modular arithmetic

Solving multi-modular integer constraint systems—comprising polynomial equalities and inequalities under distinct moduli—is notoriously difficult in cryptographic protocol verification; existing SMT solvers fail to exploit their inherent algebraic structure. Method: This paper introduces the first resolution-based decision procedure tailored for multi-modular reasoning. Contributions/Results: (1) Constraints are partitioned by modulus, and novel algebraic lifting/reduction mechanisms enable information sharing across modular subsystems; (2) Weighted Gröbner basis theory is integrated into the SMT framework for precise multi-modular algebraic reasoning—the first such incorporation; (3) A modular, embeddable solving pipeline is constructed. Evaluated on Montgomery multiplication and zero-knowledge proof implementation verification, our method substantially outperforms state-of-the-art SMT solvers: solution success rate improves by 42%, and average verification time decreases by a factor of 5.8.

This work addresses the challenge of constructing large, pairwise coprime ternary moduli sets for Residue Number Systems (RNS) under fixed bit-width constraints, where efficient modular arithmetic is critical. Method: We propose a novel coprimality criterion based on polynomial resultants and cyclotomic polynomials, model modulus pairwise coprimality as a compatibility graph, and—uniquely—apply maximum clique search algorithms to identify optimal moduli subsets. Contribution/Results: We theoretically establish an upper bound on the size of moduli sets of the form $2^n - 2^k + 1$. Experimental validation via the Chinese Remainder Theorem confirms their efficacy in fast modular reduction and scalable parallel integer arithmetic. The approach bridges theoretical rigor and hardware-oriented practicality, yielding an extensible, principled framework for RNS modulus design suitable for VLSI implementation.

This work investigates the undecidability boundary of Diophantine equations over the integers, aiming to characterize the minimal subclasses of Hilbert’s Tenth Problem (H10) that remain undecidable under joint constraints on the number of variables ν and the degree δ. Method: Departing from prior approaches reliant on reductions from the natural-number setting, we constructively derive the first tight universal bound pair (ν, δ) applicable directly over ℤ. Concurrently, we develop a formal verification framework in Isabelle/HOL that tightly integrates number-theoretic analysis, algebraic construction, and mechanized equivalence transformations. Contribution/Results: We achieve the first end-to-end, machine-checked formal verification of H10’s constrained undecidability—fully parallelizing theoretical derivation and mechanized proof. This establishes a new paradigm for decidability theory and mathematical formalization, offering both a sharp integer-specific undecidability threshold and a reproducible, logically rigorous verification methodology.

This paper addresses fundamental open problems in computational complexity—such as P vs NP and the nonexistence of polynomial-size circuits for SAT—within weak formal systems like bounded arithmetic $S^1_2$. Using a synthesis of proof complexity, model theory, recursion theory, and propositional logic simulation techniques, it establishes, for the first time, rigorous unprovability results for key complexity-theoretic statements in subexponential-strength arithmetic theories. The main contributions are: (1) proving that assertions such as “SAT has no polynomial-size circuits” are independent of $S^1_2$; (2) establishing a tight correspondence between proof complexity lower bounds and circuit lower bounds; and (3) exposing deep metatheoretic barriers preventing any feasible formal proof of P = NP, thereby offering a novel logical foundation for complexity theory.

Satisfiability checking for non-polynomial integer arithmetic—particularly involving exponential functions—remains a major challenge in program verification; existing SMT solvers support only polynomial constraints and lack systematic handling of exponential integer arithmetic. Method: This paper presents the first sound, complete, and practical SMT solver for exponential integer arithmetic. It introduces an incremental linearization framework tailored to exponential semantics, constructs exact cutting planes leveraging both convexity and discreteness, and designs a semantic-lemma-driven spurious-counterexample elimination mechanism. The solver integrates conflict-driven clause learning (CDCL), integer programming, and interval propagation for efficient reasoning. Results: Experimental evaluation on diverse program verification benchmarks shows that our approach achieves over a 3× improvement in solving success rate and reduces average solving time by 62%, significantly outperforming state-of-the-art tools.

This work presents a complete formalization of the Ramanujan–Nagell theorem in Lean 4, leveraging the Mathlib library to establish that the Diophantine equation $x^2 + 7 = 2^n$ admits exactly five integer solutions: $(n, x) = (3, \pm1), (4, \pm3), (5, \pm5), (7, \pm11), (15, \pm181)$. The proof is carried out within an interactive theorem prover by developing foundational structures from algebraic number theory, including the ring of integers, class group, and unit group of the quadratic field $\mathbb{Q}(\sqrt{-7})$. This effort not only provides the first machine-verified proof of the theorem but also significantly extends the reusable infrastructure for algebraic number theory in formalized mathematics.

Verifying large-scale arithmetic circuits for wide-word operations often incurs prohibitive computational costs due to reliance on arbitrary-precision integer arithmetic, which scales poorly with word length. This work proposes a hybrid algebraic verification approach based on polynomial reasoning that integrates both linear and nonlinear rewriting strategies. Crucially, it introduces— for the first time—a parallel multimodal homomorphic image technique that performs algebraic reasoning simultaneously over multiple prime moduli, thereby entirely eliminating the need for large-integer computations. Implemented in the TalisMan2.0 tool, the method demonstrates significant performance advantages over existing verification schemes on multiplier benchmarks, offering both high efficiency and strong scalability.

This work presents the first complete formalization of Wolstenholme’s theorem in Lean 4, establishing that for any prime \( p \geq 5 \), the congruence \( \binom{2p}{p} \equiv 2 \pmod{p^3} \) holds. The proof proceeds by expanding the shifted factorial product up to terms of order \( p^2 \), identifying its quadratic coefficient as the second elementary symmetric polynomial, and demonstrating that this coefficient is divisible by \( p \) using the fact that power sums vanish modulo \( p \). Built entirely on the Mathlib library without any unproven assumptions (i.e., no `sorry`), the formalization comprises nine lemmas and approximately 800 lines of code. It leverages a combination of relational analogy reasoning and human-guided exploration to uncover the critical proof pathway, marking the first fully verified formalization of this classical number-theoretic result in an interactive theorem prover.

This study systematically evaluates the rigorous proof reasoning and explicit construction capabilities of large language models on Olympiad-level combinatorics problems. To this end, we introduce a benchmark comprising 100 expert-annotated competition problems, categorizing tasks into analytical (proof-oriented) and constructive (implementation-oriented) types. We propose a unified evaluation protocol that integrates rubric-guided proof assessment with deterministic verification of constructions, enhanced by a Best@4 multi-solution sampling strategy. Experimental results show that the strongest model achieves an average score of 65.4% overall (75.3% under Best@4), with markedly divergent performance across the two task types, revealing current limitations in creative mathematical reasoning—particularly on existence and construction problems. This work presents the first fine-grained distinction and joint evaluation of these capabilities, offering a new benchmark and diagnostic framework for mathematical reasoning research.

This work investigates the computational complexity of verifying three-variable identities—such as distributivity—in finite algebraic structures. For the specific problem of testing distributivity between two binary operations, the paper presents the first randomized algorithm running in strongly subcubic time, namely $O(|S|^\omega)$, where $\omega$ denotes the exponent of matrix multiplication. The conditional optimality of this bound is established under conjectures related to triangle detection and 4-term arithmetic progression detection. Furthermore, the study provides a complete complexity classification for a natural class of such identities, yields nearly optimal algorithms for verifying field and ring axioms, and demonstrates that counting triples satisfying distributivity is conditionally strictly harder than merely verifying distributivity itself.