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

Abstract

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.

Cracking Factorization- Cryptography Challenges

“…In this work, we propose an innovative approach to the factorization problem, utilizing the gradient descent method, which, we hope, will open new horizons in the research of this area [4,5].…”

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

“…In the past two decades, integer factorization algorithms have evolved by improving existing ones to a great extent such that very large semi-primes of more than 250 decimal digits can be factorized with sufficient computing power [14][15][16][17]. Some of the famous integer factorization algorithms are Lenstra's elliptic curve algorithm [18], Pomerance's quadratic sieve, and the General Number Field Sieve [19,20].…”

“…We provide the practical implementation and results of our experimentation conducted using small and large semi-primes as well as RSA250, which has 250 decimal digits (829 bits). In addition, our modest contribution includes considering a general form of the Brahmagupta-Fibonacci identity [33] and applying it to the difference of squares, which extends our earlier work on the sum of squares approach for semi-prime factorization [15,17].…”

Mathematically Based Assessment of the Accuracy of Protection of Cardiac Data Realized with the Help of Cryptography and Steganography

“…The decomposition of prime numbers has been the subject of much research looking for fast and simple techniques [7,8], including the creation of polynomials generating sums of squares with a targeted application in cryptography [9,10]. The authors of [11] propose a new method for realizing a semisimple factorization based on the properties of Pythagorean triplets, proposing a new mathematical model based on the binary approach for the greatest common divisor with simple arithmetic operations to find the sum of two squares of one or both prime factors.…”

“…Another commonly used mathematical method for the encryption process is elliptic curve cryptography, an approach to provide public key encryption that is based on the algebraic structure of elliptic curves on finite fields [12]. Security encryption systems based on public key cryptography, such as RSA systems, use semiprime factorization [11], an important numerical method. Elliptic curve cryptography uses smaller-length keys to implement the same level of security as the RSA algorithm [13].…”