Integer Factorization: Another perspective

TL;DR

This paper explores novel perspectives on integer factorization by reformulating it through geometric, matrix, and algebraic approaches, aiming to inspire new algorithms and insights beyond traditional methods.

Contribution

It introduces innovative reformulations of integer factorization in terms of Lebesgue spaces, matrix decomposition, and algebraic root finding, providing fresh methodological angles.

Findings

Equivalent to perimeter finding in Lebesgue space

Matrix decomposition approaches using Gröbner basis

Small roots of bivariate polynomials via Coppersmith's method

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 $\displaystyle\left(\mathbb{Z}, \text{+}, \cdot\right)$ to the Lebesgue space $\mathcal{L}^{1}\left(X\right)$ 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 $\displaystyle\left(\mathbb{Z}, \text{+}, \cdot\right) $ to the ring of matrices $\left( M_{2}\text{(}\mathbb{Z}\text{)}, \ \text{+} \ \cdot\right)$ and show that this problem is equivalent to matrix decomposition, and therefore present some possible computing algorithms, particularly using Gr\"obner 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.