Integer Factorization: Another perspective

2025-07-09

AI Summary

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)

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Search and Optimization: Mixed Discrete/Continuous Search
• Computer Vision: Other Foundations of Computer Vision

Application Category

• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Integer factorization is a fundamental problem in algorithmic number theory and computer science. It is considered as a one way or trapdoor function in the (RSA) cryptosystem. To date, from elementary trial division to sophisticated methods like the General Number Field Sieve, no known algorithm can break the problem in polynomial time, while its proved that Shor's algorithm could on a quantum computer. In this paper, we recall some factorization algorithms and then approach the problem under different angles. Firstly, we take the problem from the ring $displaystyleleft(mathbb{Z}, ext{+}, cdot ight)$ to the Lebesgue space $mathcal{L}^{1}left(X ight)$ where $X$ can be $mathbb{Q}$ or any given interval setting. From this first perspective, integer factorization becomes equivalent to finding the perimeter of a rectangle whose area is known. In this case, it is equivalent to either finding bounds of integrals or finding primitives for some given bounds. Secondly, we take the problem from the ring $displaystyleleft(mathbb{Z}, ext{+}, cdot ight) $ to the ring of matrices $left( M_{2} ext{(}mathbb{Z} ext{)}, ext{+} cdot ight)$ and show that this problem is equivalent to matrix decomposition, and therefore present some possible computing algorithms, particularly using Gröbner basis and through matrix diagonalization. Finally, we address the problem depending on algebraic forms of factors and show that this problem is equivalent to finding small roots of a bivariate polynomial through coppersmith's method. The aim of this study is to propose innovative methodological approaches to reformulate this problem, thereby offering new perspectives.

Problem

• Reformulating integer factorization via Lebesgue space equivalence
• Linking factorization to matrix decomposition using Gröbner basis
• Transforming factorization into bivariate polynomial root-finding

Innovation

• Reformulate factorization in Lebesgue space as perimeter problem
• Transform factorization into matrix decomposition using Gröbner basis
• Apply Coppersmith's method to find bivariate polynomial roots