رمز نگاري ايمن خم هاي بيضوي

چكيده انگليسي :

Safe Elliptic Curve Cryptography Simin Esteki sesteki mathdept iut ac ir January 2019 Master of Science Thesis in Farsi Department of Mathematical Sciences Isfahan University of Technology Isfahan 84156 8311 IranSupervisor Dr Reza Rezaeian Farashahi farashahi iut ac irAdvisor Dr Omran Ahmadi http math ipm ac ir emran 2000 MSC 05C25 20B25 05C60Keywords Elliptic Curves Weierstrass form Twisted Edwards form Secure Scalar Multiplication Constant time execution Abstract This M Sc thesis is based on the following papers Selecting elliptic curves for cryp J W B C C P L M N tography an efficiency and security analysis Journal of Cryptographic Engineering vol 6 pp 259 286 Nov 2016Selecting a suitable model of an elliptic curve to have an elliptic curve cryptosystem that computes scaler multiplica tion as fast as possible depends on different factors In other words it is completely necessary to know which modelshould be selected because it directly influences on security and performance of the elliptic curves cryptosystem Inthis thesis different models of elliptic curves have been discussed Moreover I have considered and analyzed dif ferent aspects to find models of elliptic curve that satisfy both security and efficiency My suggestion based on theexperiment is the prime order elliptic curves in Weierstrass model Montgomery model and twisted Edwards modelthat have their practical advantages In addition these curves are suitable for current implementations that are sup ported by National Institute of Standards and Technology NIST over prime finite fields It could be confederatedinto existing implementations by changing the field arithmetic and the curve constant in some cases My thesis also have examined different prime finite fields to find the most efficient modular arithmetic For exam ple for modular arithmetic operations using Montgomery friendly primes and Pseudo Mersene primes enhance theperformance of the elliptic curve cryptosystem In addition to have a faster computations we require to select primesthat are congruent to 3 modulo 4 These primes are used to improve the efficiency of computation of elliptic curvecryptosystems focusing on short Weierstrass model with the curve parameter a 3 and twisted Edwards modelwith the curve parameter a 1 Algorithms and formulas that are proposed for calculating scalar Multiplication on elliptic curves are all optimum constant time and have no exceptions and also being used to increase efficiency of elliptic curve cryptosystem im plementations Results of implementing new Weierstrass curves show that variable base scalar multiplication usingthese curves at 128 bit security level is approximately 1 4 times faster than latest records of NIST curves Besides for cases that some security bits can be ignored a set of Twisted Edwards curves with efficient arithmetic has beenprovided These curves are faster up to 1 42 1 26 and 1 24 times than short Weierstrass curves for security levels at128 bit 192 bit and 256 bit respectively My thesis helps readers to understand how they can choose the best curves that are proposed to them The EllipticCurve Discrete Logarithm Problem ECDLP has full security when the Weierstrass curves is defined over primefinite fields that have fixed bitlength and also have good practical performance But nowadays Quantum computersare developing and ECDLP would not have full security in near future

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رمز نگاري ايمن خم هاي بيضوي

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