Quantum prime factorization algorithms using binary carry propagation

TL;DR

This paper introduces quantum annealing algorithms for prime factorization, modeling carry propagation in binary multiplication, and demonstrates improved scalability for factoring larger semiprimes, impacting RSA cryptography security.

Contribution

It presents a novel quantum annealing approach using HUBO and CQM formulations for integer factorization, with the CQM model enabling larger semiprimes to be factored efficiently.

Findings

HUBO model factors small semiprimes but has exponential memory growth.

CQM model successfully factors 60-bit semiprimes, including N=1152921423002469787.

Global product constraints improve factorization accuracy and consistency.

Abstract

The RSA cryptosystem, which relies on the computational difficulty of prime factorization, faces growing challenges with the advancement of quantum computing. In this study, we propose a quantum annealing based approach to integer factorization using both high order unconstrained binary optimization (HUBO) and constrained quadratic model (CQM) formulations. We begin by modeling binary multiplication with explicit carry propagation, translating this into a HUBO representation and subsequently reducing it to a quadratic unconstrained binary optimization form compatible with current quantum solvers. To address scalability limitations, we implement a CQM approach with constraint relaxation and global product consistency. While the HUBO model successfully factors small semiprimes, it exhibits exponential memory growth, making it impractical for inputs larger than 10 bits. In contrast, the CQM model achieves accurate factorization of semiprimes up to 60 bits including N = 1152921423002469787 demonstrating significantly improved scalability. Experimental results further show that applying global product constraints enhances factorization accuracy and consistency across all tested instances. This work highlights both the promise and current limitations of quantum-assisted factorization and establishes a foundation for evaluating RSA security in the emerging quantum era.