# New Semi-Prime Factorization and Application in Large RSA Key Attacks
## 2 Related Works  
Several studies on the important problem of semi-primes factorization have challenged the intractability of RSA [16; 17; 18]. Since the first RSA cryptanalytic attack by [PERSON] in 1990 using the continued fractions method [19], several methods have been explored, such as the lattice reduction approach by [PERSON] [20] and by [PERSON] and [PERSON] [21], as well as the Boneh and Durfee attack on short decryption exponents of RSA [22]. More recent works approach the problem differently, with varying purposes, such as to find weak RSA keys for transport layer security (TLS) attacks [23], LogJam [24], or to factorize RSA keys for smartcard cryptography attacks on several devices [25].  
Historically, general-purpose factorization of integers can be dated back to the continued fraction factorization method (CFRAC) introduced by [PERSON] and [PERSON] in 1931 [26], which was later implemented as a computer algorithm by [PERSON] and [PERSON] [27]. Subsequently, [PERSON] and [PERSON] devised an improved algorithm [28], and such works including [PERSON]'s \(\rho\) algorithm [29] and [PERSON]'s quadratic sieve method (1985) [30] lay the foundation for generating interests in general-purpose factorization of integers in various applications. 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 [PERSON]'s methods [31] or using a small prime difference \(p_{1}-p_{2}\) method with [PERSON]'s original method [32]. The method by [PERSON]'s elliptic-curve method also serves as the state-of-the-art proposal for several future studies [33]. A recent work has considered [PERSON]'s \(4p-1\) method [34] to provide simplified and asymptotically deterministic versions, as it is similar to the well-known [PERSON]'s methods [35]. The study analyzed existing methods as the means of a potential backdoor for the RSA primes generated.  
A survey of the literature shows that the factorization problem of prime numbers is gaining research popularity, with a recent focus towards developing efficient mathematical techniques that are computationally faster and simpler [36; 37; 38; 39]. 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 [PERSON]'s method [47; 48]. Such a factorization method focusing on the special form of primes allows for an efficient factorization of RSA moduli. Thus, an adversary is motivated to subvert the prime generation to produce such RSA keys and could serve as a backdoor. With this view, we propose a new semi-prime factorization based on unique properties of Pythagorean triples with new mathematical theories and underlying patterns that are unexplored so far, with the purpose of advancing our research further in this direction.