Normal Bases over Finite Fields
Shuhong Gao · 1993
Interest in normal bases over finite fields stems both from mathematical theory and practical applications. There has been a lot of literature dealing with various properties of normal bases (for finite fields and for Galois extension of arbitrary fields). The advantage of using normal bases to represent finite fields was noted by Hensel in 1888. With the introduction of optimal normal bases, large finite fields, that can be used in secure and efficient implementation of several cryptosystems, have recently been realized in hardware. The present thesis studies various theoretical and practical aspects of normal bases in finite fields. We first give some characterizations of normal bases. Then by using linear algebra, we prove that Fqn has a basis over Fq such that any element in Fq represented in this basis generates a normal basis if and only if some groups of coordinates are not simultaneously zero. We show how to construct an irreducible polynomial of degree 2 n with linearly independent roots over Fq for any integer n and prime power q. We also construct explicitly an irreducible polynomial in Fp[x] of degree pn with linearly independent roots for any prime p and positive integer n. We give a new characterization of the minimal polynomial of αt for any integer t when the minimal polynomial