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Irreducible polynomials over finite fields are also useful for pseudorandom number generators using feedback shift registers and discrete logarithm over F 2 n. The number of irreducible monic polynomials of degree n over F q is the number of aperiodic necklaces, given by Moreau's necklace-counting function M q (n). The closely related necklace ...
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 ...
It follows that they are roots of irreducible polynomials of degree 6 over GF(2). This implies that, over GF(2), there are exactly 9 = 54 / 6 irreducible monic polynomials of degree 6. This may be verified by factoring X 64 − X over GF(2). The elements of GF(64) are primitive n th roots of unity for some n dividing 63.
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.
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 ...
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.
Definition. For a generic degree reducible monic polynomial equation of the form , where and are polynomials and does not vanish at , the Tschirnhaus transformation is the function: Such that the new equation in , , has certain special properties, most commonly such that some coefficients, , are identically zero. [2][3]
To state it precisely, let F = GF(q) be the finite field with q elements, for some fixed q, and let N n be the number of monic irreducible polynomials over F whose degree is equal to n. That is, we are looking at polynomials with coefficients chosen from F, which cannot be written as products of polynomials of smaller degree. In this setting ...