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Paper: Index calculus for abelian varieties and the elliptic curve discrete logarithm problem

Authors:
Pierrick Gaudry
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URL: http://eprint.iacr.org/2004/073
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Abstract: We propose an index calculus algorithm for the discrete logarithm problem on general abelian varieties. 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 all elliptic curve discrete logarithm problems defined over $GF(q^3)$ in time $O(q^{10/7})$, with a reasonably small constant; and an elliptic problem over $GF(q^4)$ or a genus 2 problem over $GF(p^2)$ in time $O(q^{14/9})$ with a larger constant.
BibTeX
@misc{eprint-2004-12046,
  title={Index calculus for abelian varieties and the elliptic curve discrete logarithm problem},
  booktitle={IACR Eprint archive},
  keywords={public-key cryptography / elliptic curves, Weil descent, discrete logarithm problem},
  url={http://eprint.iacr.org/2004/073},
  note={ gaudry@lix.polytechnique.fr 12481 received 4 Mar 2004},
  author={Pierrick Gaudry},
  year=2004
}