Index calculus for abelian varieties of small dimension and the elliptic curve discrete logarithm problem

Abstract

We propose an index calculus algorithm for the discrete logarithm problem on general abelian varieties of small dimension. The main difference with the previous approaches is that we do not make use of any embedding into the Jacobian of a well-suited curve. We apply this algorithm to the Weil restriction of elliptic curves and hyperelliptic curves over small degree extension fields. In particular, our attack can solve an elliptic curve discrete logarithm problem defined over F q 3 in heuristic asymptotic running time O ̃ ( q 4 / 3 ) ; and an elliptic problem over F q 4 or a genus 2 problem over F q 2 in heuristic asymptotic running time O ̃ ( q 3 / 2 ) .