Encoding points on hyperelliptic curves over finite fields in deterministic polynomial time

Authors:
Jean-Gabriel Kammerer;Reynald Lercier;Guénaël Renault
Affiliations:
DGA MI, La Roche Marguerite, Bruz Cedex, France and Institut de recherche mathématique de Rennes, Université de Rennes 1, Rennes Cedex, France;DGA MI, La Roche Marguerite, Bruz Cedex, France and Institut de recherche mathématique de Rennes, Université de Rennes 1, Rennes Cedex, France;LIP6, Université Pierre et Marie Curie, INRIA, LIP6, Paris Cedex 05, France
Venue:
Pairing'10 Proceedings of the 4th international conference on Pairing-based cryptography
Year:
2010

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Abstract

We provide new hash functions into (hyper)elliptic curves over finite fields. These functions aim at instantiating in a secure manner cryptographic protocols where we need to map strings into points on algebraic curves, typically user identities into public keys in pairing-based IBE schemes. Contrasting with recent Icart's encoding, we start from "easy to solve by radicals" polynomials in order to obtain models of curves which in turn can be deterministically "algebraically parameterized". As a result of this strategy, we obtain a low degree encoding map for Hessian elliptic curves, and for the first time, hashing functions for genus 2 curves. More generally, we present for any genus (more narrowed) families of hyperelliptic curves with this property. The image of these encodings is large enough to be "weak" encodings in the sense of Brier et al. As such they can be easily turned into admissible cryptographic hash functions.