**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.

**Category / Keywords: **public-key cryptography / elliptic curves, Weil descent, discrete logarithm problem

**Date: **received 4 Mar 2004

**Contact author: **gaudry at lix polytechnique fr

**Available format(s): **Postscript (PS) | Compressed Postscript (PS.GZ) | BibTeX Citation

**Version: **20040304:201415 (All versions of this report)

**Short URL: **ia.cr/2004/073

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