Navigating in the Cayley graph of $SL_2(F_p)$ and applications to hashing
This addresses cryptographic security for hash functions based on group theory, offering incremental improvements by analyzing specific matrix pairs to enhance collision resistance.
The paper tackles the security of Cayley hash functions based on 2x2 matrices over finite fields, showing that for certain matrix pairs, the 'lifting attack' used previously does not produce collisions in the positive monoid, leaving no known attacks affecting security, and provides explicit lower bounds on collision lengths for specific pairs.
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 $2 \times 2$ matrices over $F_p$. Since there are many known pairs of $2 \times 2$ matrices over $Z$ that generate a free monoid, this yields numerous pairs of matrices over $F_p$, 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 Zémor in the special case where two matrices $A$ and $B$ generate (as a monoid) the whole monoid $SL_2(Z_+)$. However, in this paper we show that the situation with other, "similar", pairs of matrices from $SL_2(Z)$ 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 $SL_2(F_p)$.