Explicit Iterative Constructions of Normal Bases and Completely Free Elements in Finite Fields

✍ Scribed by Dirk Hachenberger

• Publisher: Elsevier Science
• Year: 1996
• Tongue: English
• Weight: 305 KB
• Volume: 2
• Category: Article
• ISSN: 1071-5797
• DOI: 10.1006/ffta.1996.0001

✦ Synopsis

A characterization of normal bases and complete normal bases in GF(q r n ) over GF(q), where q Ͼ 1 is any prime power, r is any prime number different from the characteristic of GF(q), and n Ն 1 is any integer, leads to a general construction scheme of series (v n ) nՆ0 in GF(q r ȍ ) :ϭ ʜ nՆ0 GF(q r n ) having the property that the partial sums w n :ϭ ͚ n i:ϭ0 v i are free or completely free in GF(q r n ) over GF(q), depending on the choice of v n .

In the case where r is an odd prime divisor of q Ϫ 1 or where r ϭ 2 and q ϵ 1 mod 4, for any integer n Ն 1, all free and completely free elements in GF(q r n ) over GF(q) are explicitly determined in terms of certain roots of unity.

In the case where r ϭ 2 and q ϵ 3 mod 4, for any n Ն 1, in terms of certain roots of unity, an explicit recursive construction for free and completely free elements in GF(q 2 n ) over GF(q) is given. As an example, for a particular series of completely free elements the corresponding minimal polynomials are given explicitly.