چكيده انگليسي :
E cient pairing computation on elliptic curves Marjan Amini Azadani marjan amini@math iut ac ir 2015 Department of Mathematical Sciences Isfahan University of Technology Isfahan 84156 83111 Iran Supervisor Dr Reza Rezaeian Farashahi farashahi@cc iut ac ir Advisor Dr Omran Ahmadi oahmadid@ipm ir 2010 MSC 68P25 14G50 11T71 94A60 Keywords Pairings Miller functions Explicit formulas Edwards curves Abstract This thesis is an extension and generalization of the work s done by Christophe Arenea TanjaLange Michael Naehrig Christophe Ritzenthaler 10 Until the late 1970 s all cryptographic messagetransmission was by symmetric key This means that someone who has enough information to encryptmessages also has enough information to decipher messages As a result any two users of the systemwho want to communicate secretly must have exchanged keys in a safe way perhaps using a trustedcourier The arena for applying mathematics to cryptography expanded dramatically when Di e andHellman invented an entirely new type of cryptography called public key The heart of this conceptis the idea of using a one way function for encryption Some of the purposes for which public keycryptography has been applied are con dential message identi cation systems authentication non repudiation key establishment electronic cash mechanisms electronic voting schemes In 1985 Koblitz and Miller independently proposed using the group of points on an elliptic curvede ned over a nite eld in discrete log cryptosystems The primary advantage that elliptic curvesystems have over systems based on the multiplicative group of a nite eld and also over systemsbased on the intractability of integer factorization is the absence of a subexponential time algorithm such as those of index calculus type that could nd discrete logs in these groups Consequently onecan use the elliptic curve group that is smaller in size while maintaining the same level of security The result is potentially smaller key sizes bandwidth savings and faster implementations featureswhich are especially attractive for security applications where computational power and integrated
