Zero-knowledge proof (ZKP) systems have surged attention and held a fundamental role in contemporary cryptography. Zero-knowledge succinct non-interactive argument of knowledge (zk-SNARK) protocols dominate the ZKP usage, implemented through arithmetic circuit programming paradigm. However, underconstrained or overconstrained circuits may lead to bugs. The former refers to circuits that lack the necessary constraints, resulting in unexpected solutions and causing the verifier to accept a bogus witness, and the latter refers to circuits that are constrained excessively, resulting in lacking necessary solutions and causing the verifier to accept no witness. This article introduces a novel approach for pinpointing two distinct types of bugs in ZKP circuits. The method involves encoding the arithmetic circuit constraints to polynomial equation systems and solving them over finite fields by the computer algebra system . The classification of verification results is refined, greatly enhancing the expressive power of the system. A tool, AC 4 , is proposed to represent the implementation of the method. Experiments show that AC 4 demonstrates an increase in the solved rate, showing a 36.7% improvement over Picus and CIVER, and a slight improvement over halo2-analyzer, a checker for halo2 circuits. Within a solvable range, the checking time has also exhibited noticeable improvement, demonstrating a magnitude increase compared to previous efforts.

The communities of blockchains and distributed ledgers have been stirred up by the introduction of zero-knowledge proofs (ZKPs). Originally designed as a solution to privacy issues, ZKPs have now evolved into an effective remedy for scalability concerns. To enable ZKPs, Rank-1 Constraint Systems (R1CSs) offer a verifier for bilinear equations. In order to accurately and efficiently represent R1CSs, several language tools, such as Circom, Noir, and Snarky, have been proposed to automate the compilation of advanced programs into R1CSs. However, due to the flexible nature of R1CS representation, there can be significant differences in the compiled R1CS forms generated from circuit language programs with the same underlying semantics. To address this issue, this article puts forth a dataflow-based R1CS paradigm algorithm, which produces a standardized format for different R1CS instances with identical semantics. Additionally, we present an R1CS benchmark, and our experimental evaluation demonstrates the efficacy of our methods.

The introduction of zero-knowledge proofs (ZKPs) has had a profound impact on the blockchain and distributed ledger communities. ZKPs require the utilization of Rank-1 Constraint Systems (R1CS), which serve as verifiers for bi-linear equations. However, the flexibility of R1CS representation leads to notable variations in the compiled R1CS forms derived from circuit language programs with identical semantics. To tackle this challenge, this paper proposes a data-flow-based R1CS paradigm algorithm, producing a standardized format for different R1CS instances with the identical semantics. By adopting the normalized R1CS format circuits, the complexity of circuits’ verification can be reduced. Furthermore, this paper presents an R1CS normalization algorithm benchmark, and our experimental evaluation demonstrates the effectiveness and accuracy of our methods.