A Cryptographic Hash Function from Markoff Triples

TL;DR

This paper introduces a new cryptographic hash function candidate based on the complex structure of Markoff triples modulo p, leveraging their hardness for path-finding problems in number theory.

Contribution

It proposes a novel hash function construction using Markoff graphs and analyzes its security based on the difficulty of path-finding in these graphs.

Findings

Path-finding in Markoff graphs is computationally hard

The proposed hash function leverages number-theoretic complexity

Analysis suggests the attack complexity exceeds O(p)

Abstract

Cryptographic hash functions from expander graphs were proposed by Charles, Goren, and Lauter in [CGL] based on the hardness of finding paths in the graph. In this paper, we propose a new candidate for a hash function based on the hardness of finding paths in the graph of Markoff triples modulo p. These graphs have been studied extensively in number theory and various other fields, and yet finding paths in the graphs remains difficult. We discuss the hardness of finding paths between points, based on the structure of the Markoff graphs. We investigate several possible avenues for attack and estimate their running time to be greater than O(p). In particular, we analyze a recent groundbreaking proof in [BGS1] that such graphs are connected and discuss how this proof gives an algorithm for finding paths