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Normal Bases over Finite Fields
, 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 repr ..."
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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...
The coefficients of primitive polynomials over finite fields
 Math. Comp
, 1996
"... 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. ..."
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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.
Constructing Normal Bases in Finite Fields
 J. Symbolic Comput
, 1990
"... 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. E ..."
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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
Primitive free cubics with specified norm and trace
 TRANSACTIONS OF AMERICAN MATHEMATICAL SOCIETY
, 2003
"... 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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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.
A Survey of Elliptic Curve Cryptosystems, Part I: Introductory
, 2003
"... The theory of elliptic curves is a classical topic in many branches of algebra and number theory, but recently it is receiving more attention in cryptography. An elliptic curve is a twodimensional (planar) curve defined by an equation involving a cubic power of coordinate x and a square power of co ..."
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The theory of elliptic curves is a classical topic in many branches of algebra and number theory, but recently it is receiving more attention in cryptography. An elliptic curve is a twodimensional (planar) curve defined by an equation involving a cubic power of coordinate x and a square power of coordinate y. One class of these curves is
The Strong Primitive Normal Basis Theorem
, 2008
"... 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 ele ..."
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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
On a Theorem of Lenstra and Schoof
, 2002
"... We give a detailed proof of Theorem 1.15 from a wellknown paper ”Primitive normal bases for finite fields ” by H.W. Lenstra Jr. and R.J. Schoof. We are not aware of any other proofs. Let L/K be a finitedimensional Galois field extension and B the set of all normal bases of this extension. Theorem ..."
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We give a detailed proof of Theorem 1.15 from a wellknown paper ”Primitive normal bases for finite fields ” by H.W. Lenstra Jr. and R.J. Schoof. We are not aware of any other proofs. Let L/K be a finitedimensional Galois field extension and B the set of all normal bases of this extension. Theorem 1.15 describes the group of all γ in the multiplicative group of L such that γB = B. Key words: Normal basis, Wedderburn’s Theorems, group ring, primitive normal basis.
Permutation Groups, ErrorCorrecting Codes
"... We replace the traditional setting for errorcorrecting codes (i.e. linear codes) with that of permutation groups, with permutations in list form as the codewords. We introduce a decoding algorithm for these codes, which uses the following notion. A base for a permutation group is a sequence of poin ..."
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We replace the traditional setting for errorcorrecting codes (i.e. linear codes) with that of permutation groups, with permutations in list form as the codewords. We introduce a decoding algorithm for these codes, which uses the following notion. A base for a permutation group is a sequence of points whose stabiliser is trivial. An uncoveringbybases (or UBB) is a set of bases such that any combination of error positions is avoided by at least one base in the set. In the case of sharply ktransitive groups, any ktuple of points forms a base, so a UBB can be formed from the complements of the blocks of a covering design. (In this case, we use the term uncovering.) A large part of the thesis (chapters 2 to 5) is concerned with constructing UBBs for groups which are basetransitive, i.e. which act transitively on their irredundant bases, which were classified by T. Maund. Various combinatorial, algebraic and numbertheoretic techniques are employed in this. Other topics include a case study of the Mathieu group M12, where we investigate ways in which