Abstract. We consider how to build an efficient compression function from a small number of random, noncompressing primitives. Our main goal is to achieve a level of collision resistance as close as possible to the optimal birthday bound. We present a 2n-to-n bit compression function based on three independent n-to-n bit random functions, each called only once. We show that if the three random functions are treated as black boxes then finding collisions requires Θ(2 n/2 /n c) queries for c ≈ 1. This result remains valid if two of the three random functions are replaced by a fixed-key ideal cipher in Davies-Meyer mode (i.e., EK(x) ⊕ x for permutation EK). We also give a heuristic, backed by experimental results, suggesting that the security loss is at most four bits for block sizes up to 256 bits. We believe this is the best result to date on the matter of building a collision-resistant compression function from non-compressing functions. It also relates to an open question from Black et al. (Eurocrypt’05), who showed that compression functions that invoke a single non-compressing random function cannot suffice. We also explore the relationship of our problem with that of doubling the output of a hash function and we show how our compression function can be used to double the output length of ideal hashes.

Abstract. In this paper, we study the security of permutation based hash functions, i.e. blockcipher based hash functions with fixed keys. SMASH is such a hash function proposed by Knudsen in 2005 and broken the same year by Pramstaller et al. Here we show that the two tweaked versions, proposed soon after by Knudsen to thwart the attack, can also be attacked in collision in time O(n2 n/3). This time complexity can be reduced to O(2 2 √ n) for the first tweak version, which means an attack against SMASH-256 in c ·2 32 for a small constant c. Then, we show that an efficient generalization of SMASH, using two permutations instead of one, can be proved secure against collision in the ideal-cipher model in Ω(2 n/4) queries to the permutations. In order to analyze the tightness of our proof, we devise a non-trivial attack in O(2 3n/8) queries. Finally, we also prove that our construction is preimage resistant in Ω(2 n/2) queries, which the best security level that can be reached for 2-permutation based hash functions, as proved in [12]. 1