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US20020099746A1 - T-sequence apparatus and method for general deterministic polynomial-time primality testing and composite factoring - Google Patents

T-sequence apparatus and method for general deterministic polynomial-time primality testing and composite factoring Download PDF

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US20020099746A1
US20020099746A1 US09/559,142 US55914200A US2002099746A1 US 20020099746 A1 US20020099746 A1 US 20020099746A1 US 55914200 A US55914200 A US 55914200A US 2002099746 A1 US2002099746 A1 US 2002099746A1
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prime
mod
factoring
numbers
sequence
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Teck Tie
Shaul Backal
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BACKAL SHAUL
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Assigned to MEGANET CORPORATION reassignment MEGANET CORPORATION ASSIGNMENT OF ASSIGNORS INTEREST (SEE DOCUMENT FOR DETAILS). Assignors: BACKAL, SHAUL O., TIE, TECK SING
Priority to AU63724/00A priority patent/AU6372400A/en
Assigned to BACKAL, SHAUL reassignment BACKAL, SHAUL ASSIGNMENT OF ASSIGNORS INTEREST (SEE DOCUMENT FOR DETAILS). Assignors: MEGANET CORPORATION
Publication of US20020099746A1 publication Critical patent/US20020099746A1/en
Priority to US10/306,072 priority patent/US9542158B2/en
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    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F7/00—Methods or arrangements for processing data by operating upon the order or content of the data handled
    • G06F7/60—Methods or arrangements for performing computations using a digital non-denominational number representation, i.e. number representation without radix; Computing devices using combinations of denominational and non-denominational quantity representations, e.g. using difunction pulse trains, STEELE computers, phase computers
    • G06F7/72—Methods or arrangements for performing computations using a digital non-denominational number representation, i.e. number representation without radix; Computing devices using combinations of denominational and non-denominational quantity representations, e.g. using difunction pulse trains, STEELE computers, phase computers using residue arithmetic
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F7/00—Methods or arrangements for processing data by operating upon the order or content of the data handled
    • G06F7/58—Random or pseudo-random number generators
    • G06F7/582—Pseudo-random number generators
    • G06F7/586—Pseudo-random number generators using an integer algorithm, e.g. using linear congruential method
    • 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
    • H04L9/06—Cryptographic mechanisms or cryptographic arrangements for secret or secure communications; Network security protocols the encryption apparatus using shift registers or memories for block-wise or stream coding, e.g. DES systems or RC4; Hash functions; Pseudorandom sequence generators
    • H04L9/065—Encryption by serially and continuously modifying data stream elements, e.g. stream cipher systems, RC4, SEAL or A5/3
    • H04L9/0656—Pseudorandom key sequence combined element-for-element with data sequence, e.g. one-time-pad [OTP] or Vernam's cipher
    • H04L9/0662—Pseudorandom key sequence combined element-for-element with data sequence, e.g. one-time-pad [OTP] or Vernam's cipher with particular pseudorandom sequence generator
    • 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
    • H04L9/30—Public key, i.e. encryption algorithm being computationally infeasible to invert or user's encryption keys not requiring secrecy
    • H04L9/3006—Public key, i.e. encryption algorithm being computationally infeasible to invert or user's encryption keys not requiring secrecy underlying computational problems or public-key parameters
    • H04L9/3033—Public key, i.e. encryption algorithm being computationally infeasible to invert or user's encryption keys not requiring secrecy underlying computational problems or public-key parameters details relating to pseudo-prime or prime number generation, e.g. primality test
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F2207/00—Indexing scheme relating to methods or arrangements for processing data by operating upon the order or content of the data handled
    • G06F2207/72—Indexing scheme relating to groups G06F7/72 - G06F7/729
    • G06F2207/7204—Prime number generation or prime number testing

Definitions

  • the present invention relates to prime and composite number computing and applications of the same, e.g., in the area of data security.
  • Prime numbers (2, 3, 5, 7, 11, 13, . . ., those positive integers divisible only by themselves or 1) are the most fundamental building blocks of math, and with the invention of the public key ciphers (RSA, El Gamal and the like), they now form the backbone of computer security.
  • the primality testing problem is about testing and determining whether a given arbitrary positive integer is a prime number or a composite (non-prime) number.
  • the factoring problem requires determining the composite number's prime factors. Practicality demands that these two problems have to be solved in polynomial time (computations being proportional to the number of digits and therefore fast), not exponential time (computations being proportional to the size of the numbers themselves and therefore too slow).
  • T-sequence Using a new mathematical technique called the T-sequence, the inventor has discovered a powerful primality testing method that meets all four conditions above. A similar approach can be applied to perform fast factoring for numerous special cases, a method that can, in all liklihood, be extended to the general case, making possible a general and fast factoring algorithm. (Researchers heretofore have been able to factor only in sub-exponential time, never in polynomial time.)
  • the same T-sequence can be used to construct a prime number formula (long sought after but never achieved) and a good random number generator.
  • the former can be used to generate infinitely many prime numbers of any size efficiently, and the latter can generate non-periodic and absolutely chaotic random numbers. These numbers are widely used in all areas of industrial and scientific simulations.
  • the T-sequence can be used to handle efficiently the fundamental problems concerning prime numbers (which include primality testing, factoring, prime number formula, infinite-pattern prime problem, etc.).
  • FIG. 1 is a block diagram of a prime number computing system
  • FIG. 2 is a flowchart illustrating a primality testing algorithm.
  • T-Sequences Definition.
  • n be a positive integer and l>3 be the order.
  • T n 1 + n 2 l T n 1 l ⁇ T n 1 - n 2 l
  • T terms can grow exponentially large, but with the above identities as well as modulo arithmetic and a type of binary decomposition method described below, testing a given integer for primality is straightforward.
  • T 31 3 T 16 3 ⁇ T 15 3 - 3
  • T 16 3 ( T 8 3 ) 2 - 2
  • T 15 3 T 8 3 ⁇ T 7 3 - 3
  • T 8 3 ( T 4 3 ) 2 - 2
  • T 7 3 T 4 3 ⁇ T 3 3 - 3
  • T 4 3 ( T 2 3 ) 2 - 2
  • T 3 3 T 2 3 ⁇ T 1 3 - 3
  • T 2 3 ( T 1 3 ) 2 - 2
  • T 0 3 2
  • T 1 3 3
  • this characteristic can also be used to do general polynomial time factoring of composites.
  • FIG. 1 a block diagram is shown of a computing system, e.g., a prime number computing system, in which T-sequences are used.
  • the computing system includes one or more processors, random-access memory, read-only (non-volative) memory, and an I/O subsystem.
  • the computing system is intended to be representative of all classes of computing systems, large and small, local or distributed.
  • Within memory is stored a routine for generating T-sequence terms.
  • the results of this routine are used by one or more other routines, e.g., a routine for primality testing, a routine for factoring, a prime number generator, a random number generator, etc.
  • routines find wide application, especially in data security, e.g., securely encrypting data or, by the opposite token, breaking a given encryption. The operation of various ones of these routines will now be described.
  • the T 3 sequence may be used to perform primality testing (any other T l sequence will do but T 3 is convenient for use here).
  • n an eligible candidate for prime
  • any n with the last digit 1 or 9 will be of the ⁇ l type in T 3
  • any n with the last digit 3 or 7 will be of the +l type in T 3 .
  • the period k(r) must be greater than 2. When the period is 1 or 2, that l value is not to be used.
  • a fast primality testing routine consists of the following three steps:
  • STEP B still misses some pseudoprimes or cofactor composites but when followed by STEP C, all possible exceptions in the form of proper cofactors or pseudoprimes will be sieved away, leaving only the genuine primes.
  • STEP C Find an l which is of opposite l type to that in STEP A in T 3 . If in STEP A the l type of n in T 3 is ⁇ , then in this STEP C, find an l for which the l type of n is + in T l and vice versa. This can be determined readily through the above-mentioned computations of small residue r or direct computations of T n - 1 l ⁇ 2 ⁇ ⁇ or ⁇ ⁇ l 2 - 2 ⁇ ⁇ ( mod ⁇ ⁇ n ) ⁇ ⁇ and ⁇ ⁇ T n l ⁇ l ⁇ ( mod ⁇ ⁇ n )
  • a variation of the foregoing algorithm uses the Jacobi to avoid blind trials seeking for opposite l types.
  • taking JACOBI(l 2 ⁇ 4, n) gives the l type.
  • Primality Testing Summary. Following the above method of computation ensures that this primality testing algorithm is 100% general, deterministic, provable and polynomial-time. It runs as follows:
  • n is a genuine prime whenever n satisfies the conditions in these three steps:
  • STEP A T n - 1 3 ⁇ 2 ⁇ ⁇ or ⁇ ⁇ 7 ⁇ ⁇ ( mod ⁇ ⁇ n ) ⁇ ⁇ and ⁇ ⁇ T n 3 ⁇ 3 ⁇ ⁇ ( mod ⁇ ⁇ n )
  • STEP C T n ⁇ 1 l ⁇ 2 or l 2 ⁇ 2 (mod n) and T n l ⁇ l (mod n) where the l type of n in T l is opposite to that in T 3 as in STEP A.
  • a promising and viable factoring method is also based on the T-sequences. This method is unlike any previous method.
  • T-sequences allow all forms of composites to be factored, without exception, in polynomial time, simply because binary decomposition modulo C is fundamentally polynomial time. So far, mathematicians have only found exponential or sub-exponential time factoring algorithms for composites less than 200 digits, in general, and no polynomial-time factoring exists for even special forms of composites like the Mersenne numbers 2 M ⁇ 1, etc. A simple extension of the T sequences, however, immediately provides just such a polynomial-time factoring algorithm PTFA) for numerous special form composites with infinite membership.
  • PTFA polynomial-time factoring algorithm
  • ⁇ 2 is used when the periods p+1 or p ⁇ 1 divides exactly into f(C) and +2 is used whenever f(C) divided by p+1 or p ⁇ 1 gives a residue of p ⁇ 1 2 ,
  • f(C) aC 3 ⁇ bC 2 ⁇ cC 1 ⁇ d where 0 ⁇ a, b, c, d ⁇ 4.
  • this method bears a strikingly close relationship to the elliptic curve method. It is general and always polynomial time. No counterexamples have so far been found. Also very effective are the above-mentioned small residue factoring sieve as well as a quadratic polynomial factoring sieve not described here. Composites of an arbitrary number of prime factors can be handled and factored too. A 100% complete and efficient PTFA should be based upon such a formula or similar one.
  • (mod n) is factored by taking a ( R n l ) 2 ⁇ bR n l ⁇ c
  • Prime Number Formula Traditionally, a prime number formula (which has never been found) has always had these requirements:
  • Prime number formula of this type can be constructed, upon reflection, it may be seen that the third requirement is inconsistent with the very definition of prime numbers, namely that they cannot be divided exactly by any other numbers other than themselves and 1.
  • the implication is that the primality of a positive integer n needs to be determined by a legitimate polynomial-time primality testing algorithm. Whether n is prime or composite cannot be ascertained right away. Rather, n must be tested for primality.
  • a prime number formula which is supposed to generate primes and not composites also needs to obey such a fundamental requirement.
  • Prime number formula is in essence one version of a primality testing algorithm; whereas the traditional formulation of a prime number formula is an NP problem, the foregoing formulation recast the problem such that NP ⁇ P.
  • a new prime number formula of the type described may be arrived at by making use of a revised version of the Fortune Conjecture, i.e., P i+1 ⁇ P 1 P 2 P 3 . . . Pi is always a prime. This can be shown to be equivalent to the conjecture that the smallest gap between two consecutive primes P i+1 , and P i is (lnP i lnlnP i ) 2 . If this gap is simplified to ln 2 P i , then following Euclid's celebrated proof for the infinity of prime numbers, one can easily show that Fortune Conjecture is equivalent to this smallest gap conjecture. The validity of these two conjectures are well substantiated empirically as well as theoretically.
  • RNG random number generator
  • the foregoing primality testing algorithm can be used generate an abundance of large primes such as cannot be generated in any other way.
  • the set of last prime digits can also be generated and arranged in all sorts of arbitrary ways.
  • the seeds can be added or subtracted in any which way too. Without a complete knowledge of the exact seeds and their mathematical operations, no one can reproduce or deduce this type of random digits of the primes.
  • These random digits of primes behave in just as chaotic fashion as the physical subatomic particles in their distribution. Therefore this method can conveniently generate any length of random digits or numbers desired to use in mathematical research or industrial simulation.
  • This generator of random digits can be implemented easily and efficiently in both hardware and software.
  • Conventional RNGs such as linear or non-linear feedback shift registers always carry period patterns which are inherent. Non-periodicity is inherent in the foregoing random prime digit generator.
  • This RNG can also be easily modified into a simple but innovative cipher: a function F 1 , (such as transposition, shuffling, etc.) that operates on the last prime digit and another function F 2 that computes and determines the seeds are both kept secret.
  • F 2 is coupled to a simple but chaotic physical system such as dice-throwing, radioactive matter, etc., for the first random input as seeds.
  • the functions F 2 and F 1 are used to generate a truly random string of digits such as 9, 7, 3, 1, 1, 9, 3, 3, 7, 3, 1, 3, 7, 9, 3, 9, 1, 1, 7, 7. This string of random digits can be used as a one-time pad for encryption.
  • the receiver who is informed only of the starting seeds (from the physical system input) can decrypt the ciphertext to obtain the plaintext since he also possesses F 1 , and F 2 as well as the relevant table of primes like the sender. As long as F 1 and F 2 are kept secret, no eavesdropper can decrypt the ciphertext.
  • the cipher can even be timed accordingly so that the functions F 1 and F 2 change according to time changes or context changes. In any event, math theory about primes guarantees that the string of random digits thus generated are absolutely chaotic. No fixed inter-relationship can be derived from among themselves.

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Cited By (41)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
WO2004001595A1 (fr) * 2002-06-21 2003-12-31 Atmel Corporation Test de nombres premiers probables pour applications cryptographiques
US20060034456A1 (en) * 2002-02-01 2006-02-16 Secure Choice Llc Method and system for performing perfectly secure key exchange and authenticated messaging
US20080198832A1 (en) * 2007-02-15 2008-08-21 Harris Corporation Low Level Sequence as an Anti-Tamper MEchanism
US20080263119A1 (en) * 2007-04-19 2008-10-23 Harris Corporation Digital Generation of a Chaotic Numerical Sequence
US20080294956A1 (en) * 2007-05-22 2008-11-27 Harris Corporation Encryption Via Induced Unweighted Errors
US20080294710A1 (en) * 2007-05-22 2008-11-27 Harris Corporation Extending a Repetition Period of a Random Sequence
US20080307024A1 (en) * 2007-06-07 2008-12-11 Harris Corporation Mixed Radix Number Generator with Chosen Statistical Artifacts
US20080307022A1 (en) * 2007-06-07 2008-12-11 Harris Corporation Mixed Radix Conversion with a Priori Defined Statistical Artifacts
US20080304666A1 (en) * 2007-06-07 2008-12-11 Harris Corporation Spread Spectrum Communications System and Method Utilizing Chaotic Sequence
US20090034727A1 (en) * 2007-08-01 2009-02-05 Harris Corporation Chaotic Spread Spectrum Communications System Receiver
US20090044080A1 (en) * 2007-05-31 2009-02-12 Harris Corporation Closed Galois Field Combination
US20090110197A1 (en) * 2007-10-30 2009-04-30 Harris Corporation Cryptographic system configured for extending a repetition period of a random sequence
US20090196420A1 (en) * 2008-02-05 2009-08-06 Harris Corporation Cryptographic system incorporating a digitally generated chaotic numerical sequence
US20090245327A1 (en) * 2008-03-26 2009-10-01 Harris Corporation Selective noise cancellation of a spread spectrum signal
US20090279688A1 (en) * 2008-05-06 2009-11-12 Harris Corporation Closed galois field cryptographic system
US20090296860A1 (en) * 2008-06-02 2009-12-03 Harris Corporation Adaptive correlation
US20090300088A1 (en) * 2008-05-29 2009-12-03 Harris Corporation Sine/cosine generator
WO2009146283A1 (fr) * 2008-05-29 2009-12-03 Harris Corporation Génération numérique d'une séquence numérique chaotique
US20090310650A1 (en) * 2008-06-12 2009-12-17 Harris Corporation Featureless coherent chaotic amplitude modulation
US20100054228A1 (en) * 2008-08-29 2010-03-04 Harris Corporation Multi-tier ad-hoc network communications
US20100091700A1 (en) * 2008-10-09 2010-04-15 Harris Corporation Ad-hoc network acquisition using chaotic sequence spread waveform
US20100165828A1 (en) * 2008-12-29 2010-07-01 Harris Corporation Communications system employing chaotic spreading codes with static offsets
US20100166041A1 (en) * 2008-12-29 2010-07-01 Harris Corporation Communications system employing orthogonal chaotic spreading codes
US20100226497A1 (en) * 2009-03-03 2010-09-09 Harris Corporation Communications system employing orthogonal chaotic spreading codes
US20100310072A1 (en) * 2009-06-08 2010-12-09 Harris Corporation Symbol duration dithering for secured chaotic communications
US20100309957A1 (en) * 2009-06-08 2010-12-09 Harris Corporation Continuous time chaos dithering
US20100316090A1 (en) * 2009-06-10 2010-12-16 Harris Corporation Discrete time chaos dithering
US20110002460A1 (en) * 2009-07-01 2011-01-06 Harris Corporation High-speed cryptographic system using chaotic sequences
US20110004792A1 (en) * 2009-07-01 2011-01-06 Harris Corporation Bit error rate reduction in chaotic communications
US20110002364A1 (en) * 2009-07-01 2011-01-06 Harris Corporation Anti-jam communications having selectively variable peak-to-average power ratio including a chaotic constant amplitude zero autocorrelation waveform
US20110019817A1 (en) * 2009-07-22 2011-01-27 Harris Corporation Permission-based tdma chaotic communication systems
US8320557B2 (en) 2008-05-08 2012-11-27 Harris Corporation Cryptographic system including a mixed radix number generator with chosen statistical artifacts
US8345725B2 (en) 2010-03-11 2013-01-01 Harris Corporation Hidden Markov Model detection for spread spectrum waveforms
US8363830B2 (en) 2008-02-07 2013-01-29 Harris Corporation Cryptographic system configured to perform a mixed radix conversion with a priori defined statistical artifacts
US8363700B2 (en) 2009-07-01 2013-01-29 Harris Corporation Rake receiver for spread spectrum chaotic communications systems
US8369377B2 (en) 2009-07-22 2013-02-05 Harris Corporation Adaptive link communications using adaptive chaotic spread waveform
US8385385B2 (en) 2009-07-01 2013-02-26 Harris Corporation Permission-based secure multiple access communication systems
US20130051552A1 (en) * 2010-01-20 2013-02-28 Héléna Handschuh Device and method for obtaining a cryptographic key
US8406352B2 (en) 2009-07-01 2013-03-26 Harris Corporation Symbol estimation for chaotic spread spectrum signal
US8428104B2 (en) 2009-07-01 2013-04-23 Harris Corporation Permission-based multiple access communications systems
US20140344319A1 (en) * 2013-05-14 2014-11-20 Daljit Singh Jandu Algorithm for primality testing based on infinite, symmetric, convergent, continuous, convolution ring group

Families Citing this family (11)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
TWI244610B (en) * 2001-04-17 2005-12-01 Matsushita Electric Ind Co Ltd Information security device, prime number generation device, and prime number generation method
US8229108B2 (en) * 2003-08-15 2012-07-24 Broadcom Corporation Pseudo-random number generation based on periodic sampling of one or more linear feedback shift registers
US7562052B2 (en) * 2004-06-07 2009-07-14 Tony Dezonno Secure customer communication method and system
CN101243389A (zh) * 2005-08-19 2008-08-13 Nxp股份有限公司 用于rsa密钥产生的电路装置和方法
US7860918B1 (en) 2006-06-01 2010-12-28 Avaya Inc. Hierarchical fair scheduling algorithm in a distributed measurement system
FR2905776B1 (fr) * 2006-09-11 2009-01-30 Bigilimwatshi Guy Boyo Dispositif electroniques et/ou informatique de factorisation des nombres entiers.
DE102006045224A1 (de) * 2006-09-26 2008-01-24 Dl-Daten Gmbh Verfahren für eine hochgradige verbindungsorientierte Authentisierung und Überprüfung der Integrität der Nutzerdatenübertragung (SynD)
US7876690B1 (en) 2007-09-26 2011-01-25 Avaya Inc. Distributed measurement system configurator tool
US7552164B1 (en) 2008-04-24 2009-06-23 International Business Machines Corporation Accelerated prime sieving using architecture-optimized partial prime product table
FR2946207A1 (fr) * 2009-05-28 2010-12-03 Proton World Internat Nv Protection d'une generation de nombres premiers pour algorithme rsa
JP2011123356A (ja) * 2009-12-11 2011-06-23 Oki Semiconductor Co Ltd 素数生成装置、素数生成方法、及び素数生成プログラム

Family Cites Families (16)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
US744041A (en) * 1901-09-21 1903-11-17 Charles G Burke Telegraphic code.
NL279100A (fr) * 1961-05-30
DE2658065A1 (de) * 1976-12-22 1978-07-06 Ibm Deutschland Maschinelles chiffrieren und dechiffrieren
US4218582A (en) * 1977-10-06 1980-08-19 The Board Of Trustees Of The Leland Stanford Junior University Public key cryptographic apparatus and method
US4351982A (en) * 1980-12-15 1982-09-28 Racal-Milgo, Inc. RSA Public-key data encryption system having large random prime number generating microprocessor or the like
US4740890A (en) * 1983-12-22 1988-04-26 Software Concepts, Inc. Software protection system with trial period usage code and unlimited use unlocking code both recorded on program storage media
US4988987A (en) * 1985-12-30 1991-01-29 Supra Products, Inc. Keysafe system with timer/calendar features
US5058160A (en) * 1988-04-29 1991-10-15 Scientific-Atlanta, Inc. In-band controller
NZ240019A (en) * 1991-09-30 1996-04-26 Peter John Smith Public key encrypted communication with non-multiplicative cipher
IL108645A (en) * 1994-02-14 1997-09-30 Elementrix Technologies Ltd Protected communication method and system
US5787172A (en) * 1994-02-24 1998-07-28 The Merdan Group, Inc. Apparatus and method for establishing a cryptographic link between elements of a system
US5663896A (en) * 1994-09-22 1997-09-02 Intel Corporation Broadcast key distribution apparatus and method using Chinese Remainder
US5594797A (en) * 1995-02-22 1997-01-14 Nokia Mobile Phones Variable security level encryption
US5724428A (en) * 1995-11-01 1998-03-03 Rsa Data Security, Inc. Block encryption algorithm with data-dependent rotations
US5771291A (en) * 1995-12-11 1998-06-23 Newton; Farrell User identification and authentication system using ultra long identification keys and ultra large databases of identification keys for secure remote terminal access to a host computer
US6219421B1 (en) * 1997-10-24 2001-04-17 Shaul O. Backal Virtual matrix encryption (VME) and virtual key cryptographic method and apparatus

Cited By (74)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
US20060034456A1 (en) * 2002-02-01 2006-02-16 Secure Choice Llc Method and system for performing perfectly secure key exchange and authenticated messaging
WO2004001595A1 (fr) * 2002-06-21 2003-12-31 Atmel Corporation Test de nombres premiers probables pour applications cryptographiques
US6718536B2 (en) * 2002-06-21 2004-04-06 Atmel Corporation Computer-implemented method for fast generation and testing of probable prime numbers for cryptographic applications
KR100938030B1 (ko) * 2002-06-21 2010-01-21 아트멜 코포레이숀 암호화 프로그램을 위한 가능성 있는 솟수들을 테스트하기 위한 방법
US20080198832A1 (en) * 2007-02-15 2008-08-21 Harris Corporation Low Level Sequence as an Anti-Tamper MEchanism
US8312551B2 (en) 2007-02-15 2012-11-13 Harris Corporation Low level sequence as an anti-tamper Mechanism
US20080263119A1 (en) * 2007-04-19 2008-10-23 Harris Corporation Digital Generation of a Chaotic Numerical Sequence
WO2008130973A1 (fr) * 2007-04-19 2008-10-30 Harris Corporation Génération numérique d'une séquence numérique chaotique
US7937427B2 (en) 2007-04-19 2011-05-03 Harris Corporation Digital generation of a chaotic numerical sequence
US20080294710A1 (en) * 2007-05-22 2008-11-27 Harris Corporation Extending a Repetition Period of a Random Sequence
US8611530B2 (en) 2007-05-22 2013-12-17 Harris Corporation Encryption via induced unweighted errors
US20080294956A1 (en) * 2007-05-22 2008-11-27 Harris Corporation Encryption Via Induced Unweighted Errors
US7921145B2 (en) 2007-05-22 2011-04-05 Harris Corporation Extending a repetition period of a random sequence
US7995757B2 (en) 2007-05-31 2011-08-09 Harris Corporation Closed galois field combination
US20090044080A1 (en) * 2007-05-31 2009-02-12 Harris Corporation Closed Galois Field Combination
US20080307024A1 (en) * 2007-06-07 2008-12-11 Harris Corporation Mixed Radix Number Generator with Chosen Statistical Artifacts
US7970809B2 (en) 2007-06-07 2011-06-28 Harris Corporation Mixed radix conversion with a priori defined statistical artifacts
US7974413B2 (en) 2007-06-07 2011-07-05 Harris Corporation Spread spectrum communications system and method utilizing chaotic sequence
US7962540B2 (en) 2007-06-07 2011-06-14 Harris Corporation Mixed radix number generator with chosen statistical artifacts
US20080307022A1 (en) * 2007-06-07 2008-12-11 Harris Corporation Mixed Radix Conversion with a Priori Defined Statistical Artifacts
US20080304666A1 (en) * 2007-06-07 2008-12-11 Harris Corporation Spread Spectrum Communications System and Method Utilizing Chaotic Sequence
US8005221B2 (en) 2007-08-01 2011-08-23 Harris Corporation Chaotic spread spectrum communications system receiver
US20090034727A1 (en) * 2007-08-01 2009-02-05 Harris Corporation Chaotic Spread Spectrum Communications System Receiver
US20090110197A1 (en) * 2007-10-30 2009-04-30 Harris Corporation Cryptographic system configured for extending a repetition period of a random sequence
US7995749B2 (en) 2007-10-30 2011-08-09 Harris Corporation Cryptographic system configured for extending a repetition period of a random sequence
US8180055B2 (en) 2008-02-05 2012-05-15 Harris Corporation Cryptographic system incorporating a digitally generated chaotic numerical sequence
US20090196420A1 (en) * 2008-02-05 2009-08-06 Harris Corporation Cryptographic system incorporating a digitally generated chaotic numerical sequence
US8363830B2 (en) 2008-02-07 2013-01-29 Harris Corporation Cryptographic system configured to perform a mixed radix conversion with a priori defined statistical artifacts
US20090245327A1 (en) * 2008-03-26 2009-10-01 Harris Corporation Selective noise cancellation of a spread spectrum signal
US8040937B2 (en) 2008-03-26 2011-10-18 Harris Corporation Selective noise cancellation of a spread spectrum signal
US20090279688A1 (en) * 2008-05-06 2009-11-12 Harris Corporation Closed galois field cryptographic system
US8139764B2 (en) 2008-05-06 2012-03-20 Harris Corporation Closed galois field cryptographic system
US8320557B2 (en) 2008-05-08 2012-11-27 Harris Corporation Cryptographic system including a mixed radix number generator with chosen statistical artifacts
WO2009146283A1 (fr) * 2008-05-29 2009-12-03 Harris Corporation Génération numérique d'une séquence numérique chaotique
KR101226781B1 (ko) 2008-05-29 2013-01-25 해리스 코포레이션 카오스 수열의 디지털 발생
US20090327387A1 (en) * 2008-05-29 2009-12-31 Harris Corporation Digital generation of an accelerated or decelerated chaotic numerical sequence
US8200728B2 (en) 2008-05-29 2012-06-12 Harris Corporation Sine/cosine generator
US8145692B2 (en) 2008-05-29 2012-03-27 Harris Corporation Digital generation of an accelerated or decelerated chaotic numerical sequence
US20090300088A1 (en) * 2008-05-29 2009-12-03 Harris Corporation Sine/cosine generator
US20090296860A1 (en) * 2008-06-02 2009-12-03 Harris Corporation Adaptive correlation
US8064552B2 (en) 2008-06-02 2011-11-22 Harris Corporation Adaptive correlation
US8068571B2 (en) 2008-06-12 2011-11-29 Harris Corporation Featureless coherent chaotic amplitude modulation
US20090310650A1 (en) * 2008-06-12 2009-12-17 Harris Corporation Featureless coherent chaotic amplitude modulation
US8325702B2 (en) 2008-08-29 2012-12-04 Harris Corporation Multi-tier ad-hoc network in which at least two types of non-interfering waveforms are communicated during a timeslot
US20100054228A1 (en) * 2008-08-29 2010-03-04 Harris Corporation Multi-tier ad-hoc network communications
US20100091700A1 (en) * 2008-10-09 2010-04-15 Harris Corporation Ad-hoc network acquisition using chaotic sequence spread waveform
US8165065B2 (en) 2008-10-09 2012-04-24 Harris Corporation Ad-hoc network acquisition using chaotic sequence spread waveform
US20100165828A1 (en) * 2008-12-29 2010-07-01 Harris Corporation Communications system employing chaotic spreading codes with static offsets
US20100166041A1 (en) * 2008-12-29 2010-07-01 Harris Corporation Communications system employing orthogonal chaotic spreading codes
US8406276B2 (en) 2008-12-29 2013-03-26 Harris Corporation Communications system employing orthogonal chaotic spreading codes
US8351484B2 (en) 2008-12-29 2013-01-08 Harris Corporation Communications system employing chaotic spreading codes with static offsets
US20100226497A1 (en) * 2009-03-03 2010-09-09 Harris Corporation Communications system employing orthogonal chaotic spreading codes
US8457077B2 (en) 2009-03-03 2013-06-04 Harris Corporation Communications system employing orthogonal chaotic spreading codes
US20100310072A1 (en) * 2009-06-08 2010-12-09 Harris Corporation Symbol duration dithering for secured chaotic communications
US20100309957A1 (en) * 2009-06-08 2010-12-09 Harris Corporation Continuous time chaos dithering
US8428102B2 (en) 2009-06-08 2013-04-23 Harris Corporation Continuous time chaos dithering
US20100316090A1 (en) * 2009-06-10 2010-12-16 Harris Corporation Discrete time chaos dithering
US8428103B2 (en) 2009-06-10 2013-04-23 Harris Corporation Discrete time chaos dithering
US8340295B2 (en) 2009-07-01 2012-12-25 Harris Corporation High-speed cryptographic system using chaotic sequences
US8406352B2 (en) 2009-07-01 2013-03-26 Harris Corporation Symbol estimation for chaotic spread spectrum signal
US20110002364A1 (en) * 2009-07-01 2011-01-06 Harris Corporation Anti-jam communications having selectively variable peak-to-average power ratio including a chaotic constant amplitude zero autocorrelation waveform
US8369376B2 (en) 2009-07-01 2013-02-05 Harris Corporation Bit error rate reduction in chaotic communications
US8379689B2 (en) 2009-07-01 2013-02-19 Harris Corporation Anti-jam communications having selectively variable peak-to-average power ratio including a chaotic constant amplitude zero autocorrelation waveform
US8385385B2 (en) 2009-07-01 2013-02-26 Harris Corporation Permission-based secure multiple access communication systems
US20110004792A1 (en) * 2009-07-01 2011-01-06 Harris Corporation Bit error rate reduction in chaotic communications
US8363700B2 (en) 2009-07-01 2013-01-29 Harris Corporation Rake receiver for spread spectrum chaotic communications systems
US8428104B2 (en) 2009-07-01 2013-04-23 Harris Corporation Permission-based multiple access communications systems
US20110002460A1 (en) * 2009-07-01 2011-01-06 Harris Corporation High-speed cryptographic system using chaotic sequences
US20110019817A1 (en) * 2009-07-22 2011-01-27 Harris Corporation Permission-based tdma chaotic communication systems
US8369377B2 (en) 2009-07-22 2013-02-05 Harris Corporation Adaptive link communications using adaptive chaotic spread waveform
US8848909B2 (en) 2009-07-22 2014-09-30 Harris Corporation Permission-based TDMA chaotic communication systems
US20130051552A1 (en) * 2010-01-20 2013-02-28 Héléna Handschuh Device and method for obtaining a cryptographic key
US8345725B2 (en) 2010-03-11 2013-01-01 Harris Corporation Hidden Markov Model detection for spread spectrum waveforms
US20140344319A1 (en) * 2013-05-14 2014-11-20 Daljit Singh Jandu Algorithm for primality testing based on infinite, symmetric, convergent, continuous, convolution ring group

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