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Isogenies and the Discrete Logarithm Problem on Jacobians of Genus 3 Hyperelliptic Curves
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| Abstract: | We describe the use of explicit isogenies to reduce Discrete Logarithm Problems (DLPs) on Jacobians of hyperelliptic genus~$3$ curves to Jacobians of non-hyperelliptic genus~$3$ curves, which are vulnerable to faster index calculus attacks. We provide algorithms which compute an isogeny with kernel isomorphic to $(\mathbb{Z}/2\mathbb{Z})^3$ for any hyperelliptic genus~$3$ curve. These algorithms provide a rational isogeny for a positive fraction of all hyperelliptic genus~$3$ curves defined over a finite field of characteristic $p > 3$. Subject to reasonable assumptions, our algorithms provide an explicit and efficient reduction from hyperelliptic DLPs to non-hyperelliptic DLPs for around $18.57\%$ of all hyperelliptic genus~$3$ curves over a given finite field. |
BibTeX
@misc{eprint-2007-13708,
title={Isogenies and the Discrete Logarithm Problem on Jacobians of Genus 3 Hyperelliptic Curves},
booktitle={IACR Eprint archive},
keywords={public-key cryptography / discrete logarithm problem, number theory},
url={http://eprint.iacr.org/2007/428},
note={ smith@lix.polytechnique.fr 13831 received 14 Nov 2007},
author={Benjamin Smith},
year=2007
}