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Explicit Iterative Constructions of Normal Bases and Completely Free Elements in Finite Fields
1996
Finite Fields and Their Applications
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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
CONSTRUCTION OF SELF-DUAL INTEGRAL NORMAL BASES IN ABELIAN EXTENSIONS OF FINITE AND LOCAL FIELDS
2010
International Journal of Number Theory
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2010 pre-print arXiv:1007.0326v1 fatcat:v3rbiq3c2bccxewyu6tb2rncla
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
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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
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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
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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
2007
Finite Fields and Their Applications
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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
1988
Journal of Pure and Applied Algebra
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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]
2019
arXiv
pre-print
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2018 pre-print arXiv:1809.01217v2 fatcat:wraq4t6qlvdttiy7kj747tuxhqOther Versions
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
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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
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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
2020
Algebra and discrete mathematics
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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
2019
Asian Research Journal of Mathematics
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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
MODIFIED CHANGE-OF-BASIS CONVERSION METHOD IN GF(2m)
2020
Radìoelektronika, Ìnformatika, Upravlìnnâ
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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
1994
Bulletin of the American Mathematical Society
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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
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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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