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

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