Generating random factorisations of polynomial values

TL;DR

This paper presents algorithms for efficiently generating random factorizations of polynomial values, enabling applications like RSA key construction and demonstrating the density of factor ratios.

Contribution

The paper introduces novel algorithms for random factorization of polynomial values and shows the density of factor ratios, with applications to cryptography.

Findings

Algorithms for random factorization of quadratic and cubic polynomial values

Construction of RSA keys with specific size and information content

Proof that ratios of factors are dense in positive real numbers

Abstract

We construct algorithms that efficiently generate random factorisations of values $P(n)$ as products of two integers, where $P\in\mathbb{Z}[x]$ is a given quadratic or cubic monic polynomial. In other words, the algorithms produce random triples $(n,d_1,d_2)\in\mathbb{Z}^3$ that solve the Diophantine equation $P(n) = d_1d_2$. In the case where $P$ is cubic, such an algorithm allows the construction of an RSA key of $k$ bits that can be described using about $k/3$ bits of information. We also show how to construct a solution $(n,d_1,d_2)$ with the ratio $d_1/d_2$ arbitrarily close to any given positive real number. This proves that among all solutions $(n,d_1,d_2)$ of $P(n) = d_1d_2$ the ratios $d_1/d_2$ are dense in $(0,+\infty)$.