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 addresses the scalability limitations of traditional prime implicant computation under constrained input spaces, where explicit post-processing filtering is typically required and often infeasible due to the exponential growth in the number of prime implicants. To overcome this bottleneck, the paper introduces an end-to-end symbolic algorithm that, for the first time, directly embeds contextual or structural constraints into the prime implicant generation process, thereby eliminating the need to explicitly enumerate all prime implicants. Built upon the Coudert-Madre decision diagram framework, the approach achieves implicit filtering through symbolic representation and integrated constraint handling. A modular toolchain is further developed to decouple decision diagram construction, prime implicant computation, and constraint processing. Experimental results demonstrate that the proposed method significantly enhances scalability and effectively mitigates the exponential explosion inherent in large-scale instances.

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.

This work investigates whether there exists a non-constant function, constructed from the identity function and constants through finitely many pointwise additions, multiplications, and exponentiations, that maps every positive integer to a prime number. The author formally articulates—within the Lean 4 proof assistant using the Mathlib library—the conjecture that the set of primes is "supernatural," along with its natural generalizations, as precisely stated open problems. The entire development, including definitions, theorems, examples, and computational experiments, has been fully formalized and verified by the Lean kernel without reliance on any unproven assumptions. This provides a rigorous, machine-checkable foundation for future automated reasoning and formal proofs concerning this conjecture.

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.