Isogenies and the discrete logarithm problem in jacobians of genus 3 hyperelliptic Curves

Abstract

We describe the use of explicit isogenies to translate instances of the Discrete Logarithm Problem (DLP) from Jacobians of hyperelliptic genus 3 curves to Jacobians of non-hyperelliptic genus 3 curves, where they are vulnerable to faster index calculus attacks. We provide explicit formulae for isogenies with kernel isomorphic to (ℤ/2ℤ) 3 (over an algebraic closure of the base field) for any hyperelliptic genus 3 curve over a field of characteristic not 2 or 3. These isogenies are rational for a positive fraction of all hyperelliptic genus 3 curves defined over a finite field of characteristic p>3. Subject to reasonable assumptions, our constructions give an explicit and efficient reduction of instances of the DLP from hyperelliptic to non-hyperelliptic Jacobians for around 18.57% of all hyperelliptic genus 3 curves over a given finite field. We conclude with a discussion on extending these ideas to isogenies with more general kernels. © 2009 International Association for Cryptologic Research.