SL2 homomorphic hash functions: Worst case to average case reduction and short collision search
This work addresses cryptographic security for hash functions, offering theoretical guarantees and practical improvements in collision search, though it is incremental as it builds on existing homomorphic proposals.
The paper tackles the security of homomorphic hash functions into SL(2,q) by providing a worst-case to average-case reduction, showing random homomorphisms are as secure as concrete ones under a number-theoretic hypothesis, and develops algorithms that find collisions of length O(log(q)) in time O(sqrt(q)), which are faster and produce shorter collisions than prior methods.
We study homomorphic hash functions into SL(2,q), the 2x2 matrices with determinant 1 over the field with $q$ elements. Modulo a well supported number theoretic hypothesis, which holds in particular for concrete homomorphisms proposed thus far, we provide a worst case to average case reduction for these hash functions: upto a logarithmic factor, a random homomorphism is as secure as _any_ concrete homomorphism. For a family of homomorphisms containing several concrete proposals in the literature, we prove that collisions of length O(log(q)) can be found in running time O(sqrt(q)). For general homomorphisms we offer an algorithm that, heuristically and according to experiments, in running time O(sqrt(q)) finds collisions of length O(log(q)) for q even, and length O(log^2(q)/loglog(q))$ for arbitrary q. While exponetial time, our algorithms are faster in practice than all earlier generic algorithms, and produce much shorter collisions.