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The cost of a polynomial greatest common divisor between two polynomials of degree at most n can be taken as O(n 2) operations in F q using classical methods, or as O(nlog 2 (n) log(log(n)) ) operations in F q using fast methods. For polynomials h, g of degree at most n, the exponentiation h q mod g can be done with O(log(q)) polynomial ...
The set of roots of L is an F q-vector space and is closed under the q-Frobenius map. Conversely, if U is any F q-linear subspace of some finite field containing F q, then the polynomial that vanishes exactly on U is a linearised polynomial. The set of linearised polynomials over a given field is closed under addition and composition of ...
A polynomial f(x) in F q [x] of degree d is called exceptional over F q if every irreducible factor (differing from x − y) or (f(x) − f(y))/(x − y)) over F q becomes reducible over the algebraic closure of F q. If q > d 4, then f(x) is exceptional if and only if f(x) is a permutation polynomial over F q. The Carlitz–Wan conjecture ...
The polynomial factors into linear factors over a field of order q. More precisely, this polynomial is the product of all monic polynomials of degree one over a field of order q. 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 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.
The splitting field of x q − x over F p is the unique finite field F q for q = p n. [2] Sometimes this field is denoted by GF(q). The splitting field of x 2 + 1 over F 7 is F 49; the polynomial has no roots in F 7, i.e., −1 is not a square there, because 7 is not congruent to 1 modulo 4. [3]
Specifically, for α ≠ 0 ∈ F q with q = p e for some prime p and any integers n ≥ 0 and 0 ≤ k < p, the n th Dickson polynomial of the (k + 1) th kind over F q, denoted by D n,k (x,α), is defined by [11]
An exceptional polynomial over GF(q) is a polynomial in F q [x] which is a permutation polynomial on GF(q m) for infinitely many m. [16] A permutation polynomial over GF(q) of degree at most q 1/4 is exceptional over GF(q). [17] Every permutation of GF(q) is induced by an exceptional polynomial. [17]