Semiprime Factorization with the Cell Method: A QUBO-Based and a Memory-Efficient Classical Cell-Decomposition Study

Factoring large integers represents a crucial challenge for computational mathematics and global cryptographic security. Although Shor's algorithm paved the way for a quantum revolution, its large-scale implementation remains theoretical, leaving a gap that requires innovative solutions. In this study, conducted in collaboration between Intesa Sanpaolo and Data Reply, we propose a critical review and reformulation of the cell method, demonstrating how unconventional approaches can redefine current limitations. Our contribution is based on two strategic axes. The first challenges current benchmarks for a full QUBO formulation in the literature with a 23-bit number and demonstrate how quantum-inspired techniques can anticipate the future of factorization. The second focuses on optimizing the method on CPU architecture using an innovative filtration procedure, capable of drastically reducing execution times and memory footprint. This combination allowed us to factor an 80-bit number, reaching a really promising result for this approach and ease outperform hybrid quantum results. The implications of these results are profound: not only they redefine the prospects of factorization in post-quantum contexts, but they also open up new scenarios for cryptographic resilience and complexity theory, marking a decisive step towards the next generation of short- to mid-term algorithms.