Deterministic Encoding and Hashing to Odd Hyperelliptic Curves

In this paper we propose a very simple and efficient encoding function from Fq to points of a hyperelliptic curve over Fq of the form H : y = f(x) where f is an odd polynomial. Hyperelliptic curves of this type have been frequently considered in the literature to obtain Jacobians of good order and pairing-friendly curves. Our new encoding is nearly a bijection to the set of Fq-rational points on H. This makes it easy to construct well-behaved hash functions to the Jacobian J of H, as well as injective maps to J(Fq) which can be used to encode scalars for such applications as ElGamal encryption. The new encoding is already interesting in the genus 1 case, where it provides a well-behaved encoding to Joux’s supersingular elliptic curves.