Advancements and Prospects in Large Integer Factorization: A Comprehensive Review of the Number Field Sieve Method

Abstract

The Number Field Sieve (NFS) stands as one of the most effective algorithms for the factorization of large integers, playing a critical role in cryptographic security, particularly in breaking RSA encryption systems. This review paper provides an in-depth analysis of NFS, highlighting its structure, key processes, and computational challenges. NFS is notably effective for integers larger than 110 digits and is recognized for its sub-exponential computational complexity, making it superior to other classical factorization algorithms. The core processes of NFSpolynomial selection, number pair sieving, matrix solving, and square root extractionare systematically examined to illustrate their roles in factorization. This study also reviews the latest advancements in NFS, including improvements in polynomial selection methods and optimizations in sieving and matrix-solving stages, which have significantly enhanced the algorithms efficiency. Moreover, the paper discusses the future prospects of NFS, emphasizing the need for further optimization to reduce computational resource demands and increase practicality in large-scale applications. The ongoing evolution of NFS continues to push the boundaries of cryptographic analysis, driving future research towards even more efficient factorization methods.