Short collision search in arbitrary SL2 homomorphic hash functions.

We study homomorphic hash functions into SL2(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 all concrete homomorphisms proposed thus far, we prove that a random homomorphism is at least 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. For any conceivable practical scenario, our algorithms are substantially faster than all earlier algorithms and produce much shorter collisions.