Automating Bitvector and Finite Field Equivalence Proofs in Lean

2026-05-14

AI Summary

This work addresses the challenge of establishing quantifier-free equivalence between bit-vector and finite-field arithmetic in zero-knowledge proof circuit verification. We introduce BitModEq, a novel tactic implemented within the Lean theorem prover, which integrates range lemmas, case analysis, and bit-blasting to enable the first verified, automatic translation and equivalence proof from finite-field operations to bit-vector representations in Lean. Evaluated on standard ZKP arithmetization benchmarks, our approach solves 19% more instances than state-of-the-art SMT solvers, effectively overcoming their scalability limitations in handling conversion operators and inequalities.

Abstract

Efforts to verify Zero-Knowledge Proof circuit encodings have highlighted the challenge of proving the correctness of quantifier-free statements that make use of both bitvector and finite field operations. Existing verification workflows are either manual or rely on SMT solvers, which scale poorly on some classes of problems for reasons that include difficulties with conversion operators and challenges reasoning about inequalities. To address these limitations, we present a novel Lean tactic BitModEq that leverages range lemmas and case analysis to produce verified translations from finite fields to bitvectors. Our approach, combined with bit-blasting, outperforms state-of-the-art SMT solvers, solving 19% more ZKP arithmetization benchmarks.

Problem

• bitvector
• finite field
• equivalence proof
• Zero-Knowledge Proof
• quantifier-free

Innovation

• BitModEq
• bitvector
• finite field
• Lean tactic
• zero-knowledge proof