Paper 2024/1482

The Power of NAPs: Compressing OR-Proofs via Collision-Resistant Hashing

Abstract

Proofs of partial knowledge, first considered by Cramer, Damgård and Schoenmakers (CRYPTO'94) and De Santis et al. (FOCS'94), allow for proving the validity of $k$ out of $n$ different statements without revealing which ones those are. In this work, we present a new approach for transforming certain proofs system into new ones that allows for proving partial knowledge. The communication complexity of the resulting proof system only depends logarithmically on the total number of statements $n$ and its security only relies on the existence of collision-resistant hash functions. As an example, we show that our transformation is applicable to the proof systems of Goldreich, Micali, and Wigderson (FOCS'86) for the graph isomorphism and the graph 3-coloring problem. Our main technical tool, which we believe to be of independent interest, is a new cryptographic primitive called non-adaptively programmable functions (NAPs). Those functions can be seen as pseudorandom functions which allow for re-programming the output at an input point, which must be fixed during key generation. Even when given the re-programmed key, it remains infeasible to find out where re-programming happened. Finally, as an additional technical tool, we also build explainable samplers for any distribution that can be sampled efficiently via rejection sampling and use them to construct NAPs for various output distributions.