Random Oracles and Non-uniformity

Abstract

We revisit security proofs for various cryptographic primitives in the auxiliary-input random-oracle model (AI-ROM), in which an attacker \(\mathcal A\) can compute arbitrary S bits of leakage about the random oracle \(\mathcal O\) before attacking the system and then use additional T oracle queries to \(\mathcal O\) during the attack. This model has natural applications in settings where traditional random-oracle proofs are not useful: (a) security against non-uniform attackers; (b) security against preprocessing. We obtain a number of new results about the AI-ROM: Unruh (CRYPTO’07) introduced the pre-sampling technique, which generically reduces security proofs in the AI-ROM to a much simpler P-bit-fixing random-oracle model (BF-ROM), where the attacker can arbitrarily fix the values of \(\mathcal O\) on some P coordinates, but then the remaining coordinates are chosen at random. Unruh’s security loss for this transformation is \(\sqrt{ST/P}\). We improve this loss to the optimal value O(ST / P), obtaining nearly tight bounds for a variety of indistinguishability applications in the AI-ROM. While the basic pre-sampling technique cannot give tight bounds for unpredictability applications, we introduce a novel “multiplicative version” of pre-sampling, which allows to dramatically reduce the size of P of the pre-sampled set to \(P=O(ST)\) and yields nearly tight security bounds for a variety of unpredictability applications in the AI-ROM. Qualitatively, it validates Unruh’s “polynomial pre-sampling conjecture”—disproved in general by Dodis et al. (EUROCRYPT’17)—for the special case of unpredictability applications. Using our techniques, we reprove nearly all AI-ROM bounds obtained by Dodis et al. (using a much more laborious compression technique), but we also apply it to many settings where the compression technique is either inapplicable (e.g., computational reductions) or appears intractable (e.g., Merkle-Damgard hashing). We show that for any salted Merkle-Damgard hash function with m-bit output there exists a collision-finding circuit of size \(\varTheta (2^{m/3})\) (taking salt as the input), which is significantly below the \(2^{m/2}\) birthday security conjectured against uniform attackers. We build two compilers to generically extend the security of applications proven in the traditional ROM to the AI-ROM. One compiler simply prepends a public salt to the random oracle, showing that salting generically provably defeats preprocessing.