Improved elliptic curve hashing and point representation

Abstract

For a large class of functions $f:\mathbb {F}_q\rightarrow E(\mathbb {F}_q)$ to the group of points of an elliptic curve $E/\mathbb {F}_q$ (typically obtained from certain algebraic correspondences between E and $\mathbb {P}^1$), Farashahi et al. (Math Comput 82(281):491–512, 2013) established that the map $(u,v)\mapsto f(u)+f(v)$ is regular, in the sense that for a uniformly random choice of $(u,v)\in \mathbb {F}_q^2$, the elliptic curve point $f(u)+f(v)$ is close to uniformly distributed in $E(\mathbb {F}_q)$. This result has several applications in cryptography, mainly to the construction of elliptic curve-valued hash functions and to the “Elligator Squared” technique by Tibouchi (in: Christin and Safavi-Naini (eds) Financial cryptography. LNCS, vol 8437, pp 139–156. Springer, Heidelberg, 2014) for representating uniform points on elliptic curves as close to uniform bitstrings. In this paper, we improve upon Farashahi et al.’s character sum estimates in two ways: we show that regularity can also be obtained for a function of the form $(u,v)\mapsto f(u)+g(v)$ where g has a much smaller domain than $\mathbb {F}_q$, and we prove that the functions f considered by Farashahi et al. also satisfy requisite bounds when restricted to large intervals inside $\mathbb {F}_q$. These improved estimates can be used to obtain more efficient hash function constructions, as well as much shorter “Elligator Squared” bitstring representations.