Cryptographic hash functions are fundamental building blocks of many computer security systems and protocols, primarily being used to ensure data integrity. Recent attacks against modern hash functions have questioned the suitability of standard hash function construction principles. In this paper we consider a hash function construction based multiplication in the group of 2 x 2 matrices over a finite field proposed by Zemor and Tillich [48, 42, 43]. We also look at how the algebraic properties of hash functions following this design can be exploited in attacks. Finally, we consider variations to the approach of Zemor and Tillich that offer some resistance to those attacks.