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Main article: Square-free factorization. The algorithm determines a square-free factorization for polynomials whose coefficients come from the finite field Fq of order q = pm with p a prime. This algorithm firstly determines the derivative and then computes the gcd of the polynomial and its derivative.
Over GF(2), x + 1 is a primitive polynomial and all other primitive polynomials have an odd number of terms, since any polynomial mod 2 with an even number of terms is divisible by x + 1 (it has 1 as a root). An irreducible polynomial F(x) of degree m over GF(p), where p is prime, is a primitive polynomial if the smallest positive integer n ...
The monic irreducible polynomial x 8 + x 4 + x 3 + x + 1 over GF(2) is not primitive. Let λ be a root of this polynomial (in the polynomial representation this would be x), that is, λ 8 + λ 4 + λ 3 + λ + 1 = 0. Now λ 51 = 1, so λ is not a primitive element of GF(2 8) and generates a multiplicative subgroup of order 51. [5]
The Cantor–Zassenhaus algorithm takes as input a square-free polynomial (i.e. one with no repeated factors) of degree n with coefficients in a finite field whose irreducible polynomial factors are all of equal degree (algorithms exist for efficiently factoring arbitrary polynomials into a product of polynomials satisfying these conditions, for instance, () / ((), ′ ()) is a squarefree ...
This implies that, if q = p n then X q − X is the product of all monic irreducible polynomials over GF(p), whose degree divides n. In fact, if P is an irreducible factor over GF(p) of X q − X, its degree divides n, as its splitting field is contained in GF(p n).
Given an n-bit message m 0,...,m n-1, we view it as a polynomial of degree n-1 over the finite field GF(2). = + + … +We then pick a random irreducible polynomial of degree k over GF(2), and we define the fingerprint of the message m to be the remainder () after division of () by () over GF(2) which can be viewed as a polynomial of degree k − 1 or as a k-bit number.
hide. In algebra, the greatest common divisor (frequently abbreviated as GCD) of two polynomials is a polynomial, of the highest possible degree, that is a factor of both the two original polynomials. This concept is analogous to the greatest common divisor of two integers. In the important case of univariate polynomials over a field the ...
Irreducibility (mathematics) In mathematics, the concept of irreducibility is used in several ways. A polynomial over a field may be an irreducible polynomial if it cannot be factored over that field. In abstract algebra, irreducible can be an abbreviation for irreducible element of an integral domain; for example an irreducible polynomial.