MFFV2 and MNQSV2: Improved Factorization Algorithms

Abstract

we propose a method to decrease processing time for two factorization algorithms. One is Modified Fermat Factorization Version 2 (MFFV2) modified from Modified Fermat Factorization (MFF). The other is Modified Non -sieving Quadratic Sieve Version 2 (MNQSV2) modified from Modified Non -Sieving Quadratic Sieve (MNQS). A key concept of this method is to decrease processing time to compute an integer's square root. This method can be used with all factorization algorithms which the modulus is written as the difference of squares. The experiments showed that the speed of MFFV2 increases when compares with MFF and the speed of MNQSV2 increases when compares with MNQS. In addition, if two primes' differences are small, the factorization speed of MNQSV2 is faster than the factorization speed of MFFV2.

Performance Analysis of Fermat Factorization Algorithms

“…Many algorithms have been proposed that are based on Fermat's factorization concept [13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31]. The goal of these algorithms is to improve the running time of the original Fermat algorithm in finding prime factors.…”

“…However, the techniques in this class cannot factor some odd composite numbers, so they cannot be considered as general methods for Fermat factorization. The second class contains algorithms [11,14,15,17,18,19,20,21,22,24,25,26,27,29] that can be applied to any odd composite number and are based on (1) replacing the high-cost operation, i.e., the perfect square in Fermat's method, with a low-cost operation or on (2) reducing the space searched to find the solution. It should also be noted that there is another strategy [13,30] that falls outside the scope of our research, which involves speeding up the running time of Fermat's algorithm that is based on a different platform such as high-performance computing [13,33].…”

“…3) The efficiency of some of these algorithms was measured based on a few data or on some examples, rather than on different values for b and ∆, see for example, [22,25]. This means that there is no exhaustive study that compares two or more integer factorization algorithms that are based on Fermat factorization concept by applying them to different data distributions in the DEF.…”

“…Ignoring the perfect square a) Sieving with modulus: One of the techniques used in sieving is the modulus operation, mod. The idea behind using the modulus arithmetic operation is to exclude all integers, x, that are definitely not perfect squares before applying the perfect squaring operation [22,24].…”

Application of the basic module's foundation for factorization of big numbers by the Fеrmаt method

“…The options of the solution of the second problem were proposed in [13,14], where a reduction in the number of operations of square root calculation is achieved by the results of analysis of least significant bits y 2 . A modified version of these algorithms is presented in [15].…”