Methods for Collisions in Some Algebraic Hash Functions

This paper focuses on devising methods for producing collisions in algebraic hash functions that may be seen as generalized forms of the well-known Z\émor and Tillich-Z\émor hash functions. In contrast to some of the previous approaches, we attempt to construct collisions in a structured and deterministic manner by constructing messages with triangular or diagonal hashes messages. Our method thus provides an alternate deterministic approach to the method for finding triangular hashes. We also consider the generalized Tillich-Z\émor hash functions over ${\mathbb{F}_p}^k$ for $p\neq 2$, relating the generator matrices to a polynomial recurrence relation, and derive a closed form for any arbitrary power of the generators. We then provide conditions for collisions, and a method to maliciously design the system so as to facilitate easy collisions, in terms of this polynomial recurrence relation. Our general conclusion is that it is very difficult in practice to achieve the theoretical collision conditions efficiently, in both the generalized Z\émor and the generalized Tillich-Z\émor cases. Therefore, although the techniques are interesting theoretically, in practice the collision-resistance of the generalized Z\émor functions is reinforced.