C23 + 23
As described above, the multivector has the form of:
Figure imgf000076_0001
[0198] Now, given the same following result multivector as for example 1 above, multivector N= 725 + 21e1 + 685 e2 + 286e3 - 721e12 + 85e13 + 601e23 + 192e123 and knowing the multivector is of the form: multivector N
Figure imgf000077_0001
then the result multivector may be rewritten to highlight the appropriate positive and negative values as: multivector N= (725) + (21)e1 + (685)e2 + (286)e3 + (-721)e12 + (85)e13 + (601)e23 +
(192)e123 and the calculation of the result numeric value N would proceed as follows:
N= C0 + Cl - C2 + C3 - C12 + 3 * CJ3 + C23 + 23
N= (725) + (21) - (685) + (286) - (-721) + 3 * (85) + (601) + 23 N= 725 + 21 - 685 + 286 + 721 + 255 + 601 + 23 such that result numeric value Nwould be:
N= 1947 where the C123 value of 192 is ignored as a dummy value not included in the homomorphic preserving mathematical relationship. Note that the third example result of 1947 does not equal the first example result of 3316 for the same result multivector. Thus, demonstrating the necessity of using the same homomorphic preserving mathematical relationship equation for all encrypted multivectors of any homomorphic operations that may involve multiple encrypted multivectors. [0199] Now, given a different result multivector that includes some negative coefficients (that is the same as the negative coefficient result multivector also used for example 1 above), such as the following vector: multivectorN= 333 - 201e1 + 248e2 + 506e3 - 71e12 + 80e13 + 211e23 - 743en3 and knowing the result multivector with the given homomorphic preserving mathematical relationship equation is of the form:
Figure imgf000078_0001
then the result multivector may be rewritten to highlight the appropriate positive and negative values as: multivector N= (333) + (-201)e1 + (248)e2 + (506)e3 + (-71)e12 + (80)e13 + (21 l)e23 + (-
743)e123 where the C123 value of -743 is a dummy value not included in the homomorphic preserving mathematical relationship. Thus, the calculation of the result numeric value N would proceed as follows:
N= CO + Cl - C2 + C3 - C12 + 3 * CI3 + C23 + 23
N= (333) + (-201) - (248) + (506) - (-71) + 3 * (80) + (211) + 23 N= 333 - 201 - 248 + 506 + 71 + 240 + 211 + 23 such that result numeric value N would be:
N= 935
Note again that the third example result of 935 does not equal the first example result of 363 for the same result multivector. Thus, again, demonstrating the necessity of using the same homomorphic preserving mathematical relationship equation for all encrypted multivectors of homomorphic operations that may involve multiple encrypted multivectors. D. Scalar Multiplicative Homomorphism in EDCHE
[0200] Scalar Multiplicative Homomorphic Encryption may be defined as ε(χ1 * k) = ε(*ι) * k, where ε is the encryption operation and k is a scalar (i.e, not a vector) value multiplied (or divided) equivalently against all portions of an encrypted vector representation of the value xi. In general terms, when the unencrypted "product" of a plaintext message and a scalar value is equal to the comparable encrypted "product" of the encrypted message multiplied by the same scalar value the operation is scalar multiplicative homomorphic. Note again that herein the term "multiplied" for scalar multiplicative homomorphic operations includes both multiplication and division operations. Thus, the plaintext message and the corresponding encrypted messages should be multiplied and divided in a comparable manner so that corresponding entities are similarly multiplied or divided in the "multiplation" operation.
General Scalar Multiplicative EDCHE example:
[0201] Described below is an example embodiment for calculating interest values presenting the scalar multiplicative homomorphic properties of an EDCHE embodiment.
[0202] Given the following secret multivectors:
Figure imgf000079_0001
[0203] Now, consider that Alice has an investment I1 of $10,500 So,
I1 = 10500 with an interest rate (ir) that is: ir = 0.008
[0204] The interest rate (ir) represents the interest rate for a particular period of time, which is usually one year for most financial transactions. However, the interest rate (ir) used to calculate compounded interest values for shorter time periods within an overall time period may be calculated by taking the overall interest rate and dividing by the number of sub- periods. For instance, the interest rate for one month of an overall year interest rate would be the yearly rate divided by twelve (the number of months in a year). For the ir = 0.008 (i.e., 0.8%), if the ir represents the yearly interest rate, the effective monthly interest rate would be 0.008/12, or 0.000667. Conversely, if the ir represents the monthly interest rate, the effective yearly interest rate would be 0.008 * 12, or 0.096.
[0205] If one wished to encrypt the monetary value above, the investment may be represented as multivector as follows:
Figure imgf000080_0002
The investment may then be encrypted by applying the triple product technique as follows:
Figure imgf000080_0001
which will generate an encrypted value of:
Figure imgf000080_0003
[0206] In order to test the scalar multiplicative homomorphic properties of EDCHE, one may apply the interest rate to the encrypted investment at the intermediary (i.e., cloud) computing system. Thus Alice, acting as the source computing device may send the encrypted value (C) of the investment Ij to the intermediary (i.e., cloud) computing system. The cloud computing system may know to automatically multiply the received encrypted value (C) by the interest rate (ir = 0.008), or Alice, now acting as the command computing device may send instructions to the intermediary/cloud computing system to multiply the encrypted value by the interest rate (ir = 0.008). The intermediary computing system may perform additional operations to achieve a desired result based on the given information. For instance, in the case of the interest rate ir = 0.008, in order to obtain the future value (FV) of the investment including the interest, the value multiplied by the present value (PV) of the investment value would necessarily be one plus the interest rate (FVir = 1 + ir = 1 + 0.008 = 1.008). The intermediary/cloud computing system may then multiply the encrypted multivector value
Figure imgf000081_0006
by the unencrypted scalar value FVir in order to obtain the future value (FV) of the investment (//) as updated encrypted value The intermediary/cloud
Figure imgf000081_0004
computing system may then send the updated encrypted value
Figure imgf000081_0005
to Alice, now acting as the destination computing system. When Alice receives the calculated encrypted information ) from the cloud, she may decrypt the information and see the updated future value (FV) of her investment.
[0207] In terms of the encrypted multivector ( representing the Alice's initial
Figure imgf000081_0002
investment (Ij), the investment rate applied on the encrypted investment in order to obtain
Figure imgf000081_0003
the future value (FV) of the investment may be calculated by the intermediary/cloud computing system as follows:
Figure imgf000081_0001
such that:
Figure imgf000081_0007
[0208] The intermediary/cloud computing system may then send the encrypted multivector to Alice and Alice may then recover the updated investment ) as follows:
Figure imgf000081_0009
Figure imgf000081_0008
such that:
Figure imgf000081_0010
and the base 10 number representation of the updated investment is then retrieved as:
Figure imgf000081_0011
[0209] Thus, Alice sent one encrypted investment to the cloud. After a while, she needed to know the updated value of her investment. The cloud applied a public interest rate on the encrypted investment and sent the result to Alice without ever decrypting the encrypted investment amount originally sent by Alice. When Alice received the result, she decrypted the result and is able to see the unencrypted updated value of her investment. The unencrypted investment value of 10584 is equal to the equivalent unencrypted operations, as follows:
/i = 10500 ir = 0.008
/'10 = h x (1 + ir) = 10500 x 1.008 = 10584
[0210] Sticking with the concept of interest calculations, it is also possible for the intermediary computing system to perform more complicated calculations as part of the unencrypted scalar data value. For instance the calculation for the future value of compound interest is FV= PV* (1 + irf, where FVv's the future value of the investment, PV\s the present value of the investment, ir is the interest rate, and n is the number of periods of compounding. Thus, for the above example, if Alice desired to see the future value (FV) of her investment after five compounding periods, the intermediary/cloud system may calculate the scalar data value to be (1 + 0.008)5 which equals 1.02525, then multiply the encrypted investment value by 1.02525 instead of 1.008 as was detailed in the example above to obtain a FV value of $10,765.14. Alice may define the interest rate, number of compounding periods, and or other information in the instructions sent to the intermediary/cloud computing system. Alternatively, Alice may instruct the intermediary/cloud computing system to look up values, such as looking up a public interest rate or other information to combine to achieve the same result. One skilled in the art will recognize that the unencrypted scalar data value that is to be multiplied by the encrypted data value may be a simple predefined data value or the unencrypted scalar data value may be the result of several operations on data provided in the multiply command instructions and/or data looked up and obtained from privately or publicly available data storage. Further, the EDCHE system may support decimal (i.e., floating point, fixed point, or other real number representation) arithmetic of the encrypted data value and/or the unencrypted scalar data value to be multiplied with the encrypted data value.
E. Hardware Implementation for Scalar Multiplicative omomorphic EDCHE
Embodiments (Fig. 8)
[0211] Fig. 8 is a block diagram 800 of the hardware implementation for a scalar multiplicative homomorphic encryption embodiment. A source computing device 802 is connected over an electronic network/bus connection 804 to an intermediary (e.g., cloud) computing device 806. In the embodiment shown in Fig. 8, the source computing device 802 sends the cryptotext multivector 810 that will be scalar multiplied with an unencrypted scalar data value through the scalar multiplicative homomorphism of an EDCHE embodiment at the intermediary computing system 806 over the network/bus connection 804 to the intermediary computing system 806. The intermediary computing system 806 receives the cryptotext multivector 810 from the network/bus connection 804.
[0212] The unencrypted scalar data value that is to be scalar-vector multiplied with the cryptotext multivector 810 may be predefined at the intermediary computing system, such as for the case of operations regarding a fixed interest rate with regard to encrypted investment values. Further, the predefined interest rate may further be used to perform additional calculations such as compounded interest calculations over multiple compounding periods. One skilled in the art will further recognize that other types of calculations outside of interest rate calculations may involve multiple arithmetic operations in order to obtain the final unencrypted scalar data value that will be scalar multiplied with the cryptotext multivector 810. While one embodiment may have predefined scalar data values stored or otherwise accessible to the intermediary computing system 806, other embodiments may receive some, or all, data necessary to determine the scalar data value for multiplication in scalar multiplication instructions 816 sent over the communications network/bus 804 from a command computing system 814. The scalar multiplication instructions may contain scalar data value information that defines the scalar data value multiplied with the cryptotext multivector 810. The scalar data value information may define the unencrypted scalar data value that is to be multiplied by the encrypted data value as a simple predefined data value, or the scalar data value may be defined to be the result of several operations on data provided in the scalar data value information of the scalar multiplication instructions 816, and/or the scalar data value information may provide instructions to look up some or all of the necessary data in privately or publicly available data storage accessible by the intermediary computing system 806. Many, maybe even most, of the time, the command computing device 814 may be the same device as the source computing device 802 as it is often the case that the provider of the encrypted data is also the entity that wishes to have operations performed on the encrypted data. Accordingly, many times the ultimate destination computing device 808 is the same as the source 802 and command 814 computing devices. While the source 802, command 814, and/or destination 808 computing devices may commonly be one and the same device, the operation of an EDCHE embodiment does not require that the functions be performed on a single computing device. Likewise, while less common in actual practice, the intermediary computing system 806 may also be the same computing device as one, two or all of the source 802, command 814, and/or destination 808 computing devices. Again, while devices may share the implementation of the functions of source 802, command 814, intermediary 806, and/or destination 808 computing systems, there is not a requirement for any such sharing for an EDCHE embodiment.
[0213] The intermediary computing system 806 may immediately perform a "scalar multiplication" of the cryptotext multivector 810 and a scalar data value using scalar-vector multiplication (as instructed by a user 816 or simply in response to receiving the cryptotext multivector 810) or the intermediary computing system 806 may store the cryptotext multivector 810 until such time that the intermediary computing system 806 is instructed 816 to perform the homomorphic scalar multiplication operation. Once the homomorphic scalar multiplication operation is completed by the intermediary computing system 806, the intermediary computing system sends the encrypted homomorphic scalar multiplicative result multivector 812 to the destination computing system 808 over the network/bus
communication connection 804. The destination computing system 808 receives the encrypted homomorphic scalar multiplicative result multivector 812 from the network/bus communication connection 804 and decrypts the encrypted homomorphic scalar
multiplicative result multivector 812 to obtain the desired plaintext scalar multiplication result.
[0214] Further, as noted in Sections 1 and 2 above, the cryptotext multivector 810 may be converted to a non-multivector cryptotext when being sent over the network/bus communication connection 804, then converted back into the cryptotext multivector at the intermediary computing system 806 for scalar multiplicative homomorphic operations. Likewise, the encrypted homomorphic scalar multiplication result multivector 812 may be converted to a non-multivector scalar multiplication result cryptotext when being sent over the network/bus communication connection 804, then converted back into the encrypted homomorphic scalar multiplication result multivector 812 at the destination computing device 808 for decryption by the destination computing device 808 into the plaintext scalar multiplicative result.
[0215] Generally, communications, including encrypted communications, are bidirectional such that the source computing device 802, the command computing device 814, the intermediary computing system 806, and/or the destination computing device 808 may change roles so as to operate as a source computing device 802, the command computing device 814, the intermediary computing system 806, and/or the destination computing device 808 as is necessary to accommodate the transfer of data back and forth between the source 802, command 814, and destination 808 computing devices as well as for computation of homomorphic scalar multiplication products at the intermediary computing system 806.
[0216] Further, as shown in Fig. 8, the source computing device 802 and command computing device 814 appear to be laptop computers and the destination computing device 808 appears to be a tablet device. Generally, any computing device capable of
communication over any form of electronic network or bus communication platform 804 may be one, multiple or all of the source computing device 802, the command computing device 814, the intermediary computing system 806, and/or the destination computing device.
Further still, the source 802, command 814, intermediary 806, and destination computing devices/systems 808 may actually be the same physical computing device communicating over an internal bus connection 804 with itself, but still desiring encrypted communication to ensure that an attacker cannot monitor the internal communications bus 804 or hack an unprotected area of the computing system (i.e., the intermediary section 806) in order to obtain sensitive data communications in an unencrypted format
[0217] Various embodiments may implement the network/bus communications channel 804 using any communications channel 804 capable of transferring electronic data between the source 802, command 814, intermediary 806, and/or destination 808 computing devices/systems. For instance, the network/bus communication connection 804 may be an Internet connection routed over one or more different communications channels during transmission from the source 802 and/or command 814 computing devices to the intermediary 806 computing system, and men onto the destination computing device 808. Likewise, the network/bus communication connection 804 may be an internal
communications bus of a computing device, or even the internal bus of a processing or memory storage Integrated Circuit (IC) chip, such as a memory chip or a Central Processing Unit (CPU) chip. The network/bus communication channel 804 may utilize any medium capable of transmitting electronic data communications, including, hut not limited to: wired communications, wireless electro-magnetic communications, fiber-optic cable
communications, light/laser communications, sonic/sound communications, etc., and any combination thereof of the various communication channels.
[0218] The various embodiments may provide the control and management functions detailed herein via an application operating on the source 802, command 814, intermediary 806, and/or destination 808 computing devices/systems. T'he source 802, command 814, intermediary 806, and/or destination 808 computing devices/systems may each be a computer or computer system, or any other electronic devices device capable of performing the communications and computations of an embodiment. The source 802, command 814, intermediary 806, and/or destination 808 computing devices/systems may include, but are not limited to: a general purpose computer, a laptop/portable computer, a tablet device, a smart phone, an industrial control computer, a data storage system controller, a CPU, a Graphical Processing Unit (GPU), an Application Specific integrated Circuit (ASI), and/or a Field Programmable Gate Array (PPGA). Notably, the first 102 and second 106 computing devices may he the storage controller of a data storage media (e.g., the controller for a hard disk drive) such that data delivered to/from the data storage media is always encrypted so as to limit the ability of an attacker to ever have access to unencrypted data. Embodiments may be provided as a computer program product which may include a computer-readable, or machine-readable, medium, having stored thereon instructions which may be used to program/operate a computer (or other electronic devices) or computer system to perform a process or processes in accordance with the various embodiments. The computer-readable medium, may include, but is not limited to, hard disk drives, floppy diskettes, optical disks, Compact Disc Read-Only Memories (CD-ROMs), Digital Versatile Disc ROMS (DVD- ROMs), Universal Serial Bus (USB) memory sticks, magneto-optical disks, ROMs, random access memories (RAMs), Erasable Programmable ROMs (EPROMs), Electrically Erasable Programmable ROMs (EEPROMs), magnetic optical cards, flash memory, or other types of media/machine-readable medium suitable for storing electronic instructions. The computer program instructions may reside and operate on a single computer/electronic device or various portions may be spread over multiple computers/devices that comprise a computer system. Moreover, embodiments may also be downloaded as a computer program product, wherein the program may be transferred from a remote computer to a requesting computer by way of data signals embodied in a carrier wave or other propagation medium via a communication link (e.g., a modem or network connection, including both wired/cabled and wireless connections).
F. Scalar Multiplicative Homomorphic EDCHE Operational Flow Charts (Figs. 9 & 10)
[0219] Fig. 9 is a flow chart 900 of the general operation for a scalar multiplicative homomorphic encryption embodiment At process 908 a shared secret numeric data value (Ss) is shared between the source 902 and destination 906. The various embodiments may share the shared secret numeric data value (Ss) between the source 902 and destination 906 via any means desired by the users. To ensure the shared secret numeric data value (Ss) is kept secret, it is likely that some type of handshaking/setup encrypted key transfer mechanism will be desired to share the shared secret numeric data value (Ss). For example, the shared secret numeric data value (Ss) may be shared between the source 906 and destination 906 by means including, but not limited to: pre-conditioning the source 902 computing device and the destination 906 computing device with the shared secret numeric value (Ss), a standard public/private key exchange technique, RSA (Rivest-Shamir-Adleman) key exchange, and/or Diffie-Hellman key exchange (disclosed in more detail herein, below). Further, the original shared secret may be an alphanumeric string in ASCII (American Standard Code for Information Exchange) or another encoding protocol that is converted to a numeric value based on the associated encoding protocol, such as ASCII. However, both the source 902 and destination 906 need to know and use the same alphanumeric text conversion into a numeric value process to ensure that results of both the source 902 and the destination 906 are the same.
[0220] At process 910, the source 902 distributes the numeric message data (M) into message multivector (M) coefficients in accord with a homomorphic mathematical relationship equation between a plaintext data value and coefficients of a multivector that represents the plaintext data value. For all operations of a single scalar multiplicative homomorphic process, the source 902 and the destination 906 should use the same homomorphic preserving mathematical relationship to preserve the homomorphism of the encrypted data. Also, the encryption system will work with just one non-zero message multivector (M) coefficient, but, the more non-zero message multivector (M) coefficients there are, the stronger the encryption will become, so it is desirable to have more than one non-zero message multivector (M) coefficient. At process 912, the source 202 distributes the shared secret numeric value (Ss) into shared secret multivector (Ss) coefficients. Again, the encryption system will work with just one non-zero shared secret multivector (Ss) coefficient, but, the more non-zero shared secret multivector (Ss) coefficients there are, the stronger the encryption will become, so, again, it is desirable to have more than one non-zero shared secret multivector
Figure imgf000088_0001
coefficient. One skilled in the art will recognize that there are many approaches for distributing numeric data into several coefficients of a multivector (see herein, above for disclosure of some example packing/distribution methods). The primary requirement for the distribution process from the numeric values of the message (M) and the shared secret (Ss) to the multivector coefficient values (A? and Ss) is that the source 902 and the destination 906 both know the processes 910/930 and 912/926 such that the destination 904 can get the proper value for the homomorphic scalar multiplicative result (SMR). As long as it is known to both the source 902 and the destination 904, the distribution of numeric data to multivector coefficients may be performed differently between the message (M )/scalar multiplicative result (SMR) pair and the shared secret (Ss).
[0221] The distributing/packing method defines, among many things, the Geometric Algebra operations permissible for EDCE and/or EDCHE embodiments. For example, the Rationalize operation on multivectors yields zero when all multivector coefficients are equal. Such multivectors having all equal coefficients have no inverse and the geometric product of such multivectors having all equal coefficients with another multivector has no inverse. For example, the Rationalize operation on multivectors yields zero when all multivector coefficients are equal. Such multivectors having all equal coefficients have no inverse and the geometric product of such multivectors having all equal coefficients with another multivector has no inverse. As discussed in more detail below, the decryption methodology for EDCE and EDCHE systems utilize the inverse of the cryptotext multivector being decrypted and of the security key(s) multivector to perform the decryption. Therefore, the cryptotext multivector being decrypted should not have all equal value coefficients. One means to ensure that the cryptotext multivector being decrypted does not have all equal value coefficients is to have the packing/coefficient distribution method ensure that not all coefficients are equal to each other (i.e., at least one coefficient should be different than the other coefficients) when creating the shared security multivector(s) and the data message multivectors. For an embodiment of the EDCE that simply transfers the data message or performs scalar multiplication operations that proportionally change the coefficients, this will ensure that the cryptotext multivector to be decrypted will not have all equivalent coefficients. [0222] For embodiments that intend to retain homomorphic properties for encrypted data messages such as an embodiment of an EDC E system, there is an additional restriction that the "packed" multivector that represents the original plaintext numeric message have a mathematical relationship (i.e., the homomorphic preserving mathematical relationship) to the original plaintext numeric message. In abstract algebra, the term homomorphism refers to a structure-preserving map between two algebraic structures, such as groups, rings, or vector spaces. An algebra homomorphism between two algebras is one that preserves the algebra structure. In order to preserve the algebra structure between arithmetic and Geometric Algebra operations, the method by which numbers are "packed" into multivector elements must remain a representation of the original number. One such relationship for packing the coefficients of the multivector that preserves homomorphic properties is to ensure that the coefficients of the multivector representation of the plaintext numeric message follow a mathematical data organization between the value of the plaintext numeric message and at least one of the values of the coefficients of the multivector representation of the plaintext numeric message where the mathematical operations incorporating the one or more values of the multivector coefficients have a result equal to the original plaintext numeric message value. The mathematical relationship may include: addition of at least one coefficient of the multivector coefficients, subtraction of at least one coefficient of the multivector coefficients, addition of a constant value, subtraction of a constant value, multiplication of at least one coefficient of the multivector coefficients by a constant value, and division of at least one coefficient of the multivector coefficients by a constant value. The location of the various mathematical operations relative to the particular locations of the coefficients in the multivector representation should also be consistently applied to all source numeric data messages converted to a multivector as well as for result multivectors converted to a result numeric data value in a particular encryption decryption pathway.
[0223] The restriction to retain homomorphic properties is only applicable to the distribution (i.e., "packing") of the message multivector (A?) coefficients and not to the distribution (i.e., "packing") of the shared secret multivector (Ss). Consequently, the distribution (i.e., "packing") of the shared secret multivector (¾) may be performed in any fashion so long as the distribution (i.e., "packing") method of the shared secret multivector (Ss) is known and used consistently by the source 902 and destination 906 computing devices as, ultimately, the shared secret multivector (Ss) used by the source 902 and destination 906 should be equal to each other to ensure that the decryption operations 928 work properly in relation to the encryption 914 operations. The number of potential coefficients is directly related to the size/dimension (N) of the multivectors such that the number of coefficients increases by a factor of 2 (i.e., 2") for each incremental increase in the size/dimension (N) of the multivector. To increase the confusion and/or diffusion of the encryption process disclosed herein, using multivectors of at least two dimensions will provide at least four coefficients to distribute the numeric data of the message (M) and the shared secret (SS). By increasing the number of dimensions (N) of multivectors beyond two-dimension
multivectors, the confusion and/or diffusion security characteristics will also be increased due to the additionally available multivector coefficients. Further, with the additionally available coefficients it is also possible to transfer more data in a single multivector message (A?) payload using the additionally available multivector coefficients.
[0224] At process 914, the source 902 encrypts a cryptotext multivector (C) as a function of at least one Geometric Algebra geometric product operation on the message multivector (M) and the shared secret multivector
Figure imgf000090_0005
Due to the nature of the geometric product operation of Geometric Algebra, there are many possible variations of the geometric product application that will provide similar degrees of confusion and diffusion. Some, but not all, of the potential geometric product calculations to encrypt the message data (M) include: a geometric product
Figure imgf000090_0003
of the message multivector
Figure imgf000090_0004
and the shared secret multivector ; geometric product "sandwich"
Figure imgf000090_0002
and multivector based Sylvester's equation
Figure imgf000090_0001
[0225] At process 916, the source 902 sends the cryptotext multivector (C) to the intermediary computing system 904. Various embodiments may optionally convert the cryptotext multivector (C) into cryptotext numeric data (C) in accord with reverse operation of a cryptotext data coefficient distribution algorithm that is known to the source 902, intermediary 904, and the destination 906 computing systems/devices. An embodiment may also skip conversion to cryptotext numeric data (C) and directly send a representation of the cryptotext multivector
Figure imgf000090_0006
) without first converting the cryptotext multivector
Figure imgf000090_0008
into cryptotext numeric data (C). The transmission may be implemented as a series of transfers of the coefficients or as some form of records/packets that define a data structure that carries the coefficient data of the cryptotext multivector
Figure imgf000090_0007
. Not converting the cryptotext multivector (C) into cryptotext numeric data (C) has the advantage of avoiding the processing time for the conversion as well as having the advantage that, for homomorphic operations performed at an intermediary computing system, the intermediary computing system need not have any knowledge of the methodology used to create the cryptotext multivector (C). If process 916 is used to convert the cryptotext multivector (C) into cryptotext numeric data (C), it is necessary for any computing device/system that wishes to operate on the cryptotext multivector (C) to have knowledge of the particular conversion methodology so that the computing device/system may properly recreate the cryptotext multivector (C). Notably, having knowledge of the process to convert back and forth between the cryptotext multivector (C) and the cryptotext numeric data (C) does not provide information that will affect the encrypted security of the plaintext message value (M). A disadvantage of not converting the cry totext multivector (C) into cryptotext numeric data (C) is that it may be possible to include additional confusion/diffusion features in conversion to cryptotext numeric data.
[0226] At process 918, the intermediary computing system 904 receives the cryptotext multivector (C) sent by the source 902. At process 920, the intermediary performs the user desired scalar multiplication of the cryptotext multivector (C) sent by the source 902 and an unencrypted scalar data value (V). At process 922, the intermediary 904 sends the scalar multiplicative result cryptotext multivector (SMRC) to the destination 906.
[0227] At process 924, the destination 906 receives the scalar multiplicative result cryptotext multivector (SMRC) sent by the intermediary 904. At process 926, the destination 906 distributes shared secret numeric value (Ss) into shared secret multivector (Ss) coefficients in the same fashion as was done for the source 902 at process 912. At process 928, the destination decrypts the scalar multiplicative result cryptotext multivector (SMRC) as a function of at least one Geometric Algebra geometric product operation on the scalar multiplicative result cryptotext multivector (SMRC) and an inverse of the shared secret multivector (Ss) into a scalar result message multivector (SMR). Again, due to the nature of the geometric product operation of Geometric Algebra there are many possible variations of the geometric product application that will provide similar degrees of confusion and diffusion. Some, but not all, of the potential geometric product calculations to decrypt the scalar multiplicative result message data (SMR) include: a geometric product
Figure imgf000091_0003
Figure imgf000091_0001
of said scalar multiplicative result cryptotext multivector (SMRC) and said inverse of said shared secret multivector geometric product "sandwich"
Figure imgf000091_0002
— _
Figure imgf000091_0004
Figure imgf000091_0005
to decrypt); and multivector based Sylvester's equation (SMR =
) to decrypt). At process 930, the
Figure imgf000091_0006
destination 906 converts the scalar multiplicative result cryptotext multivector (SMRC) into the scalar multiplicative result numeric value (SMR) in accord with reverse operation of homomorphic preserving mathematical relationship of the source 902 at process 910.
[0228] Fig. 10 is a flow chart 1000 of the operations for sending a multiply command for a scalar multiplicative homomorphic encryption embodiment. For the circumstance where the intermediary computing system 1004 does not already have a predefined scalar data value (V), the information used to determine the scalar data value (V) may be derived from scalar data value information contained in scalar multiplication instructions received from a command computing device 1002. At process 1006, the command computing device 1002 sends scalar multiplication instructions, including scalar data value information that defines the scalar data value (V) to be scalar multiplied with the cryptotext multivector (C). At process 1008, the intermediary computing system 1004 receives the scalar multiplication instructions from the command computing device 1002. At process 1010, the intermediary computing system determines the scalar data value (7) based on the scalar value information delivered as part of the scalar multiplication instructions. For example, the scalar data value information may define the unencrypted scalar data value (V) that is to be multiplied by the encrypted cryptotext multivector (C) as a simple predefined data value, or the scalar data value (V) may be defined to be the result of several operations on data provided in the scalar data value information of the scalar multiplication instructions, and/or the scalar data value information may provide instructions to the intermediary computing system 1004 to look up some or all of the necessary data in privately or publicly available data storage accessible by the intermediary computing system 1004.
[0229] A solution analogous to the results using quartenions or matrices in
[REFERENCE2] is given in [REFERENCE 1 ] as:
Figure imgf000093_0001
and is used by the destination to unpack, the cryptotext prior to the decryption process.
[0230] For more detail on general Geometric Algebra concepts, an additional discussion of an overview of general Geometric Algebra concepts may be found in
"Appendix A: Geometric Algebra Overview" of the parent patent application Serial No. 15/667,325, filed August 2, 2017, entitled "Methods and Systems for Enhanced Data-Centric Encryption Systems Using Geometric Algebra," which has been specifically incorporated herein by reference for all that it discloses and teaches. For a more exhaustive reference see [REFERENCE1]. For a more exhaustive reference regarding Sylvester's equation see
[REFERENCE2].
[0231] [REFERENCED "Functions of Multivector Variables," PLOS ONE |
DOI:10.1371/journal.pone.0116943 March 16, 2015, James M. Chappell, Azhar lqbal, Lachlan J. Gunn, Derek Abbott, School of Electrical and Electronic Engineering, University of Adelaide, Adelaide, South Australia, Australia}
[0232] [REFERENCE2] Janovsk6 D, Opfer G (2008) Linear equations in quaternionic variables. Mitt Math Ges Hamburg 27: 223-234.
[0233] The foregoing description of the invention has been presented for purposes of illustration and description. It is not intended to be exhaustive or to limit the invention to the precise form disclosed, and other modifications and variations may be possible in light of the above teachings. The embodiments were chosen and described in order to best explain the principles of the invention and its practical application to thereby enable others skilled in the art to best utilize the invention in various embodiments and various modifications as are suited to the particular use contemplated.

Claims

CLAIMS What is claimed is:
1. A method for performing homomorphie scalar multiplication on a cryptotext encrypted data representation of a corresponding plain text data value and an unencrypted scalar data value, the method comprising:
distributing by a source computing device a numeric message data value (Af ) into coefficients of a message multivector (M) representing said numeric message data value (M) in accord with a hornomorphic preserving mathematical relationship between an unencrypted numeric data value and multivector coefficietrts representing said unencrypted numeric data value that is known to said source computing device and a destination computing device;
distributing by said source computing device a shared secret numeric value (Ss) into coefficients of a shared secret multivector (Ss) in accord with a shared secret coefficient distribution algorithm that is known to said source computing device and said destination computing device, such that said shared secret numeric value (Ss) is known or knowab!e to said source competing device and said destination computing device, but is kept secret from other devices not intended to have access to said numeric message data including an intermediary computing system;
encrypting by said source computing device a cryptotext multivector (Ss) as an encryption function of at least one Geometric Algebra geometric product operation on said message multivector (M) and said shared secret multivector (Ss);
sending by said source computing device said cryptotext multivector (C) to said intermediary computing system;
receiving by said intermediary computing system said cryptotext multivector (C) sent by said source computing device;
multiplying by said intermediary computing system an unencrypted scalar data value (V) and said cryptotext multivector (C) using scalar-vector multiplication in order to obtain a scalar multiplicative result cryptotext multivector (SMRC);
sending by said intermediary computing system said scalar multiplicative result cryptotext multivector (SMRC) to said destination computing device; receiving by said destination computing device said scalar multiplicative result cryptotext multivector (SMRC) sent by said intermediary computing system;
distributing by said destination computing device said shared secret numeric value (Ss) into said shared secret multivector (Ss) in accord with said shared secret coefficient distribution algorithm;
decrypting by said destination computing device said scalar multiplicative result cryptotext multivector (SMRC) as a decryption function of at least one Geometric Algebra geometric product operation on said scalar multiplicative result cryptotext multivector (SMRC) and an inverse
Figure imgf000095_0001
of said shared secret multivector (Ss) into a scalar multiplicative result multivector (SMR) such that said decryption function provides a corresponding decryption operation for said encryption process of said cryptotext multivector (C); and
converting by said destination computing device said scalar multiplicative result multivector (SMR) into a scalar multiplicative result value (SMR) in accord with said homomorphic preserving mathematical relationship such that said scalar
multiplicative result value (SMR) is equal to a multiplication product of said unencrypted numeric message data value (M) and said unencrypted scalar data value
2. The method of claim 1 further comprising:
sending by a command computing device to said intermediary computing system scalar multiplication instructions to perform said scalar multiplication of said unencrypted scalar data value (V) and said cryptotext multivector (C) including scalar data value information that defines said unencrypted scalar data value (V);
receiving by said intermediary computing system said scalar multiplication instructions sent by said source computing device; and
determining by said intermediary computing system said unencrypted scalar data value (V) based on said scalar data value information.
3. The method of claim 2 wherein said scalar data value information that defines said
unencrypted scalar data value (V) is comprised of at least one of a group chosen from: a particular data value that is to be said unencrypted scalar data value (V), information that permits said intermediary computing system to locate said unencrypted scalar data value (V) in a data storage system available to said intermediary computing system, and information that permits said intermediary computing system to perform a calculation or algorithm to determine said unencrypted scalar data value (V).
4. The method of claim 1 wherein said unencrypted scalar data value (V) is predefined at said intermediary computing system.
5. The method of claim 1 wherein said homomorphic preserving mathematical relationship between said unencrypted numeric data value and said multivector coefficients representing said unencrypted numeric data ensures that a result of mathematical operations defined by said homomorphic preserving mathematical relationship on said multivector coefficients representing said unencrypted numeric data value is equal to said unencrypted numeric data value.
6. The method of claim 5 wherein said mathematical operations defined by said
homomorphic preserving mathematical relationship are comprised of at least one of a group chosen from: addition of at least one coefficient of said multivector coefficients, subtraction of at least one coefficient of said multivector coefficients, addition of a constant value, subtraction of a constant value, multiplication of at least one coefficient of said multivector coefficients by a constant value, and division of at least one coefficient of said multivector coefficients by a constant value.
7. The method of claim 5 wherein said mathematical operations defined by said
homomorphic preserving mathematical relationship incorporate at least one coefficient value of said multivector coefficients such that said mathematical operations defined by said homomorphic preserving mathematical relationship is comprised of one of a group chosen from: said mathematical operations defined by said homomorphic preserving mathematical relationship incorporate all coefficient values of said multivector coefficients, said mathematical operations defined by said homomorphic preserving mathematical relationship incorporate fewer than all but more than one coefficient values of said multivector coefficients, and said mathematical operations defined by said homomorphic preserving mathematical relationship incorporate one coefficient value of said multivector coefficients.
8. The method of claim 1 wherein said numeric message data value (M) and said scalar multiplicative result value (SMR) are numeric values comprised of at least one of a group chosen from: positive numbers, negative numbers, zero, integer numbers, and real numbers.
9. The method of claim 1 wherein numeric values of said coefficients of said message
multivector (M) and said coefficients of said scalar multiplicative result multivector (SMR) are comprised of at least one of a group chosen from: positive numbers, negative numbers, zero, integer numbers, and real numbers.
10. The method of claim 1 wherein scalar-vector multiplication is comprised of at least one of a group of vector operations chosen from: scalar-vector multiplication, and scalar-vector division.
11. The method of claim 1 wherein said process of multiplying said unencrypted scalar data value (V) and said cryptotext multivector (C) using scalar-vector multiplication in order to obtain a scalar multiplicative result cryptotext multivector (SMRC) supports an unlimited number of multiplication operations with different unencrypted scalar data values (Vn) in said process of multiplying using scalar-vector multiplication.
12. The method of claim 1 :
wherein said process of distributing said numeric message data value (M) into coefficients of said message multivector (M) further ensures that not all coefficients of said message multivector (M) are equal to each other; and
wherein said shared secret coefficient distribution algorithm further ensures that not all coefficients of said shared secret multivector (Ss) are equal to each other.
13. The method of claim 2 wherein said source computing device separately performs
processes of at least one of a group chosen from: said command computing device, said intermediary computing system, and said destination computing device.
14. The method of claim 2 wherein said command computing device separately performs processes of at least one of a group chosen from: said source computing device, said intermediary computing system, and said destination computing device.
15. The method of claim 2 wherein said intermediary computing system separately performs processes of at least one of a group chosen from: said source computing device, said command computing device, and said destination computing device.
16. The method of claim 2 wherein said destination computing system separately performs processes of at least one of a group chosen from: said source computing device, said command computing device, and said intermediary computing system.
17. The method of claim 1 wherein evaluation of Geometric Algebra geometric products and inverses of multivectors is implemented on said source computing device and said destination computing device using basic arithmetic operations of addition, subtraction, multiplication, and division.
18. The method of claim 17 wherein said implementation of said Geometric Algebra
geometric products and inverses of multivectors on said source computing device and said destination computing device does not include a complex operation to select a prime number, to calculate a logarithm function, and/or to calculate a natural logarithm function.
19. The method of claim 1 further comprising establishing said shared secret numeric value (Ss) between said source computing device and said destination computing device using a known shared secret technique.
20. The method of claim 19 wherein said known shared secret technique is comprised of at least one of a group chosen from: pre-conditioning said source computing device and said destination computing device with said shared secret numeric value (Ss); standard public/private key exchange technique; RSA (Rivest-Shamir-Adleman) key exchange, and Diffie- ellman key exchange.
21. The method of claim 1 wherein said eiicryption function of at least one Geometric Algebra geometric product operation and said decryption function of at least one Geometric Algebra geometric product operation is comprised of at least one of a group chosen from: a geometric product (C of a. message multivector (Af) and said
Figure imgf000099_0002
shared secret multivector (Ss) to encrypt and a geometric product of
Figure imgf000099_0003
said scalar multiplicative result cryptoiexi multivector (
Figure imgf000099_0004
said shared secret multivector (
Figure imgf000099_0006
s) to decrypt; geometric product "sandwich" (
Figure imgf000099_0007
to encrypt and S
Figure imgf000099_0005
to decrypt); and multivector based Sylvester's e uation
Figure imgf000099_0008
(
Figure imgf000099_0001
22. The method of claim 1 :
wherein said encryption function of at least one Geometric Algebra geometric product operation performed by said source computing device further comprises:
generating a second shared secret key
Figure imgf000099_0012
) as a scalar result of a 0-Blade
Reduction Operation of said shared secret multivector
Figure imgf000099_0011
;
distributing said second shared secret key into at least two non-zero
Figure imgf000099_0010
coefficients of a second shared secret multivector in accord with a second
Figure imgf000099_0009
shared secret coefficient distribution algorithm that is known to said source computing device and said destination computing device; and encrypting said cryptotext multivector (C) as a function of Geometric Algebra geometric product operations on a message multivector (M), said shared secret multivector
Figure imgf000099_0013
, and said second shared secret multivector
Figure imgf000099_0014
and
wherein said encryption function of at least one Geometric Algebra geometric product operation performed by said destination computing device further comprises:
generating said second shared secret key (¾2) as a scalar result of said G- Blade Reduction Operation of said shared secret mnltivector
Figure imgf000099_0015
distributing said second shared secret key into said second shared secret multivector (¾ ) in accord with said second shared secret coefficient distribution algorithm; and decrypting said scalar multiplicative result cryptotext multivector (SMRC) as a function of Geometric Algebra geometric product operations on said scalar multiplicative result cryptotext multivector (SMRC), an inverse
Figure imgf000100_0015
of said shared secret multivector
Figure imgf000100_0013
and an inverse of said second
Figure imgf000100_0014
shared secret multivector into said scalar multiplicative result multivector
Figure imgf000100_0012
(SMR).
23. The method of claim 22 wherein said O-Blade Reduction Operation is a geometric
product
Figure imgf000100_0003
of a geometric product of said shared secret
Figure imgf000100_0004
multivector
Figure imgf000100_0009
and a Clifford conjugate
Figure imgf000100_0008
( s) of said shared secret multivector and a
Figure imgf000100_0005
geometric reverse
Figure imgf000100_0006
f said geometric product
Figure imgf000100_0007
of said shared secret multivector (Ss) and said Clifford conjugate
Figure imgf000100_0011
of said shared secret multivector (
Figure imgf000100_0010
24. The method of claim 22 wherein said Geometric Algebra geometric product operations are comprised of at least one of a group chosen from: geometric product "sandwich"
Figure imgf000100_0002
Figure imgf000100_0001
25. The method of claim 1 wherein said processes of sending by said source computing device said first cryptotext multivector (C) to said intermediary computing system and receiving by said intermediary computing system said cryptotext multivector (C) sent by said first source computing device further comprises:
converting by said source computing device said cryptotext multivector (C) into cryptotext numeric data (C) in accord with reverse operation of a cryptotext data coefficient distribution algorithm that is known to said first source computing device and said intermediary computing system;
sending by said source computing device said cryptotext numeric data (C) to said intermediary computing system; and distributing by said intermediary computing system said cryptotext numeric data (C) into said cryptotext multivector (C) in accord with said cryptotext data coefficient distribution algorithm.
26. The method of claim 1 wherein said processes of sending by said intermediary computing system said scalar multiplicative result cryptotext multivector (SMRC) to said destination computing device and receiving by said destination computing device said scalar multiplicative result cryptotext multivector (SMRC) sent by said intermediary computing system further comprises:
converting by said intermediary computing system said scalar multiplicative result cryptotext multivector (SMRC) into corresponding scalar multiplicative result cryptotext numeric data (SMRC) in accord with reverse operation of a cryptotext data coefficient distribution algorithm that is known to said destination computing device and said intermediary computing system;
sending by said intermediary computing system said scalar multiplicative result cryptotext numeric data (SMRC) to said destination computing device;
receiving by said destination computing device said scalar multiplicative result cryptotext numeric data (SMRC) sent by said intermediary computing system; and distributing by said destination computing device said scalar multiplicative result cryptotext numeric data (SMRC) into said scalar multiplicative result cryptotext multivector (SMRC) in accord with said cryptotext data coefficient distribution algorithm.
27. A method for encrypting a numeric message data value (M) on a source computing device in order to transfer a cryptotext multivector (C) encrypted representation of said numeric message data value (M) to an intermediary computing system that will perform homomorphic scalar multiplication of said cryptotext multivector (C) and an unencrypted scalar data value (V) and deliver a result of said homomorphic scalar multiplication to a destination computing device, the method comprising:
distributing by said source computing device said numeric message data value (M) into coefficients of a message multivector (M ) in accord with a homomorphic preserving mathematical relationship between an unencrypted numeric data value and multivector coefficients representing said unencrypted numeric data value that is known to said source computing device and said destination computing device;
distributing by said source computing device a shared secret numeric value (Ss) into coefficients of a shared secret multivector (Ss) in accord with a shared secret coefficient distribution algorithm that is known to said source computing device and said destination computing device, said shared secret numeric value (Ss) being known or knowable to said source computing device and said destination computing device, but is kept secret from other devices not intended to have access to said numeric message data including said intermediary computing system;
encrypting by said source computing device said cryptotext multivector (C) as an encryption function of at least one Geometric Algebra geometric product operation on said message multivector (M ) and said shared secret multivector (Ss); and
sending by said source computing device said cryptotext multivector (C) to said intermediary computing system.
28. A method for performing homomorphic scalar multiplication on an intermediary
computer system of a cryptotext multivector (C) encrypted data representation of a corresponding plain text numeric data value received from a source computing device and an unencrypted scalar data value (7) and delivering a homomorphic scalar multiplicative result cryptotext multivector (SMRC) to a destination computing device, the method comprising:
receiving by said intermediary computing system said cryptotext multivector (C) sent by said source computing device;
multiplying by said intermediary computing system said unencrypted scalar data value (V) and said cryptotext multivector (C) using scalar-vector multiplication in order to obtain a scalar multiplicative result cryptotext multivector (SMRC); and sending by said intermediary computing system said scalar multiplicative result cryptotext multivector (SMRC) to said destination computing device.
29. A method for decrypting a scalar multiplicative result cryptotext multivector (SMRC) on a destination computing device received from an intermediary computing system that performed homomorphic scalar multiplication of a cryptotext multivector (C) originated from a source computing device and an unencrypted scalar data value (V), the method comprising:
receiving by said destination computing device said scalar multiplicative result cryptotext multivector (SMRC) sent by said intermediary computing system;
distributing by said source computing device a shared secret numeric value (Ss) into a shared secret multivector (Ss) in accord with a shared secret coefficient distribution algorithm that is known to said source computing device and said destination computing device, said shared secret numeric value (Ss) being known or knowable to said source computing device and said destination computing device, but is kept secret from other devices not intended to have access to said numeric message data including said intermediary computing system;
decrypting by said destination computing device said scalar multiplicative result cryptotext multivector (SMRC) as a decryption function of at least one Geometric Algebra geometric product operation on said scalar multiplicative result cryptotext multivector (SMRC) and an inverse (Ss -1) of said shared secret multivector (Ss) into a scalar multiplicative result multivector (SMR) such that said decryption function provides a corresponding decryption operation for an encryption process of said cryptotext multivector (C); and
converting by said destination computing device said scalar multiplicative result multivector (SMR) into a scalar multiplicative result data value (SMR) in accord with said homomorphic preserving mathematical relationship between an unencrypted numeric data value and multivector coefficients representing said unencrypted numeric data value that is known to said source computing device and said destination computing device such that said scalar multiplicative result value (SMR) is equal to a multiplication product of an unencrypted numeric message data value (M) represented by said cryptotext multivector (C) and said unencrypted scalar data value (V).
30. A scalar multiplicative homomorphic Enhanced Data-Centric Encryption (EDCE) system for scalar multiplicative homomorphic multiplication of a cryptotext encrypted data representation of a corresponding plain text data value and an unencrypted scalar data value, the scalar multiplicative homomorphic EDCE system comprising:
a source computing device, wherein said source computing device further comprises: a source numeric message distribution subsystem that distributes a numeric message data value (Ai) into coefficients of a message multivector (Ai) representing said numeric message data value (M) in accord with a homomorphic preserving mathematical relationship between an unencrypted numeric data value and multivector coefficients representing said unencrypted numeric data value that is known to said source computing device and a destination computing device;
a source numeric shared secret distribution subsystem that distributes a shared secret numeric value (Ss) into coefficients of a shared secret multivector (Ss) in accord with a shared secret coefficient distribution algorithm that is known to said source computing device and said destination computing device, such that said shared secret numeric value (Ss) is known or knowable to said source computing device and said destination computing device, but is kept secret from other devices not intended to have access to said numeric message data including an intermediary computing system;
a source encryption subsystem that encrypts a cryptotext multivector (C) as an encryption function of at least one Geometric Algebra geometric product operation on said message multivector (M ) and said shared secret multivector (¼); and
a source send subsystem that sends said cryptotext multivector (C) to said intermediary computing system;
said intermediary computing system, wherein said intermediary computing system further comprises:
an intermediary receive subsystem that receives said cryptotext multivector (C) sent by said source computing device;
an intermediary homomorphic scalar multiplication subsystem that multiplies an unencrypted scalar data value (V) and said cryptotext multivector (C) using scalar-vector multiplication in order to obtain a scalar multiplicative result cryptotext multivector (SMRC); and an intermeidary send subsystem that sends said scalar multiplicative result cryptotext multivector (SMRC) to said destination computing device; and said destination computing device, wherein said destination computing device further comprises:
a destination receive subsystem that receives said scalar multiplicative result cryptotext multivector (SMRC) sent by said intermediary computing system;
a destination numeric shared secret distribution subsystem that distributes said shared secret numeric value (Ss) into said shared secret multivector (Ss) in accord with said shared secret coefficient distribution algorithm
a des ecryption subsystem that decrypts said scalar multiplicative result cryptotext multivector (SMRC) as a decryption function of at least one Geometric Algebra geometric product operation on said scalar multiplicative result cryptotext multivector (SMRC) and an inverse of said shared secret multivector (Ss) into a scalar multiplicative result multivector (SMR) such that said decryption function provides a corresponding decryption operation for said encryption process of said cryptotext multivector (C); and
a destination convert multivector subsystem that converts said scalar multiplicative result multivector (SMR) into a scalar multiplicative result value (SMR) in accord with said homomorphic preserving mathematical relationship such that said scalar multiplicative result value (SMR) is equal to a multiplication product of said unencrypted numeric message data value (M) and said unencrypted scalar data value (V).
3 1.The scalar multiplicative homomorphic EDCE system of claim 30 further comprising: a command computing device, wherein said command computing device further omprises:
a command instruction send subsystem that sends scalar multiplication instructions to perform said scalar multiplication of said unencrypted scalar data value (V) and said cryptotext multivector (C) including scalar data value information that defines said unencrypted scalar data value (V) to said intermediary computing system; and
wherein said intermediary computing system further comprises: an intermediary receive instructions subsystem that receives said scalar multiplication instructions sent by said source computing device; and an intermediary scalar value determination subsystem that determines said unencrypted scalar data value (V) based on said scalar data value information.
32. The scalar multiplicative homomorphic EDCE system of claim 31 wherein said scalar data value information that defines said unencrypted scalar data value (V) is comprised of at least one of a group chosen from: a particular data value that is to be said unencrypted scalar data value (V), information that permits said intermediary computing system to locate said unencrypted scalar data value (V) in a data storage system available to said intermediary computing system, and information that permits said intermediary computing system to perform a calculation or algorithm to determine said unencrypted scalar data value (V).
33. The scalar multiplicative homomorphic EDCE system of claim 30 wherein said
unencrypted scalar data value (V) is predefined at said intermediary computing system.
34. The scalar multiplicative homomorphic EDCE system of claim 30 wherein said
homomorphic preserving mathematical relationship between said unencrypted numeric data value and said multivector coefficients representing said unencrypted numeric data ensures that a result of mathematical operations defined by said homomorphic preserving mathematical relationship on said multivector coefficients representing said unencrypted numeric data value is equal to said unencrypted numeric data value.
35. The scalar multiplicative homomorphic EDCE system of claim 34 wherein said
mathematical operations defined by said homomorphic preserving mathematical relationship are comprised of at least one of a group chosen from: addition of at least one coefficient of said multivector coefficients, subtraction of at least one coefficient of said multivector coefficients, addition of a constant value, subtraction of a constant value, multiplication of at least one coefficient of said multivector coefficients by a constant value, and division of at least one coefficient of said multivector coefficients by a constant value.
36. The scalar multiplicative homomorphic EDCE system of claim 34 wherein said mathematical operations defined by said homomorphic preserving mathematical relationship incorporate at least one coefficient value of said multivector coefficients such that said mathematical operations defined by said homomorphic preserving mathematical relationship is comprised of one of a group chosen from: said mathematical operations defined by said homomorphic preserving mathematical relationship incorporate all coefficient values of said multivector coefficients, said mathematical operations defined by said homomorphic preserving mathematical relationship incorporate fewer than all but more than one coefficient values of said multivector coefficients, and said mathematical operations defined by said homomorphic preserving mathematical relationship incorporate one coefficient value of said multivector coefficients.
37. The scalar multiplicative homomorphic EDCE system of claim 30 wherein said numeric message data value (M) and said scalar multiplicative result value (SMR) are numeric values comprised of at least one of a group chosen from: positive numbers, negative numbers, zero, integer numbers, and real numbers.
38. The scalar multiplicative homomorphic EDCE system of claim 30 wherein numeric values of said coefficients of said message multivector (M) and said coefficients of said scalar multiplicative result multivector (SMR) are comprised of at least one of a group chosen from: positive numbers, negative numbers, zero, integer numbers, and real numbers.
39. The scalar multiplicative homomorphic EDCE system of claim 30 wherein scalar-vector multiplication is comprised of at least one of a group of vector operations chosen from: scalar-vector multiplication, and scalar-vector division.
40. The scalar multiplicative homomorphic EDCE system of claim 30 wherein said
intermediary homomorphic scalar multiplication subsystem that multiplies said unencrypted scalar data value (V) and said cryptotext multivector (C) using scalar-vector multiplication in order to obtain a scalar multiplicative result cryptotext multivector (SMRC) supports an unlimited number of multiplication operations with different unencrypted scalar data values (Vn) in said process of multiplying using scalar-vector multiplication.
41. The scalar multiplicative homomorphic EDCE system of claim 30:
wherein said source numeric message distribution subsystem that distributes said n imeric message data value (M) into coefficients of said message multivector (M) further ensures that not all coefficients of said message multivector (ΛΪ) are equal to each other; and
wherein said shared secret coefficient distribution algorithm further ensures that ot all coefficients of said shared secret multivector (Ss) are equal to each other.
42. The scalar multiplicative homomorphic EDCE system of claim 31 wherein said source computing device separately incorporates subsystems of at least one of a group chosen from: said command computing device, said intermediary computing system, and said destination computing device.
43. The scalar multiplicative homomorphic EDCE system of claim 31 wherein said command computing device separately incorporates subsystems of at least one of a group chosen from: said source computing device, said intermediary computing system, and said destination computing device.
44. The scalar multiplicative homomorphic EDCE system of claim 31 wherein said
intermediary computing system separately incorporates subsystems of at least one of a group chosen from: said source computing device, said command computing device, and said destination computing device.
45. The scalar multiplicative homomorphic EDCE system of claim 31 wherein said
destination computing system separately incorporates subsystems of at least one of a group chosen from: said source computing device, said command computing device, and said intermediary computing system.
46. The scalar multiplicative homomorphic EDCE system of claim 30 wherein evaluation of Geometric Algebra geometric products and inverses of multivectors is implemented on said source computing device and said destination computing device using basic arithmetic operations of addition, subtraction, multiplication, and division.
47. The scalar multiplicative homomorphic EDCE system of claim 46 wherein said
implementation of said Geometric Algebra geometric products and inverses of multivectors on said source computing device and said destination computing device does not include a complex operation to select a prime number, to calculate a logarithm function, and/or to calculate a natural logarithm function.
48. The scalar multiplicative homomorphic EDCE system of claim 30 further comprising establishing said shared secret numeric value (Ss) between said source computing device and said destination computing device using a known shared secret technique.
49. The scalar multiplicative homomorphic EDCE system of claim 48 wherein said known shared secret technique is comprised of at least one of a group chosen from: preconditioning said source computing device and said destination computing device with said shared secret numeric value (Ss); standard public/private key exchange technique; RSA (Rivest-Shamir-Adleman) key exchange, and Diffie-Hellman key exchange.
50. The scalar multiplicative homomorphic EDCE system of claim 30 wherein said
encryption function of at least one Geometric Algebra geometric product operation and said decryption function of at least one Geometric Algebra geometric product operation is comprised of at least one of a group chosen from: a geometric product ) of a
Figure imgf000109_0007
message multivector (A?) and said shared secret multivector (Ss) to encrypt and a geometric produc
Figure imgf000109_0001
( S ) of said scalar multiplicative result cryptotext multivector (SMRC) and said inverse
Figure imgf000109_0002
of said shared secret multivector (Ss) to decrypt; geometric product "sandwich" to encrypt and SMR =
Figure imgf000109_0006
Figure imgf000109_0003
1 to decrypt); and multivector based Sylvester's equation
Figure imgf000109_0004
to encrypt and
Figure imgf000109_0005
decrypt).
51. The scalar multiplicative homomorphic EDCE system of claim 30:
wherein said source computing device further comprises:
a source second shared secret key generation subsystem that generates a second shared secret key (S¾) as a scalar result of a 0-Blade Reduction Operation of said shared secret multivector
Figure imgf000110_0008
and
a source second numeric shared secret distribution subsystem that distributes said second shared secret key (S¾) into at least two non-zero coefficients of a second shared secret multivector ) in accord with a second
Figure imgf000110_0007
shared secret coefficient distribution algorithm that is known to said source computing device and said destination computing device;
wherein said source encryption subsystem further encrypts said cryptotext multivector (C) as a function of Geometric Algebra geometric product operations on a message multivector (M), said shared secret multivector
Figure imgf000110_0006
, and said second shared secret multivector
Figure imgf000110_0005
wherein said destination computing device further comprises:
a destination second shared secret key generation subsystem that generates said second shared secret key (SSi) as a scalar result of said 0-Blade Reduction Operation of said shared secret multivector
Figure imgf000110_0009
and
a destination second numeric shared secret distribution subsystem that distributes said second shared secret key (SSa) into said second shared secret multivector (SS2) in accord with said second shared secret coefficient distribution algorithm; and
wherein said destination decryption subsystem further decrypts said scalar multiplicative result cryptotext multivector (SMRC) as a function of Geometric Algebra geometric product operations on said scalar multiplicative result cryptotext multivector (SMRC), an inverse ) of said shared secret multivector (Ss), and an
Figure imgf000110_0003
inverse
Figure imgf000110_0004
) of said second shared secret multivector (Ss2) into said scalar multiplicative result multivector (SMR).
52. The scalar multiplicative homomorphic EDCE system of claim 51 wherein said 0-Blade Reduction Operation is a geometric product ) of a geometric product
Figure imgf000110_0001
(SsSs) of said shared secret multivector (Ss) and a Clifford conjugate ) of said shared
Figure imgf000110_0002
108 secret multivector
Figure imgf000111_0001
and a geometric reverse
Figure imgf000111_0002
of said geometric product of said shared secret multivector and said Clifford conjugate of said
Figure imgf000111_0004
Figure imgf000111_0003
shared secret multivector
Figure imgf000111_0005
, The scalar multiplicative homomorphic EDGE system of claim 51 wherein said
Geometric Algebra geometric product operations are comprised of at. least one of a group chosen from: geometric product "sandwich"
Figure imgf000111_0006
Figure imgf000111_0007
, The scalar multiplicative homomorphic EDGE system of claim 30:
wherein said source send subsystem further converts said cryptotext multivector (C) into cryptotext numeric data (C) in accord with reverse operation of a cryptotext data coefficient distribution algorithm that is known to said first source computing device and said intermediary computing system then sends said cryptotext numeric data (C) to said intermediary computing system; and
wherein said intermediary receive subsystem further receives said cryptotext numeric data (C) sent by said sent by said source computing device, then distributes said cryptotext numeric data (C) into said cryptotext multivector (C) in accord with said cryptotext data coefficient distribution algorithm. , The scalar multiplicative homomorphic EDGE system of claim 30:
wherein said an intermediary send subsystem further converts said scalar multiplicative result cryptotext multivector (
Figure imgf000111_0008
into corresponding scalar multiplicative result cryptotext numeric data (SMRC) in accord with reverse operation of a cryptotext data coefficient distribution algorithm that is known to said destination computing device and said intermediary computing system, then sends said scalar multiplicative result cryptotext numeric data (SMRC) to said destination computing device; and
wherein said destination receive subsystem further receives said scalar multiplicative result cryptotext numeric data (SMRC) sent by said intermediary computing system, then distributes said scalar multiplicative result cryptotext numeric data (SMRC) into said scalar multiplicative result cryptotext multivector (SMRC) in accord with said cryptotext data coefficient distribution algorithm.
56. A scalar multiplicative homomorphic Enhanced Data-Centric Encryption (EDCE) system source computing device for encrypting a numeric message data value (M) in order to transfer a cryptotext multivector (C) encrypted representation of said numeric message data value (M) to an intermediary computing system that will perform homomorphic scalar multiplication of said cryptotext multivector (C) and an unencrypted scalar data value (10 and deliver
computing device, th
computing device co
a source num
message data val
a homomorphic p
numeric data valu
numeric data valu
computing devic
a source num
secret numeric value (Ss) into coefficients of a shared secret m
accord with a shared secret coefficient distribution algorithm th
source computing device and said destination computing devic
numeric value (Ss) being known or knowable to said source co
said destination computing device, but is kept secret from othe
to have access to said numeric message data including said inte
system;
a source encryption subsystem that encrypts said cryptotext multivector (C) as an encryption function of at least one Geometric Algebra geometric product operation on said message multivector (M) and said shared secret multivector (Ss); and
a source send subsystem that sends said cryptotext multivector (C) to said intermediary computing system.
57. A scalar multiplicative homomorphic Enhanced Data-Centric Encryption (EDCE) system intermediary computing system for performing homomorphic scalar multiplication of a cryptotext multivector (C) encrypted data representation of a corresponding plain text numeric data value received from a source computing device and an unencrypted scalar data value (V) and delivering a homomorphic scalar multiplicative result cryptotext multivector (SMRC) to a destination computing device, the scalar multiplicative homomorphic EDCE system intermediary computing system comprising:
an intermediary receive subsystem that receives said cryptotext multivector (C) sent by said source computing device;
an intermediary homomorphic scalar multiplication subsystem that multiplies said unencrypted scalar data value (10 and said cryptotext multivector (C) using scalar- vector multiplication in order to obtain a scalar multiplicative result cryptotext multivector (SMRC); and
an intermeidary send subsystem that sends said scalar multiplicative result cryptotext multivector (SMRC) to said destination computing device.
58. A scalar multiplicative homomorphic Enhanced Data-Centric Encryption (EDCE) system destination computing device for decrypting a scalar multiplicative result cryptotext multivector (SMRC) received from an intermediary computing system that performed homomorphic scalar multiplication of a cryptotext multivector (C) originated from a source computing device and an unencrypted scalar data value (V), the scalar multiplicative homomorphic EDCE system destination computing device comprising: a destination receive subsystem that receives said scalar multiplicative result cryptotext multivector (SMRC) sent by said intermediary computing system;
a destination numeric shared secret distribution subsystem that distributes a shared secret numeric value (Ss) into a shared secret multivector (Ss) in accord with a shared secret coefficient distribution algorithm that is known to said source computing device and said destination computing device, said shared secret numeric value (Ss) being known or knowable to said source computing device and said destination computing device, but is kept secret from other devices not intended to have access to said numeric message data including said intermediary computing system;
a destination decryption subsystem that decrypts said scalar multiplicative result cryptotext multivector (SMRC) as a decryption function of at least one Geometric
111 Algebra geometric product operation on said scalar multiplicative result cryptotext multivector (SMRC) and an inverse
Figure imgf000114_0001
( s ) of said shared secret multivector (
Figure imgf000114_0002
into a scalar multiplicative result multivector (SMR) such that said decryption function provides a corresponding decryption operation for an encryption process of said cryptotext multivector
Figure imgf000114_0003
; and
a destination convert multivector subsystem that converts said scalar
multiplicative result multivector (SMR) into a scalar multiplicative result data value (SMR) in accord with said homomorphic preserving mathematical relationship between an unencrypted numeric data value and multivector coefficients representing said unencrypted numeric data value that is known to said source computing device and said destination computing device such that said scalar multiplicative result value (SMR) is equal to a multiplication product of an unencrypted numeric message data value ( M) represented by said cryptotext multivector (C) and said unencrypted scalar data value (V).
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