Fixing cracks in the concrete: Random oracles with auxiliary input, revisited

Abstract

We revisit the security of cryptographic primitives in the random-oracle model against attackers having a bounded amount of auxiliary information about the random oracle. This situation arises most naturally when an attacker carries out offline preprocessing to generate state (namely, auxiliary information) that is later used as part of an on-line attack, with perhaps the best-known example being the use of rainbow tables for function inversion. The resulting model is also critical to obtain accurate bounds against non-uniform attackers when the random oracle is instantiated by a concrete hash function. Unruh (Crypto 2007) introduced a generic technique (called presampling) for analyzing security in this model: a random oracle for which S bits of arbitrary auxiliary information can be replaced by a random oracle whose value is fixed in some way on P points; the two are distinguishable with probability at most O(√ST/P) by attackers making at most T oracle queries. Unruh conjectured that the distinguishing advantage could be made negligible for a sufficiently large polynomial P. We show that Unruh’s conjecture is false by proving that the distinguishing probability is at least Ω(ST/P). Faced with this negative general result, we establish new security bounds, — which are nearly optimal and beat pre-sampling bounds, — for specific applications of random oracles, including one-way functions, pseudorandom functions/generators, and message authentication codes. We also explore the effectiveness of salting as a mechanism to defend against offline preprocessing, and give quantitative bounds demonstrating that salting provably helps in the context of one-wayness, collision resistance, pseudorandom generators/functions, and message authentication codes. In each case, using (at most) n bits of salt, where n is the length of the secret key, we get the same security O(T/2n) in the random oracle model with auxiliary input as we get without auxiliary input. At the heart of our results is the compression technique of Gennaro and Trevisan, and its extensions by De, Trevisan and Tulsiani.