On the discrete logarithm problem in class groups of curves

Abstract

We study the discrete logarithm problem in degree 0 class groups of curves over finite fields, with particular emphasis on curves of small genus. We prove that for every fixed g ≥ 2, the discrete logarithm problem in degree 0 class groups of curves of genus g can be solved in an expected time of O(q 2-2/g ), where F q is the ground field. This result generalizes a corresponding result for hyperelliptic curves given in imaginary quadratic representation with cyclic degree 0 class group, and just as this previous result, it is obtained via an index calculus algorithm with double large prime variation. Generalizing this result, we prove that for fixed g 0 > 2 the discrete logarithm problem in class groups of all curves C/F q of genus g ≥ g 0 can be solved in an expected time of O((q g ) 2/g 0 (1-1/g 0 ) ) and in an expected time of O( #Cl 0 (C) 2/g 0 (1- 1/g0) ). As a complementary result we prove that for any fixed n ∈ ℕ with n > 2 the discrete logarithm problem in the groups of rational points of elliptic curves over finite fields F q n, q a prime power, can be solved in an expected time of O(q 2-2/n ). Furthermore, we give an algorithm for the efficient construction of a uniformly randomly distributed effective divisor of a specific degree, given the curve and its L-polynomial.