On the Security of the Hashing Scheme Based on SL2

Tillich and Zémor proposed a hashing scheme based on the group of unimodular matrices SL2(Fq) over a finite field Fq of q = 2n elements. Charnes and Pieprzyk studied the security of this scheme. They showed that for n = 131 and for some irreducible polynomial P131(x) this scheme is weak. We show that with sufficiently high probability the polynomials Pn(x) can be chosen in such a way that this type of attack can be avoided. Futhermore, we generalize the Tillich-Zémor hashing scheme for any finite field Fq and show that the new generalized scheme has similar properties.