# Continued Fractions Applied to the One Line Factoring Algorithm for Breaking RSA
## 1 Introduction  
The RSA (Rivest-Shamir-Adleman) cryptosystems apply large semi-primes in an encryption algorithm for the generation of public keys and private keys to successfully secure an online communication [1, 2, 3]. The security of such cryptosystems depends on prime factorization of large numbers being a huge computing challenge [4, 5, 6]. Though the semi-prime factorization of large numbers has always interested mathematicians for centuries, the practical importance of preserving the security of these cryptosystems has led to exploration of the limitations of existing approaches towards the emergence of new methods, thereby redefining the cybersecurity landscape from time to time [7, 8, 9, 10, 11, 12, 13]. 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 [PERSON]'s elliptic curve algorithm [18], [PERSON]'s quadratic sieve, and the General Number Field Sieve [19, 20]. Not only mathematicians,but also computing professionals and organizations have been exploring the properties of primes for several decades. More recently, their research directions have moved towards developing efficient tests for primality with the key objective of creating several applications in the domain of information security, particularly the RSA public key cryptosystem. A sieve in this context refers to the process of searching for prime numbers among all the integers up to a prescribed bound [6; 20; 21]. The early practical building blocks of public-key cryptography are attributed to the Diffie-Hellman protocol over finite fields and the RSA cryptosystem. Though several other cryptographic primitives have entered the landscape, the RSA and finite field Diffie-Hellman methods are more prevalent in terms of their use for secure information transmission and key exchange in the computing field [22]. The key sizes in these primitives are much longer when compared to elliptic curve-based cryptosystems. Thus, finding semi-primes has been a challenge and has drawn immense attention among researchers looking to break the RSA cryptosystem [23; 24]. Our previous research work involved proposing several factorization methods including approaches using Pythagorean triples, sum of squares, and polynomials, as well as investigation of their impact on RSA attacks [25; 26; 27; 28; 29; 30]. Readers are encouraged to refer to our previous work and related published articles in the literature that provide the preliminary definitions and commonly used mathematical notations adopted in this work. In this paper, we take a modest step to further the method of continued fractions by applying one line factoring algorithms with an effect on semi-prime factorization of the RSA primitive. The main aim of this paper is to characterize k:kN as per [PERSON] [31] and apply this to [PERSON]'s one line factorization method [32]. Our research work is the first of its kind in exploring the relationship between the convergence of \(PQ\) and the attributes of \(k\), which is used in [PERSON]'s one line factorization algorithm and has applications in cryptography. Little is known about the attributes of \(k\), and our research provides a better understanding of its relationship with the convergence of a solution. This paper presents the theory of our factorization approach and illustrates the working of our method with examples for breaking keys of different sizes. Our key contribution lies in the reduction of the search space and the faster solution to the semi-prime factorization problem. 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).