A subexponential algorithm for discrete logarithms over the rational subgroup of the Jacobians of large genus hyperelliptic curves over finite fields

Abstract

There are well known subexponential algorithms for finding discrete logarithms over finite fields. However, the methods which underlie these algorithms do not appear to be easily adaptable for finding discrete logarithms in the groups associated with elliptic curves and the Jacobians of hyperelliptic curves. This has led to the development of cryptographic systems based on the discrete logarithm problem for such groups [12, 7, 8]. In this paper a Subexponential algorithm is presented for finding discrete logarithms in the group of rational points on the Jacobians of large genus hyperelliptic curves over finite fields. We give a heuristic argument that under certain assumptions, there exists a c e ℜ>0 such that for all sufficiently large g e Z>0, for all odd primes p with log p ≤ (2g + 1).98, the algorithm computes discrete logarithms in the group of rational points on the Jacobian of a genus g hyperelliptic curve over GF(p) within expected time: Lp2g+1[1/2, c] where c ≤ 2.181.