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

Benjamin Smith 1 
1 TANC - Algorithmic number theory for cryptology
LIX - Laboratoire d'informatique de l'École polytechnique [Palaiseau], Inria Saclay - Ile de France
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 $(\ZZ/2\ZZ)^3$ (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. A condensed version of this work appeared in the proceedings of the EUROCRYPT 2008 conference.
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Submitted on : Friday, November 19, 2010 - 3:22:47 PM
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Benjamin Smith. Isogenies and the Discrete Logarithm Problem in Jacobians of Genus 3 Hyperelliptic Curves. Journal of Cryptology, 2009, 22 (4), pp.505-529. ⟨10.1007/s00145-009-9038-1⟩. ⟨inria-00537851⟩



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