Possible Prime Modified Fermat Factorization: New Improved Integer Factorization to Decrease Computation Time for Breaking RSA

Somsuk, Kritsanapong; Kasemvilas, Sumonta

doi:10.1007/978-3-319-06538-0_32

Cited by 6 publications

(6 citation statements)

References 5 publications

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“…Nowadays, many improvements of FFA-1 were proposed to reduce l 1 such as [16][17][18]. However, the disadvantage of FFA-1 and it's improving algorithms are about computing square root of integer.…”

Section: Related Work 21 Fermet's Factorization Algorithm: Ffamentioning

confidence: 99%

“…In addition, the other form of modulus which is equal to the difference between two perfect square numbers is considered instead of the form of the production between two prime numbers. In fact, many improvement of FFA algorithms [16][17][18][19][20][21] were presented to reduce loops. In 2017, Specific Fermat's Factorization Algorithm Considered from X (SFFA-X) [22] was proposed to reduce the time complexity.…”

Section: Introductionmentioning

confidence: 99%

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

Somsuk¹

2020

IJECE

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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.

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Section: Related Work 21 Fermet's Factorization Algorithm: Ffamentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Somsuk¹

2020

IJECE

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“…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.…”

Section: = Modmentioning

confidence: 99%

“…In general, the improved Fermat algorithms can be classified into two classes. The first class contains algorithms based on the concept of an estimated prime factor and uses different techniques such as continued fraction method [28] or considering n as a special form 6 ± 1, where k is any integer [23]. However, the techniques in this class cannot factor some odd composite numbers, so they cannot be considered as general methods for Fermat factorization.…”

Section: = Modmentioning

confidence: 99%

Performance Analysis of Fermat Factorization Algorithms

Bahig¹,

Mohammed²,

Khaled³

et al. 2020

IJACSA

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“…A modified version of these algorithms is presented in [15]. In paper [16], the method for determining that the square root is not an integer without the procedure of root calculation was proposed.…”

Section: Literature Review and Problem Statementmentioning

confidence: 99%

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

Vynnychuk

Maksymenko

Romanenko

2018

EEJET

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Метод Ферма вважається кращим при факторизацiї чисел N=pq у випадку близьких p i q. Обчислювальна складнiсть базового алгоритму методу визначається кiлькiстю пробних значень Х при вирiшеннi рiвняння Y 2 =X 2-N, а також складнiстю арифметичних операцiй. Для її зниження запропоновано в якостi допустимих розглядати тi з пробних Х, для яких (X 2-N)modbb є квадратним залишком по модулю bb, названого базовым. При використаннi базової основи модуля bb число пробних Х зменшується в число раз, близьке до Z(N,bb)= =bb/bb*, де bb*-число елементiв множини Т коренiв рiвняння (Ymodb)2modb=((Xmodb) 2-Nmodb)modb, а Zкоефiцiєнт прискорення. Визначено, що на величину Z(N,bb) впливають значения залишкiв Nmodp (при р=2 використовуються залишки Nmod8). Запропоновано постановку задачi пошуку bb з максимальным Z(N,bb) при обмеженях на обсяг пам'ятi ЕОМ, де визначаються показники степенiв простих чисел-множникiв bb, та спосiб її вирiшення. Для зменшення числа арифметичних операцiй з великими числами попонується замiсть таких виконувати операцiї зi значеннями рiзниць мiж найближчими значеннями елементiв множини Т. Тодi арифметичнi операцiї множення i додавання з великими числами виконуються рiдко. А якщо квадратний корiнь з X 2-N визначати тiльки у випадках, коли значення (X 2-N)modb будуть квадратними залишками для багатьох рiзних основ модуля b, то обчислювальною складнiстю цiєї операцiї можна знехтувати. Встановлено, що тодi запропонований модифiкований алгоритм методу Ферма для чисел 2 1024 забезпечує зниження обчислювальної складностi в порiвняннi з базовим алгоритмом в середньому в 10 7 раз Ключовi слова: факторизацiя, метод Ферма, обчислювальна складнiсть, базова основа, прорiджування, квадратнi залишки

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Possible Prime Modified Fermat Factorization: New Improved Integer Factorization to Decrease Computation Time for Breaking RSA

Cited by 6 publications

References 5 publications

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

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

Performance Analysis of Fermat Factorization Algorithms

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

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