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Constructions for hash functions based on a block cipher are studied where the size of the hashcode is equal to the block length of the block cipher and where the key size is approximately equal to the block length. A general model is presented, and it is shown that this model covers 9 schemes that have appeared in the literature. Within this general model 64 possible schemes exist, and it is shown that 12 of these are secure; they can be reduced to 2 classes based on linear transformations of variables. The properties of these 12 schemes with respect to weaknesses of the underlying block cipher are studied. The same approach can be extended to study keyed hash functions (MAC’s) based on block ciphers and hash functions based on modular arithmetic. Finally a new attack is presented on a scheme suggested by R. Merkle..