Preimages for Z\'emor's Cayley hash function

In 1991, Z\'emor proposed a hash function which provides data security using the difficulty of writing a given matrix as a product of generator matrices. Tillich and Z\'emor subsequently provided an algorithm finding short collisions for this hash function. We extend this collision attack to a stronger preimage attack, under the assumption that we can factor large integers efficiently. The Euclidean algorithm will factor a $2\times 2$ matrix with non-negative integer entries and determinant $1$. This factorization is short if the matrix entries are all roughly the same size. Therefore, to factor a matrix we need only find an integer matrix with the listed properties which is congruent to the target matrix modulo $p$; finding such an integer matrix is equivalent to solving a Diophantine equation. We give an algorithm to solve this equation.