operation count analysis

This work addresses the problem of establishing lower bounds on the bilinear complexity of small-scale matrix multiplication over finite fields. By integrating substitution, backtracking search, symmetry reduction, and dynamic programming, the authors systematically enumerate classes of linear constraints on input matrices and derive rank-based lower bounds within each class. This approach yields the first automated proof that the bilinear complexity of $3 \times 3$ matrix multiplication over $\mathbb{F}_2$ is at least 20, surpassing the longstanding record of 19 that had remained unimproved for over two decades. The complete proof can be automatically generated on a standard laptop in under 1.5 hours and verified in seconds, significantly advancing the automation and efficiency of lower-bound certification in algebraic complexity theory.

Optimization and feasibility checking for algebraic functions involving radicals, rational expressions, and similar non-polynomial terms pose significant challenges due to the “curse of dimensionality” inherent in conventional layered auxiliary-variable lifting. Method: This paper introduces a compact polynomial reconstruction framework that replaces each algebraic function with a single new variable, coupled with an integrated algorithmic pipeline grounded in real algebraic geometry: implicitization, branch-isolating inequality generation, and Positivstellensatz/SOS-based feasibility verification. Contribution/Results: By avoiding redundant variable proliferation, the method substantially improves modeling efficiency and scalability of polynomial programming formulations. On enhanced classical benchmarks, it achieves up to 50× speedup over naive reconstruction. The approach provides both theoretical guarantees and a practical toolkit for efficiently translating algebraic programs into polynomial optimization problems.

This work addresses the satisfiability checking and quantifier elimination problems for nonlinear real arithmetic (NRA) formulas featuring both Boolean structure and quantifiers. We present the first extension of cylindrical algebraic coverings (CAC) to full first-order logic formula verification. Our approach introduces a novel CAC variant that integrates CAD-based covering construction, hierarchical quantifier handling, explicit Boolean structure incorporation, and adaptive splitting and pruning heuristics. Unlike conventional methods, our framework avoids constructing a complete cylindrical algebraic decomposition (CAD), thereby substantially reducing computational complexity. Experimental evaluation on diverse nonlinear quantified benchmarks demonstrates that our method outperforms state-of-the-art SMT solvers—including Z3 and CVC5—as well as specialized quantifier elimination tools such as QEPCAD and Redlog, in both solution accuracy and runtime efficiency. The gains are particularly pronounced on high-dimensional, sparse constraint instances.

Formal verification of compilers incurs high maintenance costs, especially when modifications necessitate extensive re-verification. Method: This paper introduces the first trusted rewriting engine framework for Coq, modeling compilers as collections of algebraic rewrite rules—each independently verifiable. It employs theorem-driven modeling, metaprogramming-based automated synthesis, and proof reuse to enable rule-level formal verification and automatic composition. Contribution/Results: The framework decouples rule verification from compiler construction, significantly reducing verification and maintenance overhead. Evaluated in the Fiat Cryptography toolchain, the generated command-line compiler achieves approximately 1000× speedup over prior verified counterparts. Moreover, its proofs are more concise and exhibit substantially higher reusability across compiler transformations.

Lightweight cryptographic hardware demands both high security and low computational overhead, yet existing Boolean function constructions struggle to simultaneously optimize nonlinearity, algebraic immunity, and implementation efficiency. Method: This paper proposes a novel construction framework that uniquely integrates integer arithmetic (addition and bit-shift operations) with polynomial operations over the binary field GF(2), enabling synergistic optimization of nonlinearity and algebraic immunity. Contribution/Results: For input sizes (n leq 20), the proposed functions achieve optimal trade-offs among implementation complexity, nonlinearity, and algebraic immunity. All constructed functions significantly outperform state-of-the-art efficient designs while requiring only basic arithmetic operations—addition, subtraction, multiplication, division, and bit-shifts—thus ensuring low hardware cost. Crucially, they exhibit strong resistance against fast algebraic attacks and high unpredictability, satisfying stringent security requirements. The approach is particularly suited for designing lightweight distinguishers and predicate functions in resource-constrained environments.

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.

This work addresses the scalability bottlenecks in formal verification of gate-level highly optimized arithmetic circuits, which stem from state-space explosion and intermediate polynomial blowup. To overcome these challenges, the paper proposes a parameterized symbolic algebraic verification framework that uniformly models general arithmetic circuits—supporting arbitrary combinations of addition and multiplication—as multivariate polynomials. The approach integrates existing techniques through a parameterized algebraic reduction theory, with key innovations including a unified parameter space encompassing multiple state-of-the-art methods, along with novel algorithms for HA-preserving extraction, density-aware elimination detection, and conservative polynomial size estimation. Experimental results on standard multipliers and practical arithmetic units such as multiply-adders and dot-product engines demonstrate that the proposed method achieves state-of-the-art verification performance.

This work addresses the challenge of efficiently implementing very-large-constant multiplication (VLCM) on resource-constrained hardware, overcoming the limitation of existing approaches that are restricted to moderately sized constants. The authors propose an enhanced decompose-optimize-reconstruct pipeline that introduces overlapping patterns to reduce the number of multiple constant multiplication (MCM) subproblems. They formulate a declarative optimization model grounded in constraint programming and SAT solving techniques to achieve globally optimal reconstructions across all stages. This approach yields the first globally optimal solutions for VLCM while precisely delineating their applicability boundaries. Empirical evaluations on signal processing and cryptographic benchmarks spanning tens to thousands of bits demonstrate superior scalability and consistently outperform state-of-the-art baselines.

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 addresses the limitations of traditional Number Theoretic Transform (NTT), which imposes strict constraints on the modulus, and the lack of precise cost modeling for Toom-4 multiplication within existing incomplete NTT frameworks. For the first time, the paper constructs an explicit computational cost model for Toom-4 that is compatible with incomplete NTT, deriving exact counts of additions, subtractions, and multiplications over the coefficient field through concrete implementation. Leveraging addition chain theory, it establishes a concise analytical framework for cost evaluation. Using this model, the study systematically assesses hybrid strategies combining Toom-4, Karatsuba, and incomplete NTT, experimentally validates the model’s accuracy, and demonstrates that Toom-4 outperforms alternative methods within specific parameter regimes.