Abstract. We propose constructing provable collision resistant hash functions from expander graphs. As examples, we investigate two specific families of optimal expander graphs for provable hash function constructions: the families of Ramanujan graphs constructed by Lubotzky-Phillips-Sarnak and Pizer respectively. When the hash function is constructed from one of Pizer’s Ramanujan graphs, (the set of supersingular elliptic curves over Fp2 with ℓ-isogenies, ℓ a prime different from p), then collision resistance follows from hardness of computing isogenies between supersingular elliptic curves. We estimate the cost per bit to compute these hash functions, and we implement our hash function for several members of the LPS graph family and give actual timings. 1

Abstract. In this paper, we explore potential mathematical principles and structures that can provide the foundation for cryptographic hash functions, and also present a simple and efficiently computable hash function based on a non-associative operation with polynomials over a finite field of characteristic 2. 1

We propose a new infinite family of cryptographic hash functions, Edon–R, based on a recently defined candidate one-way function. Edon–R is a class of hash functions with variable output lengths. It is defined using quasigroups and quasigroup string transformations.