Normal bases over GF(q)

JOURNAL OF ALGEBRA

卷:241

Article

Chang, YT ; Truong, TK ; Reed, IS

关键词: finite fields;    irreducible polynomials;    normal polynomials;    trace of an element;    trace of a polynomial;    primitive roots;   

DOI  :  10.1006/jabr.2001.8765

来源: Elsevier

【 摘 要 】

For q a power of a prime p, it is known that if m is a power of p or m itself is a prime different from p having (I as one of its primitive roots, then the roots of any irreducible polynomial of degree m and of non-zero trace are linearly independent over GF(q). As a consequence the roots of such an mth degree polynomial form a basis of GF(q(m)) over GF(q). Such a basis is called a normal basis over GF(q) and the polynomial is called normal over CF(q). Normal bases over finite fields have proved very useful for fast arithmetic computations with potential applications to coding theory and to cryptography. In this paper, we prove that for mth degree irreducible polynomials the above two conditions are indeed necessary and sufficient conditions for the equivalence between the properties of having a non-zero trace and being normal over GF(q). (C) 2001 Academic Press.

【 授权许可】

Free