Normal Bases over Finite Fields

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 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 p with linearly independent roots for any prime p and positive integer n. We give a new characterization of the minimal polynomial of α for any integer t when the minimal polynomial of α is given. When q ≡ 3 mod 4, we present an explicit complete factorization of x2n − 1 over Fq for any integer n. The principal result in the thesis is the complete determination of all optimal normal bases in finite fields, which confirms a conjecture by Mullin, Onyszchuk, Vanstone and Wilson. Finally, we present some explicit constructions of normal bases with low complexity and some explicit constructions of self-dual normal bases.