Abstract
AbstractWe present a large class of new Zémor-Tillich type hash functions whose target space is the finite group GL2(𝔽pn) for any prime p and power n. To do so, we use a novel group-theoretic approach that uses Tits’ “Ping-Pong Lemma” to outline conditions under which a set of matrices in PGL2(𝔽p((x))) generates a free group. The hash functions we form are secure against known attacks, and simultaneously preserve many of the desired features of the Zémor-Tillich hash function. In particular, our hash functions retain the mall modifications property.
Highlights
Hash functions are an essential part of many cryptographic schemes, principally as tools of message authentication and modification detection
[2] Gilles Zémor introduced the idea of building hash functions from Cayley graphs of large girth
We show that educated choices of generators produced using the Free Generators Theorem give hash functions that both are resistant to the previous attacks on the Zémor-Tillich hash function and
Summary
Hash functions are an essential part of many cryptographic schemes, principally as tools of message authentication and modification detection. Notice that for p > 2, taking c = 0, a = 1, and b, a, and bto be −1, 0, and 1 in some order satisfies the (Simplified) Free Generators Theorem for any choice of d ≥ 0 Appendix A serves as a mathematical background and is auxiliary
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