SL ₂ homomorphic hash functions: worst case to average case reduction and short collision search

We study homomorphic hash functions into SL ₂(q) , the 2 × 2 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(q). For general homomorphisms we offer an algorithm that, heuristically and according to experiments, in running time O(q) finds collisions of length O(log q) for q even, and length O(log ²q/ log log q) for arbitrary q. While exponetial time, our algorithms are faster in practice than all earlier generic algorithms, and produce much shorter collisions.