International Association for Cryptologic Research

International Association
for Cryptologic Research


Niels Ferguson


Random Oracle Combiners: Breaking the Concatenation Barrier for Collision-Resistance
Suppose we have two hash functions h1 and h2, but we trust the security of only one of them. To mitigate this worry, we wish to build a hash combiner C^{h1,h2} which is secure so long as one of the underlying hash functions is. This question has been well-studied in the regime of collision resistance. In this case, concatenating the two hash function outputs clearly works. Unfortunately for practice, a long series of works (Boneh and Boyen, CRYPTO’06; Pietrzak, Eurocrypt’07; Pietrzak, Crypto’08) showed no (noticeably) better combiner for collision resistance is possible. In this work, we revisit this pessimistic state of affairs, motivated the observation that collision-resistance is insufficient for many interesting applications of cryptographic hash functions anyway. Thus, we believe (and argue) the right formulation of the “hash combiner” is to build what we call random oracle (RO) combiners, utilizing stronger assumptions for stronger constructions. Indeed, we circumvent the previous lower bounds for collision resistance by constructing a simple length-preserving RO combiner C^{h1,h2}_{Z1,Z2} (M ) = h1(M, Z1) ⊕ h2(M, Z2), where Z1, Z2 are random salts of appropriate length. We show that this extra randomness is necessary for RO combiners, and indeed our construction is somewhat tight with this lower bound. On the negative side, we show that one cannot generically apply the composition theorem to further replace “monolithic” hash functions h1 and h2 by some simpler indifferentiable (such as the Merkle-Damgard transformation) from smaller components, such as fixed-length compression functions. Finally, despite this issue, we directly prove collision resistance of the Merkle-Damgard variant of our combiner, where h1 and h2 are replaced by iterative Merkle-Damgard hashes applied to a fixed-length compression function. Thus, we can still subvert the concatenation barrier for collision-resistance combiners while utilizing practically small fixed-length components underneath.