Hash Functions Based on Ramanujan Graphs
Cayley hash functions are a family of cryptographic hash functions constructed from Cayley graphs, with appealing properties such as a natural parallelism and a security reduction to a clean, well-defined mathematical problem. As this problem involves non-Abelian groups, it is a priori resistant to quantum period finding algorithms and Cayley hash functions may therefore be a good foundation for post-quantum cryptography. Four particular parameter sets for Cayley hash functions have been proposed in the past, and so far dedicated preimage algorithms have been found for all of them. These algorithms do however not seem to extend to generic parameters, and as a result it is still an open problem to determine the security of Cayley hash functions in general. In this chapter, we introduce how to design hash functions based on Ramanujan graphs, which can be considered as an optimal expander graphs in a sense of qualities of transmission network schemes. We introduce a polynomial time preimage attack against Cayley hash functions based on two explicit Ramanujan graphs. We suggest some possible ways to construct the Cayley hash functions that may not be affected by this type of attacks as open problems, which can contribute to a better understanding of the hard problems underlying the security of Cayley hash functions.
KeywordsExpander graphs Ramanujan graphs LPS Ramanujan graphs Cubic Ramanujan graphs Cayley graphs Cayley hash functions Lifting attacks
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