IngentaConnect Explicit Iterative Constructions of Normal Bases and Completely F...

Author: Hachenberger D.

Source: Finite Fields and Their Applications, Volume 2, Number 1, January 1996 , pp. 1-20(20)

Abstract:

A characterization of normal bases and complete normal bases in GF( q r n ) over GF( q ), where q > 1 is any prime power, r is any prime number different from the characteristic of GF( q ), and n > 1 is any integer, leads to a general construction scheme of series ( v n ) n >0 in GF( q r [infinity] ) ≔ ∪ n i >0 GF(≔ q r n ) having the property that the partial sums w n ≔ Sigma n i≔ 0 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 - 1 or where r = 2 and q = 1 mod 4, for any integer n > 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 = 2 and q = 3 mod 4, for any n > 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.

Language: English

Document Type: Research article

Affiliations: Institut fur Mathematik der Universitat Augsburg, Universitatsstrasse 14, Augsburg, D-86135, Germany:

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