  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.
