Introduction to finite fields and their applications: Preface

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.

Normal Bases over Finite Fields

We introduce the concept of the orthomorphism graph of a group. An r-clique of the orthomorphism graph of a group G corresponds to a set of r+1 mutually orthogonal latin squares based on the group G. In this paper we give constructions of nets, affine planes and cartesian groups using cliques of orthomorphism graphs and we discuss the relationship between properties of the cliques and properties of the nets constructed.

Orthomorphism Graphs of Groups

Explicit theorems on generator polynomials

Existence and properties of k-normal elements over finite fields

Let E be a finite degree extension over a finite field F = GF(q), G the Galois group of E over F and let a∈F be nonzero. We prove the existence of an element w in E satisfying the following conditions: - w is primitive in E, i.e., w generates the multiplicative group of E (as a module over the ring of integers). - the set {w

 g
 ∣g∈G} of conjugates of w under G forms a normal basis of E over F. - the (E, F)-trace of w is equal to a. This result is a strengthening of the primitive normal basis theorem of Lenstra and Schoof [10] and the theorem of Cohen on primitive elements with prescribed trace [3]. It establishes a recent conjecture of Morgan and Mullen [14], who, by means of a computer search, have verified the existence of such elements for the cases in which q≤ 97 and n≤ 6, n being the degree of E over F. Apart from two pairs (F, E) (or (q, n)) we are able to settle the conjecture purely theoretically.

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Primitive normal bases with prescribed trace

Preface. I: Introduction and Outline. 1. The Normal Basis Theorem. 2. A Strengthening of the Normal Basis Theorem. 3. Preliminaries on Finite Fields. 4. A Reduction Theorem. 5. Particular Extensions of Prime Power Degree. 6. An Outline. II: Module Structures in Finite Fields. 7. On Modules over Principal Ideal Domains. 8. Cyclic Galois Extensions. 9. Algorithms for Determining Free Elements. 10. Cyclotomic Polynomials. III: Simultaneous Module Structures. 11. Subgroups Respecting Various Module Structures. 12. Decompositions Respecting Various Module Structures. 13. Extensions of Prime Power Degree (1). IV: The Existence of Completely Free Elements. 14. The Two-Field Problem. 15. Admissibility. 16. Extendability. 17. Extensions of Prime Power Degree (2). V: A Decomposition Theory. 18. Suitable Polynomials. 19. Decompositions of Completely Free Elements. 20. Regular Extensions. 21. Enumeration. VI: Explicit Constructions. 22. Strongly Regular Extensions. 23. Exceptional Cases. 24. Constructions in Regular Extensions. 25. Product Constructions. 26. Iterative Constructions. 27. Polynomial Constructions. References. List of Symbols. Index.

Finite Fields: Normal Bases and Completely Free Elements

We study the structure of a finite group G admitting a Kantor family \kf of type (s,t) and a nontrivial normal subgroup X which is factorized by {\cal F} \cup {\cal F}^*. The most interesting cases, giving necessary conditions on the structure of G and the parameters s and t, are those where a further Kantor family is induced in X, or where a partial congruence partition is induced in the factor group G/X. Most of the known finite generalized quadrangles can be constructed as coset geometries with respect to a Kantor family. We show that the parameters of a skew translation generalized quadrangle necessarily are powers of the same prime. Furthermore, the structure of nonabelian groups admitting a Kantor family consisting only of abelian members is considered.

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Groups Admitting a Kantor Family and a Factorized NormalSubgroup

http://opus.bibliothek.uni-augsburg.de/opus4/files/1019/Pub_Hachenberger_2000_2.pdf

Primitivity, freeness, norm and trace

Let F = GF(q). To any polynomial G element of F[x] there is associated a mapping on the set I_F of monic irreducible polynomials over F. We present a natural and effective theory of the dynamics of the mapping for the case in which G is a monic q-linearized polynomial. The main outcome is the following theorem. Assume that G is not of the form x^(q^l), where l>= 0 (in which event the dynamics is trivial). Then, for every integer n >= 1 and for every integer k >= 0, there exist infinitely many mu element of I_F having preperiod k and primitive period n with respect to the mapping. Previously, Morton, by somewhat different means, had studied the primitive periods of the mapping when G = x^q - ax, a a non-zero element of F. Our theorem extends and generalizes Morton's result. Moreover, it establishes a conjecture of Morton for the class of q-linearized polynomials.

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Dirk Hachenberger

Papers

Primitive normal bases with prescribed trace

Finite Fields: Normal Bases and Completely Free Elements

Groups Admitting a Kantor Family and a Factorized NormalSubgroup

Primitivity, freeness, norm and trace

The dynamics of linearized polynomials