Isogenies and the Discrete Logarithm Problem in Jacobians of Genus 3 Hyperelliptic Curves,

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. This paper was solicted by the Editors-in-Chief as one of the best papers from EUROCRYPT 2008, based on the recommendation of the program committee. A condensed version of this work appeared in the proceedings of the EUROCRYPT 2008 conference.