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Factorization of polynomials. In mathematics and computer algebra, factorization of polynomials or polynomial factorization expresses a polynomial with coefficients in a given field or in the integers as the product of irreducible factors with coefficients in the same domain. Polynomial factorization is one of the fundamental components of ...
In mathematics, factorization (or factorisation, see English spelling differences) or factoring consists of writing a number or another mathematical object as a product of several factors, usually smaller or simpler objects of the same kind. For example, 3 × 5 is an integer factorization of 15, and (x – 2) (x + 2) is a polynomial ...
In mathematics, particularly computational algebra, Berlekamp's algorithm is a well-known method for factoring polynomials over finite fields (also known as Galois fields ). The algorithm consists mainly of matrix reduction and polynomial GCD computations. It was invented by Elwyn Berlekamp in 1967. It was the dominant algorithm for solving the ...
Cantor–Zassenhaus algorithm. In computational algebra, the Cantor–Zassenhaus algorithm is a method for factoring polynomials over finite fields (also called Galois fields). The algorithm consists mainly of exponentiation and polynomial GCD computations. It was invented by David G. Cantor and Hans Zassenhaus in 1981.
A finite field or Galois field is a field with a finite order (number of elements). The order of a finite field is always a prime or a power of prime. For each prime power q = pr, there exists exactly one finite field with q elements, up to isomorphism. This field is denoted GF ( q) or Fq.
On a quantum computer, to factor an integer , Shor's algorithm runs in polynomial time, meaning the time taken is polynomial in , the size of the integer given as input. [6] Specifically, it takes quantum gates of order using fast multiplication, [7] or even utilizing the asymptotically fastest multiplication algorithm currently known due to ...
Fermat's factorization method, named after Pierre de Fermat, is based on the representation of an odd integer as the difference of two squares : That difference is algebraically factorable as ; if neither factor equals one, it is a proper factorization of N . Each odd number has such a representation. Indeed, if is a factorization of N, then.
Assume that p − 1, where p is the smallest prime factor of n, can be modelled as a random number of size less than √ n. By Dixon's theorem, the probability that the largest factor of such a number is less than (p − 1) 1/ε is roughly ε −ε; so there is a probability of about 3 −3 = 1/27 that a B value of n 1/6 will yield a factorisation.
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