Factoring integer using elliptic curves over rational number field $\mathbb{Q}$

TL;DR

This paper presents a method to factor integers that are products of two primes using elliptic curves over the rational numbers, under certain conjectures, by analyzing the valuations of points on these curves.

Contribution

It introduces a novel elliptic curve construction for factoring integers and demonstrates how to recover prime factors assuming the parity and Riemann hypotheses.

Findings

Under the parity conjecture, the elliptic curve has rank one.

The valuations of points on the curve distinguish the prime factors.

Conditional bounds on the parameter r are established under GRH.

Abstract

For the integer $ D=pq$ of the product of two distinct odd primes, we construct an elliptic curve $E_{2rD}:y^2=x^3-2rDx$ over $\mathbb Q$, where $r$ is a parameter dependent on the classes of $p$ and $q$ modulo 8, and show, under the parity conjecture, that the elliptic curve has rank one and $v_p(x([k]Q))\not=v_q(x([k]Q))$ for odd $k$ and a generator $Q$ of the free part of $E_{2rD}(\mathbb Q)$. Thus we can recover $p$ and $q$ from the data $D$ and $ x([k]Q))$. Furthermore, under the Generalized Riemann hypothesis, we prove that one can take $r<c\log^4D$ such that the elliptic curve $E_{2rD}$ has these properties, where $c$ is an absolute constant.