Revisiting Fermat's Factorization for the RSA Modulus

TL;DR

This paper analyzes Fermat's factorization method for RSA moduli, introduces two variants, compares their effectiveness, and discusses implications for cryptographic security, highlighting areas for future research.

Contribution

It presents two new variants of Fermat's factorization method and compares their effective regions, offering insights into potential vulnerabilities of RSA moduli.

Findings

Two variants of Fermat's method are proposed and analyzed.

Comparison shows differing effective regions for the variants.

The study highlights open questions for further analysis of factorization techniques.

Abstract

We revisit Fermat's factorization method for a positive integer $n$ that is a product of two primes $p$ and $q$. Such an integer is used as the modulus for both encryption and decryption operations of an RSA cryptosystem. The security of RSA relies on the hardness of factoring this modulus. As a consequence of our analysis, two variants of Fermat's approach emerge. We also present a comparison between the two methods' effective regions. Though our study does not yield a new state-of-the-art algorithm for integer factorization, we believe that it reveals some interesting observations that are open for further analysis.