Abstract This paper presents a new refutation procedure for multimodular systems of integer constraints that commonly arise when verifying cryptographic protocols. These systems, involving polynomial equalities and disequalities modulo different constants, are challenging for existing solvers due to their inability to exploit multimodular structure. To address this issue, our method partitions constraints by modulus and uses lifting and lowering techniques to share information across subsystems, supported by algebraic tools like weighted Gr bner bases. Our experiments show that the proposed method outperforms existing state-of-the-art solvers in verifying cryptographic implementations related to Montgomery arithmetic and zero-knowledge proofs.

Abstract Satisfiability modulo finite fields enables automated verification for cryptosystems. Unfortunately, previous solvers scale poorly for even some simple systems of field equations, in part because they build a full Gröbner basis (GB) for the system. We propose a new solver that uses multiple, simpler GBs instead of one full GB. Our solver, implemented within the cvc5 SMT solver, admits specialized propagation algorithms, e.g., for understanding bitsums. Experiments show that it solves important bitsum-heavy determinism benchmarks far faster than prior solvers, without introducing much overhead for other benchmarks.

The Gilbert-Varshamov bound guarantees the existence of families of codes over the finite field Fℓwith good asymptotic parameters. We show that this bound can be improved for all nonprime fields Fℓwith ℓ ≥ 49 , except possibly ℓ = 125. We observe that the same improvement even holds within the class of transitive codes and within the class of self-orthogonal codes.