Abstract. Weil’s character sum estimate is used to study the problem of constructing generators for the multiplicative group of a finite field. An application to the distribution of irreducible polynomials is given, which confirms an asymptotic version of a conjecture of Hansen-Mullen. 1.

Abstract. For n ≥ 7, we prove that there always exists a primitive polynomial of degree n over a finite field Fq (q odd) with the first and second coefficients prescribed in advance. 1.

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 e#cient 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 F q n has a basis over F q such that any element in F q 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 i...

This paper addresses the question: how can we find a normal element efficiently? More generally, we consider how to find an element of any given additive order. Hensel (1888) pioneered the study of normal bases for finite fields and proved that they always exist. We use his algorithm in Section 2. Eisenstein (1850) had already noted that normal bases always exist. Hensel, and also Ore (1934), determine exactly the number of these bases, and Ore develops the more general concept of additive order. Ore's approach is developed into more constructive proofs of the normal basis theorem in several textbooks (for example, van der Waerden 1966, Section 67, and Albert 1956, Section 4.15); these all use some linear algebra calculations. Schwarz (1988) has given a new proof along these lines, and several recent papers have translated this approach into algorithms. Sidel'nikov (1988) deals with the case where n divides one of p (the characteristic of F q ), q + 1, or

The existence of a primitive free (normal) cubic x 3 − ax 2 + cx − b over a finite field F with arbitrary specified values of a (�=0)andb (primitive) is guaranteed. This is the most delicate case of a general existence theorem whose proof is thereby completed.

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Finite fields play an important role for many applications (e.g. coding theory, cryptography). There are different ways to construct a finite field for a given prime power. The paper describes the different constructions implemented in AXIOM. These are polynomial basis representation, cyclic group representation, and normal basis representation. Furthermore, the concept of the implementation, the used algorithms and the various datatype coercions between these representations are discussed. Address of authors: Vangerowstr. 18, Postfach 10 30 68, D-6900 Heidelberg, Germany, email: grabm@dhdibm1.bitnet resp. adscheer@dhdibm1.bitnet Contents 1 Introduction 4 2 Basic theory and notations 5 3 Categories for finite field domains 7 4 General finite field functions 8 4.1 E as an algebra of rank n over F : : : : : : : : : : : : : : : : : : 8 4.2 The F [X]-module structure of E : : : : : : : : : : : : : : : : : : 10 4.3 The cyclic group E : : : : : : : : : : : : : : : : : : : : : : : : ...

Abstract. In this note, we prove that the order of x p − x − 1 ∈ Fp[x] is pp−1, where p is a p−1 prime and Fp is the finite field of size p. As a consequence, it is shown that x p − x − a ∈ Fp[x] is primitive if and only if a is a primitive element in Fp. 1.

Arithmetic of finite fields is not only important for other branches of mathematics but also widely used in applications such as coding and cryptography. A primitive element of a finite field is of particular interest since it enables one to represent all other elements of the field. Therefore an extensive research has been done on primitive elements, especially those satisfying extra conditions. We are interested in the existence of primitive elements in extensions of finite fields with prescribed trace value. This existence problem can be settled by means of two important theories. One is character sums and the other is the theory of algebraic function fields. The aim of this thesis is to introduce some important properties of these two topics and to show how they are used in answering the existence problem mentioned above.

An element α of the extension E of degree n over the finite field F = GF(q) is called free over F if {α, αq,...,α qn−1} is a (normal) basis of E/F. The primitive normal basis theorem, first established in full by Lenstra and Schoof (1987), asserts that for any such extension E/F, there exists an element α ∈ E such that α is simultaneously primitive (i.e., generates the multiplicative group of E) and free over F. In this paper we prove the following strengthening of this theorem: aside from five specific extensions E/F, there exists an element α ∈ E such that both α and α −1 are simultaneously primitive and free over F. 1 1