integer factorization

Standard neural embeddings struggle to capture the algebraic structure of integers, particularly in modular arithmetic, where relational patterns must be learned from scratch. This work proposes Prime Fourier Embeddings (PFE), the first approach that integrates group representation theory with harmonic analysis to encode integers as cosine–sine pairs indexed by prime numbers, thereby reducing modular operations to selection over corresponding prime-indexed channels. Theoretically, PFE exhibits a block-diagonal structure under linear equivariant mappings, with channel selection guided by the Chinese Remainder Theorem. Empirically, PFE achieves 100% in-distribution accuracy across all square-free composite modulus tasks, and demonstrates a specialization ratio exceeding 500× between task-relevant and task-irrelevant channels.

This work presents the first formalization in Lean 4’s Mathlib library of Nagata’s criterion for unique factorization domains (UFDs). Addressing an implicit degeneracy condition in the classical statement, we replace the traditional assumption involving “prime elements or units” with a more robust formulation based on submonoids generated by prime elements. This refinement uniformly handles both concrete and abstract localizations. The revised criterion not only corrects and strengthens the original mathematical statement but also enables a successful application to proving that polynomial rings \( R[X] \) and their iterated extensions \( R[X][Y] \) are UFDs. These case studies demonstrate the generality and rigor of our approach, significantly enhancing the expressiveness and scope of formalized algebra within proof assistants.

Integer factorization underpins the security of RSA, yet no classical polynomial-time algorithm is known. This paper departs from conventional algebraic approaches by establishing the first systematic reduction of integer factorization to problems in the Lebesgue space $L^1(X)$ and the matrix ring $M_2(mathbb{Z})$, yielding equivalent formulations as integral boundary solving, matrix decomposition, and bivariate polynomial small-root finding. Leveraging this equivalence, we develop a novel hybrid algebraic–analytic framework integrating integral geometry, matrix diagonalization, Gröbner basis theory, and the Coppersmith method. Our work provides the first rigorous functional-space and noncommutative-ring characterizations of integer factorization, thereby enabling structural transfer across mathematical domains. It opens a new paradigm for algorithm design grounded in cross-domain structural mapping and furnishes both theoretical foundations and multidimensional feasibility evidence for classical speedups beyond the number field sieve. (149 words)

Efficient factorization of RSA moduli (N = pq) remains a fundamental challenge in computational number theory. Method: This paper introduces a novel algorithm based on the family (mathcal{E}_2) of elliptic curves of even order. Its core innovation lies in the first integration of 2-adic point-order separation with the property that the minimal prime-order of a point (Q) on curve (E) modulo (N) satisfies (l_{min}(E, Q) leq 2). This shifts smoothness analysis from the global group order (as in ECM) to the local order structure of individual point–curve pairs ((E, Q)) modulo (N), thereby circumventing ECM’s reliance on group-order smoothness. Contribution/Results: The algorithm achieves subexponential time complexity (L_{sqrt{2}+o(1)}(min(p,q))) in the classical computation model and, under plausible heuristic assumptions, yields deterministic factorization in (t^{1+o(1)}) time. Empirical evaluation confirms that a substantial fraction of pairs ((E, Q)) satisfy (l_{min} leq 2), significantly boosting success probability and reducing overall computational cost.

This study investigates the algebraicity and arithmetic properties of hypergeometric functions over the rational numbers, finite fields, and p-adic fields. Leveraging the SageMath computer algebra system, the work integrates techniques from algebraic number theory, finite field theory, and p-adic analysis to systematically implement, for the first time in an open-source framework, algorithms capable of determining algebraicity, computing valuations, and solving for minimal polynomials in positive characteristic. This implementation fills a critical gap in existing computational toolchains by enabling uniform arithmetic analysis of hypergeometric functions across multiple number-theoretic domains, thereby substantially enhancing SageMath’s capacity for algebraic manipulation of such functions.

This work exposes security vulnerabilities in RSA key generation arising from insufficient entropy in embedded device random number generators, leading to flawed prime selection. We identify two practical threats: (i) excessively small prime gaps—enabling Fermat factorization—and (ii) prime reuse or sharing—enabling greatest-common-divisor (GCD) attacks. For the first time, we systematically establish the full causal chain: “hardware entropy deficiency → prime collisions or proximity → scalable private-key recovery.” Through large-scale TLS certificate scanning (covering over 64,000 vulnerable hosts), statistical analysis of prime distributions, entropy evaluation, and multiple cryptanalytic techniques—including Fermat factorization and GCD-based key recovery—we empirically demonstrate that entropy-induced key weaknesses remain widespread. We further propose a lightweight entropy enhancement mechanism and a prime robustness verification scheme, both designed for practical deployment in resource-constrained environments.

This work addresses the integer factorization problem by introducing a novel “unoperation” paradigm grounded in quantum reversible computation. Methodologically, it constructs a quantum circuit for the inverse of addition (unaddition) and leverages it to design the first dedicated quantum unmultiplier, thereby recasting factorization as an input-inversion problem over a reversible mapping. The key contribution is the systematic introduction of a reversible-operation framework: the unmultiplier coherently maps a given product state back to a superposition of all possible factor pairs, enabling parallel inversion-based factor search. Crucially, the architecture requires only O((log N)²) qubits—achieving state-of-the-art resource efficiency—and offers a conceptually distinct, hardware-efficient alternative to Shor’s algorithm for quantum integer factorization.

Defining mathematical concepts formally remains a critical bottleneck in interactive theorem proving: steep learning curves hinder newcomers, and undergraduate-level formalization progresses slowly. This paper investigates the generality, readability, and type-system compatibility of definitions, using Lean’s mathlib as an empirical foundation. We systematically analyze hundreds of equivalent definitions across diverse mathematical domains, evaluating them via usability metrics—theorem verification success rate, proof conciseness, and interface orthogonality. We identify three key determinants of definition quality: abstraction level, constructive strength, and interface granularity; from these, we distill reusable design principles. Furthermore, we contrast definition strategies in computer algebra systems (CAS) and, for the first time, establish a cross-system formal definition design guide. Our framework significantly improves the efficiency of standardized knowledge construction and long-term collaborative sustainability in libraries such as mathlib.

Integer factorization lacks exact, closed-form analytical equations, posing a fundamental challenge in number theory and cryptography. Method: This paper introduces the first analytically tractable modeling framework for factorization based on tensor networks. It reformulates factorization as an exactly solvable tensor network problem and proposes MeLoCoToN—a structured architecture that systematically enumerates and filters all candidate factor pairs. To enhance efficiency, the method jointly optimizes tensor count, tensor dimensions, and contraction order. Furthermore, it integrates classical tensor network formalism, the MeLoCoToN architecture, and tensor train compression techniques to support both exact and high-fidelity approximate solutions. Results: Experimental evaluation demonstrates feasibility and effectiveness on small- to medium-scale integers. The approach achieves exact factorization where feasible and delivers accurate approximations otherwise, establishing a novel computational paradigm for number-theoretic problems grounded in tensor algebra.

This work addresses the high computational complexity of cylindrical algebraic decomposition (CAD) in quantifier elimination and SMT solving. We introduce “projective separability”—a weakened yet rigorously provable property that replaces classical projection boundedness—ensuring decomposition completeness while substantially reducing the size of projection operators and simplifying single-cell construction. Methodologically, we integrate this new notion into the CAD framework and synergistically combine it with heuristic search strategies such as NLSAT and NuCAD to enhance single-cell generation efficiency. We formally prove correctness and evaluate the approach on standard benchmarks: results show an average 32% reduction in the number of projection polynomials and approximately 27% speedup in single-cell construction time. This work establishes a novel theoretical foundation and practical optimization pathway for guided CAD algorithms.