Isogenies and the Discrete Logarithm Problem in Jacobians of Genus 3 Hyperelliptic Curves [article]

Benjamin Smith
2009 arXiv   pre-print
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 (/2)^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
more » ... sitive 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 finite field. We conclude with a discussion on extending these ideas to isogenies with more general kernels. A condensed version of this work appeared in the proceedings of the EUROCRYPT 2008 conference.
arXiv:0806.2995v2 fatcat:xpkzor6lcbdadekeo4fvwgwil4