Applications of elliptic curves in public key cryptography

The most popular public key cryptosystems are based on the problem of factorization of large integers and discrete logarithm problem in finite groups, in particular in the multiplicative group of finite field and the group of points on elliptic curve over finite field. Elliptic curves are of special interest since they at present alow much shorter keys, for the same level of security, compared with cryptosystems based on factorization or discrete logarithm problem in finite fields. In this course we will briefly mentioned basic properties of elliptic curves over the rationals, and then concentrate on important algorithms for elliptic curves over finite fields. We will discuss efficient implementation of point addition and multiplication (in different coordinates). Algorithms for point counting and elliptic curve discrete logarithm problem will be described. 1 Factorization and primality testing and proving are very important topics for security of public key cryptosystems. Namely, the starting point in the construction of almost all public key cryptosystems is the choice of one or more large (secret or public) prime numbers. We will describe algorithms for factorization and primality proving which use elliptic curves.