Discrete logarithms in finite fields and their cryptographic significance
Proc. of the EUROCRYPT 84 workshop on Advances in cryptology: theory and application of cryptographic techniques
Journal of Cryptology
Computing Riemann---Roch spaces in algebraic function fields and related topics
Journal of Symbolic Computation
Isogenies and the discrete logarithm problem in Jacobians of genus 3 hyperelliptic curves
EUROCRYPT'08 Proceedings of the theory and applications of cryptographic techniques 27th annual international conference on Advances in cryptology
An index calculus algorithm for plane curves of small degree
ANTS'06 Proceedings of the 7th international conference on Algorithmic Number Theory
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The discrete logarithm is an important crypto primitive for public key cryptography. The main source for suitable groups are divisor class groups of carefully chosen curves over finite fields. Because of index-calculus algorithms one has to avoid curves of genus ≥4 and non-hyperelliptic curves of genus 3. An important observation of Smith [17] is that for "many" hyperelliptic curves of genus 3 there is an explicit isogeny of their Jacobian variety to the Jacobian of a non-hyperelliptic curve. Hence divisor class groups of these hyperelliptic curves are mapped in polynomial time to divisor class groups of non-hyperelliptic curves. Behind his construction are results of Donagi, Recillas and Livné using classical algebraic geometry. In this paper we only use the theory of curves to study Hurwitz spaces with monodromy group S4 and to get correspondences for hyperelliptic curves. For hyperelliptic curves of genus 3 we find Smith's results now valid for ground fields with odd characteristic, and for fields with characteristic 2 one can apply the methods of this paper to get analogous results at least for curves with ordinary Jacobian.