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\'emor and Tillich-Z\'emor 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\'emor 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\'emor and the generalized Tillich-Z\'emor cases. Therefore, although the techniques are interesting theoretically, in practice the collision-resistance of the generalized Z\'emor functions is reinforced.

翻译：本文专注于设计针对代数哈希函数产生碰撞的方法，这些函数可被视为著名的Zémor函数及Tillich-Zémor函数的广义形式。与先前部分方法不同，我们尝试通过构造具有三角或对角哈希值的消息，以结构化和确定性的方式构建碰撞。因此，我们的方法为寻找三角哈希提供了一种替代的确定性途径。同时，我们考虑了在${\mathbb{F}_p}^k$上（其中$p\neq 2$）的广义Tillich-Zémor哈希函数，将生成矩阵与多项式递推关系相关联，并推导出生成矩阵任意幂次的封闭形式。随后，基于此多项式递推关系，我们给出了碰撞的条件，以及一种恶意设计系统以方便产生碰撞的方法。总体结论是，在广义Zémor和广义Tillich-Zémor两种情形下，实践中均难以高效满足理论上的碰撞条件。因此，尽管这些技术有趣的理论价值，但在实践中广义Zémor函数的抗碰撞性得到了强化。