ABSTRACT. 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 linear functions of one variable over F p . The corresponding hash functions are very efficient. In particular, we show that hashing a bit string of length n with our method requires, in general, at most 2n multiplications in F p , but with particular pairs of linear functions that we suggest, one does not need to perform any multiplications at all. We also give explicit lower bounds on the length of collisions for hash functions corresponding to these particular pairs of linear functions over F p .

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The hash function proposed by Shpilrain and Sosnovski (2016), based on affine maps in one variable over a finite field, was proven insecure. This paper shows that the variation proposed by Ghaffari and Mostaghim (2018) that uses Shpilrain and Sosnovski's hash is also insecure. We demonstrate its security vulnerability by constructing collisions.