Factorization and malleability of RSA modules, and counting points on elliptic curves modulo N

TL;DR

This paper establishes a connection between RSA module factorization, elliptic curve point counting, and demonstrates the malleability of RSA factoring using elliptic curves and Coppersmith's algorithm.

Contribution

It shows that factoring RSA modules is equivalent to counting points on elliptic curves and addresses RSA malleability through elliptic curve point counts and Coppersmith's algorithm.

Findings

Factoring RSA is equivalent to counting points on elliptic curves modulo N.

RSA malleability can be analyzed using elliptic curve point counts.

Coppersmith's algorithm can be used to influence RSA factorization.

Abstract

In this paper we address two different problems related with the factorization of an RSA module N. First we can show that factoring is equivalent in deterministic polynomial time to counting points on a pair of twisted Elliptic curves modulo N. Also we settle the malleability of factoring an RSA module, as described in [9], using the number of points of a single elliptic curve modulo N, and Coppersmith's algorithm.