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

Pierrick Gaudry 1, 2, 3
2 TANC - Algorithmic number theory for cryptology
Inria Saclay - Ile de France, LIX - Laboratoire d'informatique de l'École polytechnique [Palaiseau]
3 CACAO - Curves, Algebra, Computer Arithmetic, and so On
INRIA Lorraine, LORIA - Laboratoire Lorrain de Recherche en Informatique et ses Applications
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 GF(q^3) in heuristic asymptotic running time O~(q^(4/3)); and an elliptic problem over GF(q^4) or a genus 2 problem over GF(q^2) in heuristic asymptotic running time O~(q^(3/2)).
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Pierrick Gaudry. Index calculus for abelian varieties of small dimension and the elliptic curve discrete logarithm problem. Journal of Symbolic Computation, Elsevier, 2009, 44 (12), pp.1690-1702. ⟨10.1016/j.jsc.2008.08.005⟩. ⟨inria-00337631⟩

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