Computing Riemann---Roch spaces in algebraic function fields and related topics
Journal of Symbolic Computation
An index calculus algorithm for plane curves of small degree
ANTS'06 Proceedings of the 7th international conference on Algorithmic Number Theory
Evolution of hyperelliptic curve cryptosystems
ICDCIT'10 Proceedings of the 6th international conference on Distributed Computing and Internet Technology
Correspondences on hyperelliptic curves and applications to the discrete logarithm
SIIS'11 Proceedings of the 2011 international conference on Security and Intelligent Information Systems
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We describe the use of explicit isogenies to translate instances of the Discrete Logarithm Problem 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.