Small Private Exponent Attacks on RSA Using Continued Fractions and Multicore Systems

Bahig, Hatem M.; Nassr, Dieaa I.; Mahdi, Mohammed A.; Bahig, Hazem M.

doi:10.3390/sym14091897

Cited by 6 publications

(3 citation statements)

References 27 publications

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“…Public key encryption uses Euler's totient function [56,57], which provides the basis for factoring algorithms that have a major impact on cybersecurity. Hence, past works on number theory, primality testing, and the implementation of algorithms [58][59][60][61][62][63] have been the backbone of this research work to arrive at a new method. Further, the speeding up of existing methods using more efficient methods and the adoption of more computing power, including parallelism [64][65][66][67], have fueled this ongoing research.…”

Section: Improvements and Future Researchmentioning

confidence: 99%

Continued Fractions Applied to the One Line Factoring Algorithm for Breaking RSA

Overmars,

Venkatraman

2024

JCP

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The RSA (Rivest–Shamir–Adleman) cryptosystem is an asymmetric public key cryptosystem popular for its use in encryptions and digital signatures. However, the Wiener’s attack on the RSA cryptosystem utilizes continued fractions, which has generated much interest in developing competitive factoring algorithms. A general-purpose integer factorization method first proposed by Lehmer and Powers formed the basis of the well-known Continued Fraction Factorization (CFRAC) method. Recent work on the one line factoring algorithm by Hart and its connection with Lehman’s factoring method have motivated this paper. The emphasis of this paper is to explore the representations of PQ as continued fractions and the suitability of lower ordered convergences as representations of ab. These simpler convergences are then prescribed to Hart’s one line factoring algorithm. As an illustration, we demonstrate the working of our approach with two numbers: one smaller number and another larger number occupying 95 bits. Using our method, the fourth convergence finds the factors as the solution for the smaller number, while the eleventh convergence finds the factors for the larger number. The security of the RSA public key cryptosystem relies on the computational difficulty of factoring large integers. Among the challenges in breaking RSA semi-primes, RSA250, which is an 829-bit semi-prime, continues to hold a research record. In this paper, we apply our method to factorize RSA250 and present the practical implementation of our algorithm. Our approach’s theoretical and experimental findings demonstrate the reduction of the search space and a faster solution to the semi-prime factorization problem, resulting in key contributions and practical implications. We identify further research to extend our approach by exploring limitations and additional considerations such as the difference of squares method, paving the way for further research in this direction.

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Section: Improvements and Future Researchmentioning

confidence: 99%

Continued Fractions Applied to the One Line Factoring Algorithm for Breaking RSA

Overmars,

Venkatraman

2024

JCP

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“…The complexity of the integer factorization issue affects the security of several public key cryptosystems such as [11,20,23,25,36], while the exponentiation problem determines the effectiveness of such cryptosystems [9,10,34].…”

Section: Introductionmentioning

confidence: 99%

Accelerating the Wheel Factoring Techniques

Zaki¹,

Bakr²,

Alsahangiti³

et al. 2023

Preprint

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The efficiency with which an integer may be factored into its prime factors determines several public key cryptosystems′ security in use today. Although there is a quantum-based technique with a polynomial time for integer factoring, on a traditional computer, there is no polynomial time algorithm. We investigate how to enhance the wheel factoring technique in this paper. Current wheel factorization algorithms rely on a very restricted set of prime integers as a base. In this study, we intend to adapt this notion to rely on a greater number of prime integers, resulting in a considerable improvement in the execution time. The experiments on composite numbers n reveal that the proposed algorithm improves on the existing wheel factoring algorithm by about 75%

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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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Small Private Exponent Attacks on RSA Using Continued Fractions and Multicore Systems

Cited by 6 publications

References 27 publications

Continued Fractions Applied to the One Line Factoring Algorithm for Breaking RSA

Continued Fractions Applied to the One Line Factoring Algorithm for Breaking RSA

Accelerating the Wheel Factoring Techniques

Efficient Sequential and Parallel Prime Sieve Algorithms

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