An Attack Bound for Small Multiplicative Inverse of φ(N) mod e with a Composed Prime Sum p + q Using Sublattice Based Techniques

Kameswari, P. Anuradha; Lambadi, Jyotsna

doi:10.3390/cryptography2040036

Cited by 2 publications

(3 citation statements)

References 9 publications

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“…Some early studies have considered polynomial equations to study low exponent RSA vulnerabilities and their variants [61,62]. Recent number theory-based studies in this domain have focused on lattice reduction (LR) techniques with prime number theory to study RSA key attacks [31,38,39]. Such LR based techniques come under the general category of Coppersmith attacks.…”

Section: Discussionmentioning

confidence: 99%

“…Most of the recent research has also been focusing on extending the number range upper bound of d and e in the RSA private and public keys by working on the limitation of the Wiener and Coppersmith methods by approaching the problem differently. One recent work [39] considers the prime sum p 1 + p 2 using sublattice reduction techniques and Coppersmith's methods for finding small roots of modular polynomial equations, achieving slight improvement in the upper bound with reduced lattice dimension. Another work [17] uses a small prime difference p 1 − p 2 method which is then developed into a continuous fraction as per Wiener's original method.…”

Section: Introductionmentioning

confidence: 99%

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Mathematical Attack of RSA by Extending the Sum of Squares of Primes to Factorize a Semi-Prime

Overmars¹,

Venkatraman²

2020

MCA

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The security of RSA relies on the computationally challenging factorization of RSA modulus with being a large semi-prime consisting of two primes for the generation of RSA keys in commonly adopted cryptosystems. The property of both congruent to 1 mod 4, is used in Euler’s factorization method to theoretically factorize them. While this caters to only a quarter of the possible combinations of primes, the rest of the combinations congruent to 3 mod 4 can be found by extending the method using Gaussian primes. However, based on Pythagorean primes that are applied in RSA, the semi-prime has only two sums of two squares in the range of possible squares . As becomes large, the probability of finding the two sums of two squares becomes computationally intractable in the practical world. In this paper, we apply Pythagorean primes to explore how the number of sums of two squares in the search field can be increased thereby increasing the likelihood that a sum of two squares can be found. Once two such sums of squares are found, even though many may exist, we show that it is sufficient to only find two solutions to factorize the original semi-prime. We present the algorithm showing the simplicity of steps that use rudimentary arithmetic operations requiring minimal memory, with search cycle time being a factor for very large semi-primes, which can be contained. We demonstrate the correctness of our approach with practical illustrations for breaking RSA keys. Our enhanced factorization method is an improvement on our previous work with results compared to other factorization algorithms and continues to be an ongoing area of our research.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Mathematical Attack of RSA by Extending the Sum of Squares of Primes to Factorize a Semi-Prime

Overmars¹,

Venkatraman²

2020

MCA

View full text Add to dashboard Cite

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“…In the context of RSA cryptography applications, recent works have considered such known factorization methods to focus on security parameters such as the length of the prime factors p 1 and p 2 , of the RSA modulus n = p 1 p 2 , or other structural properties of the primes. Modifications of existing methods have become popular recently, such as using the prime sum p 1 + p 2 with sublattice reduction techniques and Coppersmith's methods [31] or using a small prime difference p 1 − p 2 method with Wiener's original method [32]. The method by Lenstra's elliptic-curve method also serves as the state-of-the-art proposal for several future studies [33].…”

Section: Related Workmentioning

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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An Attack Bound for Small Multiplicative Inverse of φ(N) mod e with a Composed Prime Sum p + q Using Sublattice Based Techniques

Cited by 2 publications

References 9 publications

Mathematical Attack of RSA by Extending the Sum of Squares of Primes to Factorize a Semi-Prime

Mathematical Attack of RSA by Extending the Sum of Squares of Primes to Factorize a Semi-Prime

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

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