Correlated-Input Secure Hash Functions

We undertake a general study of hash functions secure under

correlated inputs

, meaning that security should be maintained when the adversary sees hash values of many related high-entropy inputs. Such a property is satisfied by a random oracle, and its importance is illustrated by study of the “avalanche effect,” a well-known heuristic in cryptographic hash function design. One can interpret “security” in different ways: e.g., asking for one-wayness or that the hash values look uniformly and independently random; the latter case can be seen as a generalization of correlation-robustness introduced by Ishai et al. (CRYPTO 2003). We give specific applications of these notions to password-based login and efficient search on encrypted data. Our main construction achieves them (without random oracles) for inputs related by

polynomials

over the input space (namely ℤ

p

), based on corresponding variants of the

q

-Diffie Hellman Inversion assumption. Additionally, we show relations between correlated-input secure hash functions and cryptographic primitives secure under related-key attacks. Using our techniques, we are also able to obtain a host of new results for such related-key attack secure cryptographic primitives.