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

Overmars, Anthony; Venkatraman, Sitalakshmi

doi:10.3390/mca25040063

Cited by 11 publications

(9 citation statements)

References 45 publications

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“…The RSA algorithm is a commonly used solution in various types of security applications. Despite the difficulty in breaking RSA keys, RSA algorithms still suffer varied attacks [12,13].…”

Section: The Rsa Algorithmmentioning

confidence: 99%

Small Prime Divisors Attack and Countermeasure against the RSA-OTP Algorithm

Sarna

Czerwiński

2021

Electronics

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One-time password algorithms are widely used in digital services to improve security. However, many such solutions use a constant secret key to encrypt (process) one-time plaintexts. A paradigm shift from constant to one-time keys could introduce tangible benefits to the application security field. This paper analyzes a one-time password concept for the Rivest–Shamir–Adleman algorithm, in which each key element is hidden, and the value of the modulus is changed after each encryption attempt. The difference between successive moduli is exchanged between communication sides via an unsecure channel. Analysis shows that such an approach is not secure. Moreover, determining the one-time password element (Rivest–Shamir–Adleman modulus) can be straightforward. A countermeasure for the analyzed algorithm is proposed.

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“…The RSA algorithm is a commonly used solution in various types of security applications. Despite the difficulty in breaking RSA keys, RSA algorithms still suffer varied attacks [12,13].…”

Section: The Rsa Algorithmmentioning

confidence: 99%

Small Prime Divisors Attack and Countermeasure against the RSA-OTP Algorithm

Sarna

Czerwiński

2021

Electronics

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“…Recent research interest in polynomials, which generate sums of squares, has featured applications to cryptography [40][41][42][43][44][45]. In this context, our previous works leverage the semi-prime representation as the sum of four squares [1,46] with an enhancement to Euler's method [47,48]. Such a factorization method focusing on the special form of primes allows for an efficient factorization of RSA moduli.…”

Section: Related Workmentioning

confidence: 99%

“…Table 1 demonstrates the application of our proposed semi-prime factorization method for a key length of 768 decimal digits, denoted as RSA-768. The sums of squares and polynomials have been explored for semi-prime factorization in previous research works [1,46,69]. However, there are more than 50 properties of Pythagorean triples that have been reported and new patterns yet to be explored [15,70].…”

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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“…The most efficient of factorization attack to standard RSA algorithm is the General Number Field Sieve (GNFS) method [10], [16], [62]- [65]. Granger et al [66] provide an in-depth report in 2009 on the factorization of the 768-bit number RSA-768 by the number field sieve (NSF) factoring method and provide some implications for the RSA [8], [65]. The largest such semi-prime yet factored was RSA-250, an 829bit number with 250 decimal digits, in February 2020.…”

Section: B Security Comparisonmentioning

confidence: 99%

Eight Prime Numbers of Modified RSA Algorithm Method for More Secure Single Board Computer Implementation

Hermawan¹,

Winarko²,

Ashari³

2021

International Journal on Advanced Science, Engineering and Information Technology

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RSA is the most popular public-key cryptography. The main strength of the algorithm is based on the difficulty of factoring in a large integer number. RSA has also been applied in a system with limited resource environments like single-board computers (SBC). To ensure data security, a recommendation to use a key size longer than 2048 bits generates challenges for implementing RSA in the SBC. This research proposes an EPNR (Eight Prime Numbers of Modified RSA) method, a modified double RSA based on eight prime numbers combined with the CRT method, to speed up the random key generation and decryption mechanism. The method is implemented in a Raspberry Pi 4 Model B+. The running time and security performances of the EPNR were analyzed and compared to the other models. Compared to the others model based on the standard RSA scheme, the proposed model is faster 21.78 times in a random key generation, 9.03 times in encryption and decryption processing. The EPNR has resistance to Wiener, statistical, and factorization attacks (GNFS and Fermat). Using standard RSA in the second encryption mechanism, the GNFS is not yet effective for attacking the proposed model. The modified Fermat Factorization algorithm is more difficult and needed more extra times for factoring a large composite number into eight prime numbers correctly. The method will be useful for implementing certificates authentication and distribution of the secret key. It is very suitable to enhance more secure RSA implementation in an SBC environment.

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

Cited by 11 publications

References 45 publications

Small Prime Divisors Attack and Countermeasure against the RSA-OTP Algorithm

Small Prime Divisors Attack and Countermeasure against the RSA-OTP Algorithm

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

Eight Prime Numbers of Modified RSA Algorithm Method for More Secure Single Board Computer Implementation

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