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 isoge...

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Bibliographic Details
Published inJournal of cryptology Vol. 22; no. 4; pp. 505 - 529
Main Author Smith, Benjamin
Format Journal Article
LanguageEnglish
Published New York Springer-Verlag 01.10.2009
Springer
Springer Nature B.V
Springer Verlag
Subjects
Applied sciences
Calculus
Coding and Information Theory
Combinatorics
Communications Engineering
Computational Mathematics and Numerical Analysis
Computer Science
Cryptography
Cryptography and Security
Exact sciences and technology
Fields (mathematics)
Information, signal and communications theory
Jacobians
Kernels
Mathematics
Networks
Number Theory
Probability Theory and Stochastic Processes
Signal and communications theory
Telecommunications and information theory
Genus 3
Discrete logarithm problem
Hyperelliptic curve cryptography
Isogeny
Finite field
Logarithmic function
Vulnerability
Jacobi matrix
Kernel method
Discrete function
Discrete logarithm
Security of data
Cryptography
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Summary: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.
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ISSN:0933-2790
1432-1378
DOI:10.1007/s00145-009-9038-1