We study adaptive security of delayed-input Sigma protocols and non-interactive zero-knowledge (NIZK) proof systems in the common reference string (CRS) model. Our contributions are threefold:-We exhibit a generic compiler taking any delayed-input Sigma protocol and returning a delayed-input Sigma protocol satisfying adaptiveinput special honest-verifier zero knowledge (SHVZK). In case the initial Sigma protocol also satisfies adaptive-input special soundness, our compiler preserves this property.-We revisit the recent paradigm by Canetti et al. (STOC 2019) for obtaining NIZK proof systems in the CRS model via the Fiat-Shamir transform applied to so-called trapdoor Sigma protocols, in the context of adaptive security. In particular, assuming correlation-intractable hash functions for all sparse relations, we prove that Fiat-Shamir NIZKs satisfy either: (i) Adaptive soundness (and non-adaptive zero knowledge), so long as the challenge is obtained by hashing both the prover's first round and the instance being proven; (ii) Adaptive zero knowledge (and non-adaptive soundness), so long as the challenge is obtained by hashing only the prover's first round, and further assuming that the initial trapdoor Sigma protocol satisfies adaptive-input SHVZK.-We exhibit a generic compiler taking any Sigma protocol and returning a trapdoor Sigma protocol. Unfortunately, this transform does not preserve the delayed-input property of the initial Sigma protocol (if any). To complement this result, we also give yet another compiler taking any delayed-input trapdoor Sigma protocol and returning a delayed-input trapdoor Sigma protocol with adaptive-input SHVZK. An attractive feature of our first two compilers is that they allow obtaining efficient delayed-input Sigma protocols with adaptive security, and efficient Fiat-Shamir NIZKs with adaptive soundness (and non-adaptive zero knowledge) in the CRS model. Prior to our work, the latter was only possible using generic NP reductions.

Significantly extending the framework of (Couteau and Hartmann, Crypto 2020), we propose a general methodology to construct NIZKs for showing that an encrypted vector χ belongs to an algebraic set, i.e., is in the zero locus of an ideal I of a polynomial ring. In the case where I is principal, i.e., generated by a single polynomial F , we first construct a matrix that is a "quasideterminantal representation" of F and then a NIZK argument to show that F (χ) = 0. This leads to compact NIZKs for general computational structures, such as polynomial-size algebraic branching programs. We extend the framework to the case where I is non-principal, obtaining efficient NIZKs for R1CS, arithmetic constraint satisfaction systems, and thus for NP. As an independent result, we explicitly describe the corresponding language of ciphertexts as an algebraic language, with smaller parameters than in previous constructions that were based on the disjunction of algebraic languages. This results in an efficient GL-SPHF for algebraic branching programs.

No abstract