An Efficient Variant of Pollard’s p − 1 for the Case That All Prime Factors of the p − 1 in B-Smooth

• Published in: Symmetry (Basel) Vol. 14; no. 2; p. 312
• Main Author: Somsuk, Kritsanapong
• Format: Journal Article
• Language: English
• Published: Basel MDPI AG 01.02.2022
• Subjects: Algorithms; Approximation; Cryptography; Factorization; Integers; Numbers; Polynomials; Quantum computers; Quantum computing; Run time (computers)
• ISSN: 2073-8994; 2073-8994
• DOI: 10.3390/sym14020312

Due to the computational limitations at present, there is no efficient integer factorization algorithm that can break at least 2048 bits of RSA with strong prime factors in polynomial time. Although Shor’s algorithm based on a quantum computer has been presented, the quantum computer is still in its early stages of the development. As a result, the integer factorization problem (IFP) is a technique that is still being refined. Pollard’s p − 1 is an integer factorization algorithm based on all prime factors of p − 1 or q − 1, where p and q are two distinct prime factors of modulus. In fact, Pollard’s p − 1 is an efficient method when all prime factors of p − 1 or q − 1 are small. The aim of this paper is to propose a variant of Pollard’s p − 1 in order to decrease the computation time. In general, the proposed method is very efficient when all prime factors of p − 1 or q − 1 are the members of B-smooth. Assuming this condition exists, the experimental results demonstrate that the proposed method is approximately 80 to 90 percent faster than Pollard’s p − 1. Furthermore, the proposed technique is still faster than Pollard’s p − 1 for some values of modulus in which at least one integer is a prime factor of p − 1 or q − 1 while it is not a member of B-smooth. In addition, it is demonstrated that the proposed method’s best-case running time is O(x),where x is represented as bits length of n.