Determination of a Good Indicator for Estimated Prime Factor and Its Modification in Fermat’s Factoring Algorithm

Tahir, Rasyid Redha Mohd; Asbullah, Muhammad Asyraf; Ariffin, Muhammad Rezal Kamel; Mahad, Zahari

doi:10.3390/sym13050735

Cited by 7 publications

(4 citation statements)

References 16 publications

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“…The various sizes of n included in this experiment are 64 bits, 128 bits, 256 bits 512 bits, 1024 bits, and 2048 bits. However, all prime factors of p − 1 or q − 1 are the members of B = {2, 3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59, 61, 67, 71, 73, 79, 83, 89, 97}, a prime number which is between 1 and 100. In addition, IPP1_V2 is compared to Pollard's p − 1 [26] and Pollard's p − 1 [28].…”

Section: Resultsmentioning

confidence: 99%

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

Somsuk

2022

Symmetry

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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)

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Section: Resultsmentioning

confidence: 99%

“…However, this method, which is called the estimated prime factor (EPF) [10], cannot be chosen to solve n derived from the balanced primes. In 2021, EPF was improved [11] to solve n which is generated from unbalanced primes or balanced primes. Therefore, the modified method can be chosen to recover p and q without the mistake.…”

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confidence: 99%

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

Somsuk

2022

Symmetry

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“…In addition, a technique known as an estimated prime factor (EPF) was improved by Tahir et. al [7] to solve N generated from balanced or unbalanced primes p and q. Furthermore, Pollard [8] showed that N with a small size is easily factored since the complexity of the factorization algorithm depends on the size of √ N. Subsequently, research undertaken by many others [9][10][11] extended this complexity using the number field sieve method, which has dominated efforts to factor the RSA modulus ever since.…”

Section: Introductionmentioning

confidence: 99%

On (Unknowingly) Using Near-Square RSA Primes

et al. 2022

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The invention in 1978 of the first practical asymmetric cryptosystem known as RSA was a breakthrough within the long history of secret communications. Since its inception, the RSA cryptosystem has become embedded in millions of digital applications with the objectives of ensuring confidentiality, integrity, authenticity, and disallowing repudiation. However, the generation of the RSA modulus, N=pq which requires p and q to be random primes, may accidentally entail the choice of a special type of prime called a near-square prime. This structure of N may be used unknowingly en masse in real-world applications since no current cryptographic implementation prevents its generation. In this study, we show that use of this type of prime will potentially lead to total destruction of RSA. We present three cases of near-square primes used as RSA primes, set in the form of (i) N=pq=(am−ra)(bm−rb); (ii) N=pq=(am+ra)(bm−rb); and (iii) N=pq=(am−ra)(bm+rb). Although (ii) and (iii) are quite similar, p and q must be within the same size range of n-bits, which results in different conditions for both cases. We formulate attacks using three different algorithms to better understand their feasibility. We also provide an efficient countermeasure that it is recommended is adopted by current cryptographic libraries with RSA implementation.

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“…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.…”

Section: Introductionmentioning

confidence: 99%

Efficient Sequential and Parallel Prime Sieve Algorithms

et al. 2022

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Generating prime numbers less than or equal to an integer number m plays an important role in many asymmetric key cryptosystems. Recently, a new sequential prime sieve algorithm was proposed based on set theory. The main drawback of this algorithm is that the running time and storage are high when the size of m is large. This paper introduces three new algorithms for a prime sieve based on two approaches. The first approach develops a fast sequential prime sieve algorithm based on set theory and some structural improvements to the recent prime sieve algorithm. The second approach introduces two new parallel algorithms in the shared memory parallel model based on static and dynamic strategies. The analysis of the experimental studies shows the following results. (1) The proposed sequential algorithm outperforms the recent prime sieve algorithm in terms of running time by 98% and memory consumption by 80%, on average. (2) The two proposed parallel algorithms outperform the proposed sequential algorithm by 72% and 67%, respectively, on average. (3) The maximum speedups achieved by the dynamic and static parallel algorithms using 16 threads are 7 and 4.5, respectively. As a result, the proposed algorithms are more effective than the recent algorithm in terms of running time, storage and scalability in generating primes.

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Determination of a Good Indicator for Estimated Prime Factor and Its Modification in Fermat’s Factoring Algorithm

Cited by 7 publications

References 16 publications

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

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

On (Unknowingly) Using Near-Square RSA Primes

Efficient Sequential and Parallel Prime Sieve Algorithms

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