有限域上线性化多项式与正规基

Linearized polynomials and normal bases in finite fields

设F_(q)为q阶有限域, F_(q)^(n)为F_(q)的n次扩域. F_(q)^(n)上形如{α,α^(q),…,α^(q)^(n-1)}的一组基称为F_(q)^(n)/F_(q)的正规基,此时α称为F_(q)^(n)/F_(q)上的正规元.设k为n的一个正因子,用F_(q)^(k)表示F_(q)的正规元集合,用G_(n,k)表示F_(q)^(n)中满足F_(q)^(k)(α)=F_(q)^(n)且Σn/k-1 j-0 α^(q)^(kj)∈n_(k)的元素α的集合.本文利用有限域上线性化多项式的性质,给出了n_(n)与G_(n,k)的计数公式.

Let F_(q) be the finite field of q elements, and F_(q)^(n) be its extension of degree n. A normal basis of F_(q)^(n) over F_(q) is a basis of the form {α,α^(q),...,α^(q)^(n-1)} and α is called a normal element of F_(q)^(n)/F_(q). For a positive divisor k of n, let m_(k) denote the set of all the normal elements of F_(q)^(k)/F_(q), G_(n,k)denote the set of α∈F_(q) satisfying F_(q)^(k)(α)=F_(q)^(n) and Σn/k-1 j-0 α^(q)^(kj)∈n_(k). In this paper, we provide the counting formulae for n_(n) and G_(n,k)using properties of linearized polynomials in finite fields.