# New Semi-Prime Factorization and Application in Large RSA Key Attacks
## 1 Introduction  
Prime numbers have caught the attention of mathematicians since the work of [PERSON] due to their unfathomable structural properties. This paper leverages some elegant properties of prime numbers beyond their basic definition of being divisible by themselves and one only. They also possess the property of being randomly distributed, which is not exploited fully [1, 2, 3]. The current perception is that there appears to be only a limited understanding of their underlying structure, and several mathematicians are constantly trying to uncover the mysteries behind these prime numbers. There is still much to be carried out, and areas of further interest are channeled towards a better understanding of the structure of primes for arriving at faster prime number generating algorithms and faster solutions to the prime factorization problem [4, 5, 6, 7]. There is a need for generating more robust primes that are less susceptible to known factorization methods. In this paper, we draw attention to creating new approaches for the prime factorization of large primefactors or the semi-prime factorization problem. The focus of this research problem has an impact worldwide due to its practical application in digital communication, and in particular, the associated information security attacks and privacy challenges of today [8, 9]. In information security, large semi-primes have applications in encryption algorithms. They are used for generating public keys and private keys, such as in the Rivest-Shamir-Adleman (RSA) cryptosystems [10]. The property that the prime factorization of large numbers is a challengingly difficult task is well utilized in RSA-based encryption algorithms. Due to the dominant application of the RSA public-key primitive in cryptography, the security of RSA has been extensively analyzed for various attack scenarios [11]. In this paper, we take a modest step further by proposing new methods of semi-prime factorization of the RSA primitive. Previous research [1] proved that the semi-primes can be represented as the sum of four squares. A new factorization method, by exploiting the relationship among the four squares, was proposed as a faster alternative to [PERSON]'s method, as given in [12]. The purpose of this paper is to explore the topic of factorization further with a computationally simple approach for applications in RSA cryptography. In earlier work [1], we showed that a semi-prime \(N\) could be constructed from two primes, \(p_{1}\) and \(p_{2}\), in accordance with [PERSON]'s Christmas theorem, as given in [13]. In other words, [PERSON] stated that an odd prime \(p\) can be represented as the sum of two squares of integers \(x_{1}\) and \(x_{2}\), if and only if \(1\pmod{4}\equiv p=x_{1}^{2}+x_{2}^{2}\). Hence, this determined which numbers can be represented as the sum of two squares and was later proved by [PERSON] [12]. It is also mathematically represented that the semi-prime product is congruent to \(1\pmod{4}\equiv p_{1}p_{2}\). This paper advances further by applying the Brahmagupta-Fibonacci identity uniquely to establish that the product of these two primes (which is a sum of four squares) could be mathematically reduced to two sums of two squares [14]. In other words, we prove mathematically how the Brahmagupta-Fibonacci identity reduces these four Pythagorean quadruples to two Pythagorean triples. Hence, our proposed factorization method based on this property would be computationally successful whenever the semi-prime, a product of two primes, is such that both factors are congruent to \(1\) modulus \(4\). In this work, we leverage on the gaps found in the literature towards providing a novel proposal for semi-prime factorization. While there are several properties of Pythagorean triples, new patterns based on these properties are yet to be researched in the context of semi-prime factorization [15]. The main contributions of the paper are envisaged via the key features of our proposed factorization method, as listed below:  
1.