Zero-Knowledge Accumulators and Set Algebra.

Cryptographic accumulators allow to succinctly represent a set by an accumulation value with respect to which short non-membership proofs about the set can be efficiently constructed and verified. Traditionally, their security captures soundness but offers no privacy: Convincing proofs reliably encode set membership, but they may well leak information about the accumulated set. In this paper we put forward a strong privacy-preserving enhancement by introducing and devising zero-knowledge accumulators that additionally provide hiding guarantees: Accumulation values and proofs leak nothing about a dynamic set that evolves via element insertions/deletions. We formalize the new property using the standard real-ideal paradigm, namely demanding that an adaptive adversary with access to query/update oracles, cannot tell whether he interacts with honest protocol executions or a simulator fully ignorant of the set even of the type of updates on it. We rigorously compare the new primitive to existing ones for privacy-preserving verification of set membership or other relations and derive interesting implications among related security definitions, showing that zero-knowledge accumulators offer stronger privacy than recent related works by Naor et al. [TCC﾿2015] and Derler et al. [CT-RSA﾿2015]. We construct the first dynamic universal zero-knowledge accumulator that we show to be perfect zero-knowledge and secure under the q-Strong Bilinear Diffie-Hellman assumption. Finally, we extend our new privacy notion and our new construction to provide privacy-preserving proofs also for an authenticated dynamic set collection--a primitive for efficiently verifying more elaborate set operations, beyond set-membership. We introduce a primitive that supports a zero-knowledge verifiable set algebra: Succinct proofs for union, intersection and set difference queries over a dynamically evolving collection of sets can be efficiently constructed and optimally verified, while--for the first time--they leak nothing about the collection beyond the query result.