finite field arithmetic

This work addresses the inefficiency of basic arithmetic operations over odd prime fields $\mathbb{F}_p$ and the high computational cost of determining the minimum Hamming distance of linear codes. To overcome these challenges, the authors propose an efficient arithmetic framework based on the binary field $\mathbb{F}_2$, leveraging pure bitwise operations to emulate $\mathbb{F}_p$ computations. Novel techniques such as isometric addition are introduced to accelerate distance calculations. Implemented in highly optimized C code, the approach demonstrates substantial performance gains over state-of-the-art systems like Magma and GAP/Guava across single-core, multi-core, and shared-memory multiprocessor architectures, offering a scalable high-performance solution for large-scale coding theory 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.

This work addresses the long-standing absence of a truly quasi-linear time, output-sensitive algorithm for multiplying sparse polynomials with integer coefficients. By integrating modular black-box interpolation with sparse interpolation techniques, the authors present the first algorithm achieving rigorous quasi-linear bit complexity in this setting. Their approach refutes a prior claim of having resolved the problem and establishes output-sensitive quasi-linear bit complexity for integer-coefficient sparse polynomial multiplication. Moreover, over finite fields, the method further optimizes the bit complexity to be linear in the number of terms, the logarithm of the degree, and the logarithm of the field size.

This work addresses the problem of efficient polynomial multiplication over arbitrary finite fields $mathbb{F}_q$. We propose a general acceleration framework based on additive Fourier transforms—extending the Gao–Mateer algorithm, previously restricted to fields of characteristic two, to all finite fields $mathbb{F}_q$. Leveraging the additive group structure of $mathbb{F}_q$ and character theory, we construct a $q$-adaptive additive Fourier transform and design corresponding divide-and-conquer and reordering strategies. Our method achieves a time complexity of $O(n log n log log n)$ for multiplying degree-$n$ polynomials, improving upon both classical Karatsuba multiplication and state-of-the-art number-theoretic transform–based approaches. This result provides a theoretically superior and broadly applicable algorithmic foundation for core applications relying on finite-field polynomial arithmetic, including public-key cryptography and error-correcting codes.

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.

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 inefficiency of solving structured sparse polynomial systems over finite fields by proposing an efficient resultant-based algorithm. Leveraging the inherent sparsity and structure of the system, the method iteratively computes resultants to eliminate variables and ultimately derive a univariate polynomial, thereby circumventing the high computational complexity of brute-force search and conventional Gröbner basis approaches. The paper presents the first systematic application of resultant techniques to this class of problems and introduces ResultantSolver, a parallelizable algorithmic framework tailored for such systems. Experimental evaluation on benchmark instances from the GMV 2025 competition demonstrates that the proposed method significantly outperforms existing solvers in both speed and scalability, confirming its effectiveness and practical utility.

This work investigates, under the assumption that P ≠ NP, whether NP-complete problems admit algorithms substantially faster than naïve brute-force search and whether current best-known algorithms are already optimal. By integrating fine-grained complexity theory, algebraic techniques, extremal and additive combinatorics, cryptography, and conditional hypotheses such as the Strong Exponential Time Hypothesis (SETH), the project establishes a unified framework for deriving conditional time lower bounds for NP-complete problems. Through a systematic synthesis of classical and recent results, and by leveraging reductions and combinatorial analyses, the study provides strong evidence for the hardness of improving existing algorithms for several canonical NP-complete problems, thereby advancing our understanding of the fine-grained structure of computational complexity.

This work addresses the challenge of constructing seed-length-optimal pseudorandom generators (PRGs) for low-degree polynomials over small finite fields. Building upon the Derksen–Viola framework, the authors introduce a novel pseudorandom object that replaces traditional hitting set generators, substantially relaxing the requirement on field size. Under the assumption of sufficiently large characteristic, they achieve—for the first time—a PRG with optimal seed length for degree-$d$ polynomials over fields of size $q \approx d^4$. Moreover, the paper uncovers a threshold phenomenon linking field size to PRGs over the binary field and demonstrates that this threshold is inherent: if a PRG without characteristic restrictions could be constructed over sublinear field sizes (e.g., $q = d^{0.99}$), it would directly yield a near-optimal PRG over the binary field.

Existing methods for solving quadratic polynomial equations over binary extension fields rely on complex exponentiation or case-by-case analyses based on the parity or power structure of the field dimension \( m \), lacking both uniformity and efficiency. This work proposes a unified solution framework that leverages Reed–Muller matrices to represent function evaluations, thereby transforming the problem into a system of binary linear equations solvable using only XOR operations. For the first time, this approach yields a general formula applicable to any positive integer \( m \) without requiring case distinctions. The algorithm demands at most \( m^2 - 2m + 1 \) XOR operations and achieves a parallel time delay as low as \( \lceil \log_2 m \rceil \), significantly enhancing its generality, hardware friendliness, and suitability for low-power, low-latency applications.