Estimating the Size of the Image of Deterministic Hash Functions to Elliptic Curves

Abstract

Let E be a non-supersingular elliptic curve over a finite field \(\mathbb{F}_{\!q}\). At CRYPTO 2009, Icart introduced a deterministic function \(\mathbb{F}_{\!q}\to E(\mathbb{F}_{\!q})\) which can be computed efficiently, and allowed him and Coron to define well-behaved hash functions with values in \(E(\mathbb{F}_{\!q})\). Some properties of this function rely on a conjecture which was left as an open problem in Icart’s paper. We prove this conjecture below as well as analogues for other hash functions.KeywordsElliptic CurvesFunction FieldsHash Functions