New applications of elliptic curves and function fields in cryptography

Abstract

Public key cryptography based on elliptic curves over finite fields was proposed by Miller and Koblitz in 1985. Elliptic curves over finite fields have been used to implement the Di:ffie-Hellm.an key passing scheme and the ElGamal, Schnorr and NIST signature schemes. Elliptic curves have also been used over the ring Zn to implement an RSA type scheme. In the first part of this thesis however, we propose using elliptic curves over the ring ~n in a new way. In this system the information is carried in the exponent space and not in the group itself. Also security depends on the difficulty of factoring a 150 digit number in order to trapdoor the discrete logarithm problem.

The continued fraction expansion and infrastructure for quadratic congruence function fields of odd characteristic have been well studied. Recently, these ideas have even been used to produce cryptosystems. Much less is known concerning the continued fraction expansion and infrastructure for quadratic function fields of even characteristic. In the second part of this thesis we will explore these ideas, and show that the situation is very similar to the odd characteristic case. This exploration will result in a method for computing the regulator for quadratic function fields of characteristic 2. We will also be able to show that cryptosystems proposed for the infrastructure of function fields of odd characteristic can be implemented in even characteristic and give a possible attack. Most importantly we will be able to show that the elliptic curve discrete logarithm problem is equivalent to a discrete logarithm problem in the infrastructure of certain quadratic function fields. This is a modification of a result by Stein for fields of odd characteristic.