Multi-Collision Resistance: A Paradigm for Keyless Hash Functions

We introduce a new notion of multi-collision resistance for keyless hash functions. This is a natural relaxation of collision resistance where it is hard to find multiple inputs with the same hash in the following sense. The number of colliding inputs that a polynomial-time non-uniform adversary can find is not much larger than its advice. We discuss potential candidates for this notion and study its applications. Assuming the existence of such hash functions, we resolve the long-standing question of the round complexity of zero knowledge protocols --- we construct a 3-message zero knowledge argument against arbitrary polynomial-size non-uniform adversaries. We also improve the round complexity in several other central applications, including a 3-message succinct argument of knowledge for NP, a 4-message zero-knowledge proof, and a 5-message public-coin zero-knowledge argument. Our techniques can also be applied in the keyed setting, where we match the round complexity of known protocols while relaxing the underlying assumption from collision-resistance to keyed multi-collision resistance. The core technical contribution behind our results is a domain extension transformation from multi-collision-resistant hash functions for a fixed input length to ones with an arbitrary input length and a local opening property. The transformation is based on a combination of classical domain extension techniques, together with new information-theoretic tools. In particular, we define and construct a new variant of list-recoverable codes, which may be of independent interest

Classical Cryptographic Protocols in a Quantum World

Cryptographic protocols, such as protocols for secure function evaluation (SFE), have played a crucial role in the development of modern cryptography. The extensive theory of these protocols, however, deals almost exclusively with classical attackers. If we accept that quantum information processing is the most realistic model of physically feasible computation, then we must ask: what classical protocols remain secure against quantum attackers? Our main contribution is showing the existence of classical two-party protocols for the secure evaluation of any polynomial-time function under reasonable computational assumptions (for example, it suffices that the learning with errors problem be hard for quantum polynomial time). Our result shows that the basic two-party feasibility picture from classical cryptography remains unchanged in a quantum world.Comment: Full version of an old paper in Crypto'11. Invited to IJQI. This is authors' copy with different formattin

Predictable arguments of knowledge

We initiate a formal investigation on the power of predictability for argument of knowledge systems for NP. Specifically, we consider private-coin argument systems where the answer of the prover can be predicted, given the private randomness of the verifier; we call such protocols Predictable Arguments of Knowledge (PAoK). Our study encompasses a full characterization of PAoK, showing that such arguments can be made extremely laconic, with the prover sending a single bit, and assumed to have only one round (i.e., two messages) of communication without loss of generality. We additionally explore PAoK satisfying additional properties (including zero-knowledge and the possibility of re-using the same challenge across multiple executions with the prover), present several constructions of PAoK relying on different cryptographic tools, and discuss applications to cryptography