چكيده انگليسي :
Isomorphism Classes of Elliptic Curves over Finite Fields Mehran Hosseini mehran hosseini@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 2010 MSC 14H52 12Y05 11T71 Keywords Elliptic Curves Isomorphism Classes of Elliptic Curves Arithmetic Geometry Cryp tography AbstractElliptic curves play important roles in mathematics as well as many other areas of science such ascomputer science and cryptography In cryptography public key cryptosystems using elliptic curvesare advantageous over those using finite fields for shorter key length The shorter key length increasesspeed and efficiency The security of these systems is based on the hardness of problems like discrete logarithm problem DLP and Diffie Hellman problem DHP for the group of points on elliptic curves over finite fieldsFq of prime power order q Using Pohlig Hellman attack the problems are shown not to be hard ifthe group of points of an elliptic curve over the finite field splits to subgroups of small prime power or equivalently the group order is multiple of small primes This motivated researchers to study thegroup structure of elliptic curves Curves over finite fields with order divisible by a large prime resist against Pohlig Hellman attack Hendrik Lenstra in his paper Factoring integers with elliptic curves showed that the number of curves over a prime finite field Fp that order is divisible by a prime is almost 1 O p E W Howe extended Lenstra s result to arbitrary integers instead of a prime and elliptic curves overextension fields Wouter Castryck and Hendrik Hubrechts generalized Howe s result and obtainedformulae for the distribution of number of pints on elliptic curves over extension fields modulo anarbitrary integer n Another factor cryptographers look for is the efficiency of scalar multiplication in an elliptic curve
