Cayley hash functions are based on a simple idea of using a pair of
(semi)group elements, A and B, to hash the 0 and 1 bit, respectively, and
then to hash an arbitrary bit string in the natural way, by using
multiplication of elements in the (semi)group. In this paper, we focus on
hashing with 2×2 matrices over Fp. Since there are many known pairs
of 2×2 matrices over Z that generate a free monoid, this yields
numerous pairs of matrices over Fp, for a sufficiently large prime p, that
are candidates for collision-resistant hashing. However, this trick can
"backfire", and lifting matrix entries to Z may facilitate finding a
collision. This "lifting attack" was successfully used by Tillich and Z\'emor
in the special case where two matrices A and B generate (as a monoid) the
whole monoid SL2(Z+). However, in this paper we show that the situation
with other, "similar", pairs of matrices from SL2(Z) is different, and the
"lifting attack" can (in some cases) produce collisions in the group generated
by A and B, but not in the positive monoid. Therefore, we argue that for
these pairs of matrices, there are no known attacks at this time that would
affect security of the corresponding hash functions. We also give explicit
lower bounds on the length of collisions for hash functions corresponding to
some particular pairs of matrices from SL2(Fp).Comment: 10 page