The Integer Factorization Algorithm With Pisano Period

Wu, Liangshun; Cai, Hengjin; Gong, Zexi

doi:10.1109/access.2019.2953755

Cited by 10 publications

(6 citation statements)

References 16 publications

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“…Moreover, if 𝑘 = 1 and 𝑏 = −1, then 𝑡 𝑛 is a Mersenne number 𝑀 𝑛 . 22 This test is considered a deterministic primality test for Mersenne numbers. It is based on the recursive Lucas sequence in the special case {𝑃, 𝑄} = {4,1}.…”

Section: Remarkmentioning

confidence: 99%

Comparative Study Between a Novel Deterministic Test for Mersenne Primes and the Well-Known Primality Tests

2023

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In this article, a new deterministic primality test for Mersenne primes is presented. It also includes a comparative study between well-known primality tests in order to identify the best test. Moreover, new modiﬁcations are suggested in order to eliminate pseudoprimes. The study covers random primes such as Mersenne primes and Proth primes. Finally, these tests are arranged from the best to the worst according to strength, speed, and effectiveness based on the results obtained through programs prepared and operated by Mathematica, and the results are presented through tables and graphs.

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Section: Remarkmentioning

confidence: 99%

Comparative Study Between a Novel Deterministic Test for Mersenne Primes and the Well-Known Primality Tests

2023

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“…A recent experimental study employed the representation of primes in the form p = 6 • x ± 1 and applied the theory to the RSA factorization problem [73]. Another work shows the decomposition of the two prime numbers with the Pisano period factorization method, which has been proven to be a subexponential complexity method [74]. Several integer factorization methods have also suggested direct application to cryptanalysis of RSA by applying different genetic algorithms [75].…”

Section: Application Example 1 Let Us Consider Case Example 2 With N = 377mentioning

confidence: 99%

New Semi-Prime Factorization and Application in Large RSA Key Attacks

Overmars¹,

Venkatraman²

2021

JCP

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Semi-prime factorization is an increasingly important number theoretic problem, since it is computationally intractable. Further, this property has been applied in public-key cryptography, such as the Rivest–Shamir–Adleman (RSA) encryption systems for secure digital communications. Hence, alternate approaches to solve the semi-prime factorization problem are proposed. Recently, Pythagorean tuples to factor semi-primes have been explored to consider Fermat’s Christmas theorem, with the two squares having opposite parity. This paper is motivated by the property that the integer separating these two squares being odd reduces the search for semi-prime factorization by half. In this paper, we prove that if a Pythagorean quadruple is known and one of its squares represents a Pythagorean triple, then the semi-prime is factorized. The problem of semi-prime factorization is reduced to the problem of finding only one such sum of three squares to factorize a semi-prime. We modify the Lebesgue identity as the sum of four squares to obtain four sums of three squares. These are then expressed as four Pythagorean quadruples. The Brahmagupta–Fibonacci identity reduces these four Pythagorean quadruples to two Pythagorean triples. The greatest common divisors of the sides contained therein are the factors of the semi-prime. We then prove that to factor a semi-prime, it is sufficient that only one of these Pythagorean quadruples be known. We provide the algorithm of our proposed semi-prime factorization method, highlighting its complexity and comparative advantage of the solution space with Fermat’s method. Our algorithm has the advantage when the factors of a semi-prime are congruent to 1 modulus 4. Illustrations of our method for real-world applications, such as factorization of the 768-bit number RSA-768, are established. Further, the computational viabilities, despite the mathematical constraints and the unexplored properties, are suggested as opportunities for future research.

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“…To give access to the S1 robot, we modified the classical El Gamal algorithm with a split private key [ 34 ]. The description of this algorithm is further presented.…”

Section: Proposed Solutionmentioning

confidence: 99%

Method to Increase Dependability in a Cloud-Fog-Edge Environment

Stan

Enyedi

Corches

et al. 2021

Sensors

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Robots can be very different, from humanoids to intelligent self-driving cars or just IoT systems that collect and process local sensors’ information. This paper presents a way to increase dependability for information exchange and processing in systems with Cloud-Fog-Edge architectures. In an ideal interconnected world, the recognized and registered robots must be able to communicate with each other if they are close enough, or through the Fog access points without overloading the Cloud. In essence, the presented work addresses the Edge area and how the devices can communicate in a safe and secure environment using cryptographic methods for structured systems. The presented work emphasizes the importance of security in a system’s dependability and offers a communication mechanism for several robots without overburdening the Cloud. This solution is ideal to be used where various monitoring and control aspects demand extra degrees of safety. The extra private keys employed by this procedure further enhance algorithm complexity, limiting the probability that the method may be broken by brute force or systemic attacks.

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The Integer Factorization Algorithm With Pisano Period

Cited by 10 publications

References 16 publications

Comparative Study Between a Novel Deterministic Test for Mersenne Primes and the Well-Known Primality Tests

Comparative Study Between a Novel Deterministic Test for Mersenne Primes and the Well-Known Primality Tests

New Semi-Prime Factorization and Application in Large RSA Key Attacks

Method to Increase Dependability in a Cloud-Fog-Edge Environment

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