The new integer factorization algorithm based on Fermat’s Factorization Algorithm and Euler’s theorem

Abstract

Although, Integer Factorization is one of the hard problems to break RSA, many factoring techniques are still developed. Fermat’s Factorization Algorithm (FFA) which has very high performance when prime factors are close to each other is a type of integer factorization algorithms. In fact, there are two ways to implement FFA. The first is called FFA-1, it is a process to find the integer from square root computing. Because this operation takes high computation cost, it consumes high computation time to find the result. The other method is called FFA-2 which is the different technique to find prime factors. Although the computation loops are quite large, there is no square root computing that included into the computation. In this paper, the new efficient factorization algorithm is introduced. Euler’s theorem is chosen to apply with FFA to find the addition result between two prime factors. The advantage of the proposed method is that almost of square root operations are left out from the computation while loops are not increased, they are equal to the first method. Therefore, if the proposed method is compared with the FFA-1, it implies that the computation time is decreased, because there is no the square root operation and the loops are same. On the other hand, the loops of the proposed method are less than the second method. Therefore, time is also reduced. Furthermore, the proposed method can be also selected to apply with many methods which are modified from FFA to decrease more cost.

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].…”

“…1) The experimental studies for most of these algorithms were implemented when the size of an odd composite was less than 64 bits [15,20,21,26,32], for example, n = 84449 × 21121 =1783647329. This number of bits is small compared to that required in cryptography.…”

“…This means that any comparison between Fermat's algorithm and these algorithms is not realistic because it has been proved that the Fermat factorization method is not efficient outside the DEF. For example,n=19710741 × 531349691=1047329636821139813 [20].…”

“…The second algorithm in this group is based on removing the perfect squaring operation by using modular multiplication. The method of modification is based on Euler's theorem which is given by the following formula [20]:…”