A Cryptographic Hash Function from Markoff Triples
This addresses the need for secure cryptographic hash functions, but it appears incremental as it builds on prior work with expander graphs.
The paper tackles the problem of constructing a cryptographic hash function by proposing a new candidate based on the hardness of finding paths in Markoff triple graphs modulo p, with estimated attack running times greater than O(p).
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