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

Pierrick Gaudry

Paper 2004/073

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

Pierrick Gaudry

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.

Metadata

Available format(s): PS
Category: Public-key cryptography
Publication info: Published elsewhere. Unknown where it was published
Keywords: elliptic curves Weil descent discrete logarithm problem
Contact author(s): gaudry @ lix polytechnique fr
History: 2004-03-04: received
Short URL: https://ia.cr/2004/073
License: CC BY

BibTeX

@misc{cryptoeprint:2004/073,
      author = {Pierrick Gaudry},
      title = {Index calculus for abelian varieties and the elliptic curve discrete logarithm problem},
      howpublished = {Cryptology {ePrint} Archive, Paper 2004/073},
      year = {2004},
      url = {https://eprint.iacr.org/2004/073}
}