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Explicit Iterative Constructions of Normal Bases and Completely Free Elements in Finite Fields

Dirk Hachenberger
1996 Finite Fields and Their Applications  
A characterization of normal bases and complete normal bases in GF(q r n ) over GF(q), where q Ͼ 1 is any prime power, r is any prime number different from the characteristic of GF(q), and n Ն 1 is any  ...  Such a basis is called a normal basis in GF(q m ) over GF(q).  ...  So, we are able to iteratively find normal bases or complete normal bases in any finite subfield of GF(q r ȍ ) over GF(q), if we can solve the following main problem. Problem 2.6.  ... 
doi:10.1006/ffta.1996.0001 fatcat:smmdbtjtrnh23g35b5c627nhey


2010 International Journal of Number Theory  
In this paper we construct self-dual normal basis generators for finite extensions of finite fields whenever they exist.  ...  It is known that a self-dual integral normal basis exists for A_L/K if and only if L/K is weakly ramified. Assuming p≠ 2, we construct such bases whenever they exist.  ...  Finite Fields In this section we describe how to construct self-dual normal bases for any finite extension of a finite field for which they exist.  ... 
doi:10.1142/s1793042110003654 fatcat:uhs3licaira7rcogp2c7ojg3jm

Page 1556 of Mathematical Reviews Vol. , Issue 99c [page]

1991 Mathematical Reviews  
Summary: “GauB periods have been used successfully as a tool for constructing normal bases in finite fields.  ...  Since normal bases in finite fields have been proved in the last two decades to be very useful for doing arithmetic computations, the algorithmic and explicit construction of (certain) such bases has become  ... 

Page 3702 of Mathematical Reviews Vol. , Issue 94g [page]

1994 Mathematical Reviews  
Normal bases; 4. Dual bases; 5. Normal bases and duality; 6. Shift register sequences; 7. Characters and Gaussian sums.  ...  The book emphasizes the constructive and computational sides of finite field theory, and is divided into the following seven chapters: 1. Basic results; 2. Explicit construction of finite fields; 3.  ... 

Page 4606 of Mathematical Reviews Vol. , Issue 96h [page]

1996 Mathematical Reviews  
The aim of the paper is to construct self-complementary normal ’ 11 NUMBER THEORY 4606 bases, also called self-dual normal bases, in finite fields with characteristic 2.  ...  In the second part the author constructs a self-complementary normal basis over GF(q), where g = 2’, with the help of a normal basis B in GF(q").  ... 

Finite field arithmetic using quasi-normal bases

Christophe Negre
2007 Finite Fields and Their Applications  
For some extension fields F q n over F q , we construct quasi-normal bases of low density.  ...  Efficient multiplication in finite fields F q n requires F q -bases of low density, i.e., such that the products of the basis elements have a sparse expression in the basis.  ...  They are a new family of F q -bases of finite field F q n which generalizes the notion of normal bases.  ... 
doi:10.1016/j.ffa.2006.09.008 fatcat:6pm54fwepfbv5ku52j3q53jsc4

A proper base change theorem for non-torsion sheaves in étale cohomology

Ch. Deninger
1988 Journal of Pure and Applied Algebra  
We show that under certain additional hypotheses the proper base change theorem holds for arbitrary abelian sheaves.  ...  abelian group and r: Y* Y is a finite morphism with Y normal.  ...  In particular this is the case for the proper base change theorem as is shown by the example in [l, XII, 21.  ... 
doi:10.1016/0022-4049(88)90102-8 fatcat:6n765dntjbfd3nfdzywtd5ey4q

The gamma construction and asymptotic invariants of line bundles over arbitrary fields [article]

Takumi Murayama
2019 arXiv   pre-print
To do so over imperfect fields, we prove a scheme-theoretic version of the gamma construction of Hochster and Huneke to reduce to the setting where the ground field is F-finite.  ...  We also extend Nakayama's description of the restricted base locus to klt or strongly F-regular varieties over arbitrary fields.  ...  Since k(x 1/p ∞ ) contains the infinite perfect field F p (x 1/p ∞ ) and applying the gamma construction to k(x 1/p ∞ ) results in an F -finite field (Construction 3.1 and Lemma 3.2(i)), it therefore suffices  ... 
arXiv:1809.01217v3 fatcat:wcaeoon4wvbyvdyjm32vz5ram4

Page 99 of Mathematical Reviews Vol. , Issue 98A [page]

1998 Mathematical Reviews  
If F = GF(q) is a finite field and n > | an integer, then it is one of the major problems in the theory of finite fields to (explicitly) construct an irreducible polynomial of degree n over F (possibly  ...  Dirk Hachenberger (D-AGSB-MI; Augsburg) 98a:11172 11T06 Chapman, Robin (4-EXTR; Exeter) Completely normal elements in iterated quadratic extensions of finite fields.  ... 

Page 3841 of Mathematical Reviews Vol. , Issue 95g [page]

1995 Mathematical Reviews  
For similar values of n and q, self-dual normal bases are also constructed and these too can have complexity 3n — 2 when n = p, the novel feature here being constructibility.  ...  “optimal normal bases” [see X.  ... 

Normal high order elements in finite field extensions based on the cyclotomic polynomials

R. Popovych, Lviv Polytechnic National University, R. Skuratovskii, Igor Sikorsky Kiev Polytechnic Institute
2020 Algebra and discrete mathematics  
We consider elements which are both of high multiplicative order and normal in extensions F q m of the field F q .  ...  If the extension is defined by a cyclotomic polynomial, we construct such elements explicitly and give explicit lower bounds on their orders. 2010 MSC: 11T30.  ...  The current paper enriches the list of normal high order elements in finite field extensions based on cyclotomic polynomials.  ... 
doi:10.12958/adm1117 fatcat:jbvkgochgrcjxig2gpmlvqnxci

Construction of Irreducible Polynomials in Galois elds, GF(2m) Using Normal Bases

Abraham Aidoo, Kwasi Baah Gyam
2019 Asian Research Journal of Mathematics  
This thesis is about Construction of Polynomials in Galois fields Using Normal Bases in finite fields.  ...  We found the actual construction of polynomials in GF(2m) with degree less than or equal to m − 1 and also illustrated how this construction can be done using normal bases.  ...  Acknowledgement The authors are grateful to the referees for their careful reading, constructive criticisms, comments and suggestions, which have helped us to improve this work significantly.  ... 
doi:10.9734/arjom/2019/v14i330131 fatcat:ip2oqqrdrjd6ddo5iry237stpm


I. A. Dychka, V. P. Legeza, M. V. Onai, A. I. Severin
2020 Radìoelektronika, Ìnformatika, Upravlìnnâ  
The subject of the study is the methods and algorithms for calculations in the Galois fields in polynomial and normal bases. Objective.  ...  the conversion between bases of binary fields gives an increase in speed over 15 times for the parameter m > 14; the proposed approach for constructing a conversion matrix gives an increase in the speed  ...  It can be implemented by the construction of digital filters that are based on computations in finite fields.  ... 
doi:10.15588/1607-3274-2020-2-12 fatcat:42mt3dujwrdedd7ealnsonkwru

Book Review: Computational and algorithmic problems in finite fields

Daqing Wan
1994 Bulletin of the American Mathematical Society  
It is an excellent textbook and reference for people interested in normal bases and in the applications of finite fields to combinatorics.  ...  Therefore, they should be of interest to mathematicians working on areas related to finite fields. Jungnickel's book essentially focuses on normal bases in finite fields, namely, a basis of F?  ... 
doi:10.1090/s0273-0979-1994-00466-5 fatcat:lpl6spoff5b7hop6pbbrlxhxsa

Page 6790 of Mathematical Reviews Vol. , Issue 2004i [page]

2004 Mathematical Reviews  
Included too is an application to the existence of trace-compatible sequences of primitive normal bases for certain primary closures of Galois fields. S. D.  ...  Let K be a field and F a finite Galois extension of K with Ga- lois group G.  ... 
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