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Research on Several Problems of Elliptic Curves Related to Cryptography

Author: ZhaoLong
Tutor: HanWenBao
School: PLA Information Engineering University
Course: Cryptography
Keywords: Elliptic curve cryptosystem Elliptic curve decomposition algorithm Elliptic curve discrete logarithm problem Elliptic curve linear congruential sequence Index and SIMD instructions Graphics processing unit
CLC: TN918.1
Type: PhD thesis
Year: 2011
Downloads: 66
Quote: 0
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Abstract


Elliptic curve as a mathematical study has 100 years of history, with the development of cryptography, especially the emergence of public-key cryptography, elliptic curve began to move from theory to practical applications, including elliptic curve cryptography (ECC) decomposition algorithm and elliptic curve (ECM) is the most far-reaching impact, the most widely used. ECC bit security strength is the highest public-key cryptosystem, has short key length, encryption speed, mobile devices and high-speed applications environment has gradually become the standard public key algorithm. ECM is the decomposition of the best medium-sized prime factor algorithm has important applications in the remaining factorization of RSA Number Field Sieve. ECC and ECM algorithm has been password scholars are concerned about the hot issues, on the one hand, the the research optimization algorithm in hardware and software platforms help to improve the efficiency of ECC and RSA attack, reduce the attack cost, and enhance the security of ECC and RSA sexual understanding and awareness; On the other hand, in the implementation of these algorithms optimized it also contains many interesting theoretical questions, such as the distribution of isomorphism classes of rational points on elliptic curve conducive to choose good elliptic curve research elliptic curve The pseudo-random sequence can be used for ECC in a pseudo-random number generation, and a new perspective to find the potential weaknesses of ECC. This paper studies the ECC and ECM some theoretical and engineering issues, including the isomorphism classes of elliptic curves with rational points distribution, elliptic curve (EC-LCG) linear congruential sequence distribution and attack, SIMD instruction attacks in ECC in applications on the GPU, ECM efficiency analysis and optimization to achieve the main achievements are as follows: 1. constructed of representatives of the two groups of elliptic curve isomorphism classes using the index and gives minimum coefficient of isomorphism classes of elliptic curve theoretical upper bound of the value from the point of view of the theory of probability, the minimum value is actually bounded analysis; calculated when the elliptic curve in a fixed coefficient, a coefficient traveled all over? q or? quadratic residue of q class the Rational Points mean and variance calculated when the two factors are equal, and at the same time running all over? rational points of the mean quadratic residue class of q? q; divisor polynomial given rational points primes charge necessary condition divisibility of rational points decision problem into Equations decision problem, and then study a coefficient fixed, another coefficient traveled all over? q or? subgroup of q rational points 2 and 3 divisible probability. 2 system investigated prime field EC-LCG sequence continuous period of bits distributed nature, prove the For the general form of the primes p-, EC-LCG sequence low half of the bit segment has a good distribution properties, when the prime number p-= 2n? c, where c is a small integer, EC-LCG sequence bit segment in half portion and the intermediate portion of the bit segment having a good distribution properties; calculated EC-LCG sequence stars bit sequence in binary fields consistent distribution measure and the order k related measure, the balance of the multi-dimensional sub-bit sequence and linear complexity, and is based on the sub-bit sequence constructed two types of binary pseudo-random sequence clusters; given several EC-LCG sequence of attack methods, in the case of the unknown parameters, if leaked enough the x coordinate, then the sequence may be partially restored or restore all. Three Montgomery modular multiplication algorithm was designed based on the SIMD instruction in the choice of the prime field ECC attacks on implementations and experimental results are given on the basis of the the optimal modular multiplication algorithm; in the non-bitsliced, and bitsliced ??two data structure respectively binary polynomial multiplication based on the SIMD instruction, and based on the bit exchange thought gives rapid conversion algorithm of the two data structures, and finally the binary domain is given to a combination of the two data structures ECC attack implementation and experimental results; calculated using probability theory continuous solving the multiple elliptic curve discrete logarithm of the mean and variance of the complexity of solving the discrete logarithm is more, the overall complexity is relatively more stable, the average per complexity of the discrete logarithm is lower. Established algorithm performance assessment model for GPU platform, through the analysis of the main factors affecting the performance of the algorithm, comprehensive instruction throughput, instruction latencies, memory bandwidth and memory access latency, given a specific algorithm consumption when the assessment formula, choose the best algorithm for the GPU platform implementations provide a theoretical basis; designed three Montgomery modular multiplication algorithm based on the floating point and integer arithmetic on the GPU, and algorithm performance assessment model established three algorithms efficiency analysis and performance comparison, the last in the choice the optimal modular multiplication algorithm based on the ECM implementations and experimental results are given.

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