Decomposition of RSA modulus applying even order elliptic curves

2025-03-02

AI Summary

Efficient factorization of RSA moduli (N = pq) remains a fundamental challenge in computational number theory. Method: This paper introduces a novel algorithm based on the family (mathcal{E}_2) of elliptic curves of even order. Its core innovation lies in the first integration of 2-adic point-order separation with the property that the minimal prime-order of a point (Q) on curve (E) modulo (N) satisfies (l_{min}(E, Q) leq 2). This shifts smoothness analysis from the global group order (as in ECM) to the local order structure of individual point–curve pairs ((E, Q)) modulo (N), thereby circumventing ECM’s reliance on group-order smoothness. Contribution/Results: The algorithm achieves subexponential time complexity (L_{sqrt{2}+o(1)}(min(p,q))) in the classical computation model and, under plausible heuristic assumptions, yields deterministic factorization in (t^{1+o(1)}) time. Empirical evaluation confirms that a substantial fraction of pairs ((E, Q)) satisfy (l_{min} leq 2), significantly boosting success probability and reducing overall computational cost.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Machine Learning: Other Foundations of Machine Learning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems

Abstract

An efficient integer factorization algorithm would reduce the security of all variants of the RSA cryptographic scheme to zero. Despite the passage of years, no method for efficiently factoring large semiprime numbers in a classical computational model has been discovered. In this paper, we demonstrate how a natural extension of the generalized approach to smoothness, combined with the separation of $2$-adic point orders, leads us to propose a factoring algorithm that finds (conjecturally) the prime decomposition $N = pq$ in subexponential time $L(sqrt 2+o(1), min(p,q))$. This approach motivated by the papers cite{Len}, cite{MMV} and cite{PoZo} is based on a more careful investigation of pairs $(E,Q)$, where $Q$ is a point on an elliptic curve $E$ over $ _N$. Specifically, in contrast to the familiar condition that the largest prime divisor $P^+(ord Q_p)$ of the reduced order $ord Q_p$ does not divide $#E(F_q)$ we focus on the relation between $P^+(ord Q_r)$ and the smallest prime number $l_{min}(E,Q)$ separating the orders $ord Q_p$ and $ord Q_q$. We focus on the ${calE}_2$ family of even order elliptic curves over $_N$ since then the condition $l_{min}(E,Q)le 2$ holds true for large fraction of points $(x,y)in E(_N)$. Moreover if we know the pair $(E,Q)$ such that $P^+(ord Q_r)le t<l_{min}(E,Q)$ and $d=max_{rin {p,q}}(ord Q_r)$ is large in comparison to $min_{rin {p,q}}|a_r(E)| eq 0$ then we can decompose $N$ in deterministic time $t^{1+o(1)}$ by representing $N$ in base $d$.

Problem

• Proposes a subexponential-time algorithm for RSA modulus factorization.
• Focuses on elliptic curves and 2-adic point orders for factorization.
• Explores conditions for efficient prime decomposition using elliptic curves.

Innovation

• Uses even order elliptic curves for RSA modulus decomposition.
• Focuses on 2-adic point orders and smoothness conditions.
• Deterministic subexponential time algorithm for prime factorization.