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

Abstract

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)

On (Unknowingly) Using Near-Square RSA Primes

“…Before RSA was introduced, prior results had shown that p − 1 and q − 1 that have small factors cause p • q to be vulnerable when factored in polynomial time using the Pollard p − 1 algorithm [5]. Pollard's p − 1 algorithm is exceptionally efficient whenever all prime factors of p − 1 and q − 1 are small [6]. In addition, a technique known as an estimated prime factor (EPF) was improved by Tahir et.…”

“…From [24], we know that the classical modular reduction of modulo σ works at O(2 log 2 σ). Since σ is the potential value of (ab) m/2 , the maximum integers to find it are less than N γ 2 m/2 +1 2 , as shown in Equation (6). Based on this computation, we have the complexity of Attack I presented in Algorithm 2 to be…”

Efficient Sequential and Parallel Prime Sieve Algorithms

“…Many problems in number theory and computer arithmetic play important roles in cryptography. Examples of such problems are the generation of prime numbers [1][2][3], primality testing [4,5], modular exponentiation [6], addition chains and sequences [7,8] and integer factorization [9][10][11][12]. Developing fast algorithms that address these problems is one of the main challenges of algorithm complexity and leads to significant improvements in various applications.…”