An Original Numerical Factorization Algorithm

This paper presents a mathematical algorithm that supports the identification of prime numbers the product of which results in a given number. The associated research has found several dissimilar factorization algorithms. Considering that, none of the found algorithms resembles the one presented here; it is the author's belief that the presented algorithm is original. Using the algorithm presented in this paper, factorization is not a trial and error process, but a finite step by step process that can be very easily programmed and executed. Consequently, the ability to easily factorize a given number will disrupt existing numerical methodologies, while opening up possibilities for new cryptosyntesis and cryptanalysis areas of research and applications.

less than . . .(the number to be factorized, and until all of the primes are found which) . . .when multiplied together . .." produce the number to be factorized [1].
In cryptography and in other numerical applications, factorization -because of its difficulty -has been a major asset.The development of this algorithm will force cryptographers to work harder, while opening new horizons in numerical analysis, especially in data compression.

Review of Related Literature
Considering the abundance of applications that employ the factorization of large integers, especially in cryptography, there are similarly numerous factorization methods.The most commonly known are briefly described in

Williams' p + 1
The method is limited by a number of parametric constraints, explained in the reference

Elliptic Curve
It is believed that this method ". . .turns out to be good enough in some cases . .." not supporting the factorization of any integer number.[2 p7].

Continued Fractions
For the factorization of large numbers ". . .we need to go through high precision calculations, which could also be costly."[2 p8]

Quadratic Sieve
In the factorization mathematics ". . . the matrix we need to solve, in the linear algebra part, is sparse," posing a variety of problems.Furthermore, ". . .we note that there are also theoretic sub-cubic methods . . .but these aren't efficient in practice."[2 p10] A very thorough study on integer factorization algorithms concludes with the statement: "There are no known algorithms which can factor arbitrary large integers efficiently" [5].With the factorization algorithm presented in this paper we may say that, now there is an algorithm that can factor arbitrary large integers efficiently.

The Developed Algorithm
The basic principle of the factorization algorithm, presented in this paper, is to treat the given integer, N, as the product of two numbers X and Y, where N=X*Y, or Y=N/X.The algorithm tracks the Y=X/N curve in unity steps, and stops when X*Y=N.
The algorithm uses the square root of N as a starting point climbing upward along the curve.
Our aim in this climb is to reach either X*Y=N or X=1.In the first case, we stop and we evaluate each X and Y value separately to determine if either is a prime number.In the second case, we have successfully reached the end of the process having determined that Y is a prime number.Fig. 1 illustrates the general flowchart of the presented factorization algorithm.
In the climbing up on the Y=N/X curve, when the X&Y point falls in the X*Y<N area, Y is incremented Y in order for the X&Y point to get closer to the curve, when the X&Y point falls in the X*Y>N area, X is decremented to get closer to the curve.
The presented algorithm is an iterative process using a step by step approach.Each iteration produces two factors, identified as X and Y.If X*Y=P, we stop.If X=1, then Y=N and N is a prime number.If X<>1, no conclusion is drawn, and the X and Y values are individually analyzed.It should be noted that, at the starting point of the iteration, the values of the X&Y pair are the integers most close to R=sqrt(N), the value of the square root of N, where X<R and Y>R.
Should R be an integer itself, then X=Y and X*Y=N, with N being a perfect square.In this case, R is being evaluated to determine if in itself is a prime number.From this initial point on, the value of X is steadily decremented and that of Y is steadily incremented.The X*Y product is continuously tested against the value of N, and when X*Y=N and the X value reaches unity, then we conclude that the corresponding Y is a prime number.

An Example
In this section, there is an example demonstrating the use of the above factorization algorithm and the logic behind it.Here, N is assumed to be 19.Table Two lists the coordinates of the points, as we climb around curve X*Y=N.
Fig. 2 shows the path of the X&Y pair ascending along the N=19=X*Y or Y=19/X curve, where the square root of N is R=4.35889...., identified as Point Zero.The starting point, or Point One, is the X&Y integral pair nearest to the X=Y=sqrt(N)=R point, where X<R and Y>R.Thus, in this case, the starting coordinates become X<R or X=4 and Y>R or Y=5.
At Point One, P=X*Y=4*5=20.Because the point falls in the X*Y>N (right hand side) area, the next step is to decrement X creating Point Two.The zig-zag continues ending up in one of two options..One is the case where N is a prime number, as in Fig. 1, where N=19.The other case is where product X*Y=N but X<>1, and we need to analyze X and Y individually.

Conclusion
The development of a simple and reliable algorithm for the factorization of any integer, as presented here, creates new research opportunities in numerical analysis, cryptography and in data compression and data representation.Factorization is the backbone of most cryptographic protocols.The availability of the presented factorization algorithm bases cryptography less on computing power, and more on cryptographic ingenuity.
______________________________________________________________________________________________________________ ______________ George Kostopoulos (2016), Journal of Information Assurance & Cybersecurity, DOI: 10.5171/2016.775081 If in the current step product X*Y is than N, in the following step the value of X is decremented.If in the current step product X*Y is less than N, in the following step the value of Y is incremented.When X*Y reaches N, the process stops.If X=1, as it is in this case, Y is a prime number.If X>1, N is the product of X and Y, and X and Y are individually assessed to determine their prime status.Journal of Information Assurance & Cyber security ______________________________________________________________________________________________________________ ______________ George Kostopoulos (2016), Journal of Information Assurance & Cybersecurity, DOI: 10.5171/2016.775081

Table 1 : Most Known Factorization Methods Algorithm Name Comments Pollard
's rho ". . .given N ≈ √ 2p random integers, we expect there to be a pair x, y which leads to a factorization of p.The