Abstract
We construct the first non-interactive zero-knowledge (NIZK) proof systems in the fine-grained setting where adversaries’ resources are bounded and honest users have no more resources than an adversary. More concretely, our setting is the \(\mathsf {NC^1}\)-fine-grained setting, namely, all parties (including adversaries and honest participants) are in \(\mathsf {NC^1}\).
Our NIZK systems are for circuit satisfiability (SAT) under the worst-case assumption, \(\mathsf {NC^1}\subsetneq \mathsf{\oplus L/poly}\) . As technical contributions, we propose two approaches to construct NIZKs in the \(\mathsf {NC^1}\)-fine-grained setting. In stark contrast to the classical Fiat-Shamir transformation, both our approaches start with a simple \(\varSigma \)-protocol and transform it into NIZKs for circuit SAT without random oracles. Additionally, our second approach firstly proposes a fully homomorphic encryption (FHE) scheme in the fine-grained setting, which was not known before, as a building block. Compared with the first approach, the resulting NIZK only supports circuits with constant multiplicative depth, while its proof size is independent of the statement circuit size.
Extending our approaches, we obtain two NIZK systems in the uniform reference string model and two non-interactive zaps (namely, non-interactive witness-indistinguishability proof systems in the plain model). While the previous constructions from Ball, Dachman-Soled, and Kulkarni (CRYPTO 2020) require provers to run in polynomial-time, our constructions are the first one with provers in \(\mathsf {NC^1}\).
Y. Wang—Supported by the National Natural Science Foundation for Young Scientists of China under Grant Number 62002049 and the Fundamental Research Funds for the Central Universities under Grant Number ZYGX2020J017.
J. Pan—Supported by the Research Council of Norway under Project No. 324235.
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Change history
25 May 2022
In an older version of this paper, there was an erroneous insertion of an equation on page 315. This has been removed.
Notes
- 1.
Recall that any circuit can be converted to one consisting only of NAND gates, and \(1-\mathsf {w}_{i}\mathsf {w}_{j}=0\) is equivalent to \(\mathsf {w}_{i}+\mathsf {w}_{j}+2\mathsf {w}_{k}-2\in \{0, 1\}\) in \(\mathbb {Z}_p\) for a large number p.
- 2.
Notice that all the computations are performed in GF(2) and thus addition and subtraction are equivalent.
- 3.
As remarked in Sect. 7.3, we can make the CRS of \({\mathsf {NCNIZK}}^*\) a single matrix in \({\mathsf {OneSamp}}(\lambda )\).
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Wang, Y., Pan, J. (2022). Non-Interactive Zero-Knowledge Proofs with Fine-Grained Security. In: Dunkelman, O., Dziembowski, S. (eds) Advances in Cryptology – EUROCRYPT 2022. EUROCRYPT 2022. Lecture Notes in Computer Science, vol 13276. Springer, Cham. https://doi.org/10.1007/978-3-031-07085-3_11
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