Journal of Cryptology

, Volume 22, Issue 4, pp 505–529 | Cite as

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.


Hyperelliptic curve cryptography Discrete logarithm problem Isogeny Genus 3 


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© International Association for Cryptologic Research 2009

Authors and Affiliations

  1. 1.Laboratoire d’Informatique de l’École polytechnique (LIX)INRIA Saclay–Île-de-FrancePalaiseau CedexFrance

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