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WO2018187604A1 - Procédés et systèmes destinés à des systèmes améliorés de chiffrement homomorphe multiplicatif scalaire centré sur les données utilisant l'algèbre géométrique - Google Patents

Procédés et systèmes destinés à des systèmes améliorés de chiffrement homomorphe multiplicatif scalaire centré sur les données utilisant l'algèbre géométrique 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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WIPO (PCT)
Prior art keywords
multivector
scalar
computing device
cryptotext
shared secret
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Application number
PCT/US2018/026305
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English (en)
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/fr

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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

L'invention concerne des procédés et des systèmes permettant de chiffrer un message numérique à l'aide de l'algèbre géométrique sur un dispositif informatique source, d'effectuer une multiplication scalaire-vectorielle sur le message numérique chiffré et sur une valeur scalaire de données non chiffrées en vue d'obtenir un résultat de multiplication scalaire chiffré sans déchiffrer le message numérique chiffré sur un système informatique intermédiaire qui n'a pas connaissance des clés de sécurité de chiffrement, et de déchiffrer à l'aide de l'algèbre géométrique le résultat de multiplication scalaire chiffré sur un dispositif informatique de destination de sorte que le résultat déchiffré est égal à la multiplication du message numérique non chiffré et de la valeur scalaire de données. Des opérations de chiffrement utilisent le produit géométrique (produit de Clifford) de multivecteurs créés à partir de texte en clair/de données des messages de données numériques, et un ou plusieurs autres multivecteurs comprenant des clés de chiffrement. Pendant l'opération de déchiffrement le résultat multiplicatif scalaire est déchiffre à l'aide d'opérations d'algèbre géométrique telles que l'inverse d'un multivecteur, le conjugué de Clifford, et d'autres, conjointement avec le produit géométrique.
PCT/US2018/026305 2017-04-07 2018-04-05 Procédés et systèmes destinés à des systèmes améliorés de chiffrement homomorphe multiplicatif scalaire centré sur les données utilisant l'algèbre géométrique WO2018187604A1 (fr)

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US15/667,325 US10728227B2 (en) 2016-08-02 2017-08-02 Methods and systems for enhanced data-centric encryption systems using geometric algebra
US15/667,325 2017-08-02
US201762572970P 2017-10-16 2017-10-16
US201762572955P 2017-10-16 2017-10-16
US62/572,970 2017-10-16
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