SHOR’S ALGORITHM Another way of factorizing 15 appears to have been demonstrated by Scientists at IBM’s Almaden Research Center in 2001. They built a small quantum computer and used the following algorithm due to Peter Shor. Assume N is composite. Choose a < N , gcd(a,N) = 1. Find r, the period of the function f : x 7→ a mod N ; that is, find the smallest r > 0 for which a ≡ 1 (mod N). Assume r is even and a 6≡ ±1 (mod N); otherwise start again with a new a. Then, since (a − 1)(a + 1) = a − 1 ≡ 0 (mod N), gcd(a − 1, N) is a non-trivial factor of N . For example, N = 15, a = 2 gives r = 4, gcd(2 − 1, 15) = 3 and gcd(2 + 1, 15) = 5. The determination of r is particularly suited to quantum computing. For details, see [2, section 8.5.2], or look up ‘Shor’s algorithm’ in Wikipedia.

In the following article, we propose an integer factorization algorithm, based on sieving quadradic numbers. The algorithm does not use Fermat's factorization method based on finding a congruence of squares modulo the integer N which we intend to factor. The present approach is based on the difference of squares which are "close" to the integer N. The proposed method could be used to attack the RSA public-key cryptosystem. Method Suppose we are trying to factor the composite number N = pq with p and q prime numbers. We start to look for a number M such that í µí±€ 2 =(í µí±+í µí±ž 2) 2 , (1) on this purpose we select a square number í µí±€ 2 as close as possible to N, but bigger than N according to condition (1). Now we make the difference with the adjacent, lower square number considering that every quadradic number can be expressed as the sum of the first 2í µí±€ − 1 odd numbers: í µí±€ 2 − (í µí±€ −1) 2 = 2í µí±€ − 1 (2) the difference (2) will be always equal to the odd number 2í µí±€ − 1. We are looking for a solution of this simple equation: 2í µí±€ − 1 = í µí± + í µí±ž − 1 = í µí±í µí±ž − φ(í µí±) (3) Where φ(N) is the Euler totient function φ(N) = (p − 1)(q − 1). We have to verify if we have found the right difference: 2í µí±€ − 1 = í µí± + í µí±ž − 1 so we make a verification based on Euler's totient theorem. Given an integer í µí±Ž coprime with N , we have to check if we have found Euler totient function φ(í µí±) : í µí±Ž φ(í µí±) ≡ 1 mod N (4) Or similarly í µí±Ž pq+1 ≡ í µí±Ž p+q mod N (4.1) To simplify calculations, í µí±Ž ,in our examples, will be equal to: í µí±Ž = 2. If condition (4) is verified we have found φ(í µí±) and also p+q; therefore, we are able to factorize N. If p and q are close to each other, and their difference is small, then equation (3) will soon be valid.

In number theory, integer factorization or prime factorization is the decomposition of a composite number into smaller non-trivial divisors, which when multiplied together equals the original integer. A prime factor can be visualized by understanding Euclid's geometric position. He saw a whole number as a line segment, which has a smallest line segment greater than 1 that can divide equally into it. By the fundamental theorem of arithmetic, every positive integer has a unique prime factorization. However, the fundamental theorem of arithmetic gives no insight into how to obtain an integer's prime factorization; it only guarantees its existence.

The factorization of integers is far harder than the reversal of it called multiplication, even for computers if the factors and their product are big enough. For the very reason that the two inverse operations are the kind of "trapdoor" useful in cryptography, as in the RSA public key encryption method, there is already a considerable body of published research on integer factorization. A corpus of 2,500 relevant titles downloaded from Dimensions (metadata not fulltexts) was subjected it to analysis with R, mainly using the R package called "bibliometrix". Some findings relate to articles in the corpus, others to authors within it. Heuristics to guide those wishing to read or publish within the field were sought, such as the most-cited journal sources. Intriguingly, the timeline of citations spiked in 1985 and, to a somewhat lesser extent 1997. Some productive and highly cited authors were identified, with information about their "distance" from each other as determined from a co-citation analysis.