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

翻译：Charles、Goren和Lauter在 [CGL] 中根据查找图中路径的难度,从扩展图中提出了加密的散列函数。在本文中,我们根据Markoff三重模型模型图中查找路径的难度,提出一个新的散列函数候选人。这些图表在数量理论和其他各个领域中已经进行了广泛的研究,但在图表中找到路径仍然困难。我们根据Markoff 图表的结构讨论了找到点之间的路径的难度。我们调查了几种可能的攻击途径,估计了它们的运行时间比O(p)还要大。特别是,我们分析了最近在[BGS1]中找到的证据,即这些图表是连接起来的,并讨论了这一证据如何为寻找路径提供算法。