RSA cryptanalysis — Fermat factorization exact bound and the role of integer sequences in factorization problem

Abstract

Large composite integer factorization algorithms still need to be more efficient and, thus, not operate linearly. The RSA popular cryptographic algorithm strength solely depends on factoring the modulus N. This study proposes a technique based on the Fermat factorization theorem to factor modulus N and decreases processing time, obtaining greater algorithmic and storage complexities. We analyse the Fermat factorization theorem and show why it can factorize numbers in constant time complexity (i.e. O(1)) if the difference is slight. We also derive the exact factorization bound for the Fermat factorization method. Furthermore, we link the Fermat factorization with integer sequences and raise further research questions on how our technique can be generalized to factoring N regardless of the difference between the two prime numbers p and q by linking the Fermat factorization theorem with integer sequences. In the end, We address each thought and raise further research questions showing that identifying formulas for some integer sequences can be the key to factoring any composite integer in a linear time.