**Isogenies and the Discrete Logarithm Problem on Jacobians of Genus 3 Hyperelliptic Curves**

*Benjamin Smith*

**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.

**Category / Keywords: **public-key cryptography / discrete logarithm problem, number theory

**Date: **received 14 Nov 2007

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

**Available format(s): **PDF | BibTeX Citation

**Version: **20071118:222826 (All versions of this report)

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