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Finite Fields and Their Applications
[chapter]

1996
*
Handbook of Algebra
*

Acknowledgments We would like to thank the anonymous referee for the very useful suggestions

doi:10.1016/s1570-7954(96)80013-1
fatcat:noprxntirvchplkwjbue6amsma
*and*comments. The second author was partially supported by TÜBİTAK under Grant No. TBAG-107T826. ... Özbudak /*Finite**Fields**and**Their**Applications*••• (••••) •••-••• ... Özbudak /*Finite**Fields**and**Their**Applications*••• (••••) •••-••• 26 ) 26 is the disjoint union over all monic polynomials d ∈ F q [x] dividing f . ...##
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A decade of Finite Fields and Their Applications

2005
*
Finite Fields and Their Applications
*

The journal

doi:10.1016/j.ffa.2005.05.009
fatcat:rgii7n2uanc27ohm3tse4tcbhi
*Finite**Fields**and**Their**Applications*(FFTA) began publication in 1995,*and*in this special issue, we provide a flavor of some of the huge volume of*finite*fieldrelated activity that has taken ... On behalf of the entire*finite**field*community, I would like to take this opportunity as Editor-in-Chief to sincerely thank you for making*Finite**Fields**and**Their**Applications*the great success that it ...##
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Binary Representations of Finite Fields and Their Application to Complexity Theory

1996
*
Finite Fields and Their Applications
*

Binary representations of

doi:10.1006/ffta.1996.0022
fatcat:pevl73mnfndhrbugi2urgia6s4
*finite**fields*are defined as an injective mapping from a*finite**field*to l-tuples with components in ͕0, 1͖ where 0*and*1 are elements of the*field*itself. ... The standard representation of a*finite**field*is shown to be among the two-way easiest representations of this*field*. ... In particular, it would follow that the algebraic complexity*and*the circuit complexity are equivalent for every*finite**field*. ...##
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Finite Upper Half Planes over Finite Fields

1996
*
Finite Fields and Their Applications
*

*Finite*upper half planes have been studied by Terras, Poulos, Celniker, Trimble,

*and*Velasquez. ... Since this is not possible with a

*finite*

*field*of even characteristic, this paper will introduce a modification which will enable all

*finite*

*fields*to be considered. ... about

*their*collapsed adjacency matrices. ...

##
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On group automata over finite fields

2004
*
Finite Fields and Their Applications
*

We investigate the structure of incidence rings of group automata over

doi:10.1016/s1071-5797(03)00043-1
fatcat:vjmc4zi6avelrkt5m7mr3qfn5i
*finite**fields*. One of the major tools used in research on the structure of ring constructions is the Jacobson radical. ... Our main theorem gives a complete description of the Jacobson radicals of incidence rings of (possibly nondeterministic) group automata over*finite**fields*in the important special case where the input ... Kelarev /*Finite**Fields**and**Their**Applications*10 (2004) 65-71 ...##
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On Diagonal Equations over Finite Fields

1997
*
Finite Fields and Their Applications
*

We get an explicit formula for the number of solutions of a diagonal equation over

doi:10.1006/ffta.1996.0173
fatcat:3yyi5dlkezhlfcka2oeliv3q7m
*finite**fields*, under a certain natural restriction on the exponents. ... Since 2 ͉ n, thus G nϪ2 (, ) ϭ ((Ϫ1)q) (nϪ2)/2*and*G( nϪ2 j , ) ϭ G( j , ). ... Let g be a fixed primitive element of F*and*a ϭ g h , where a ʦ F *, 0 Յ h Ͻ q Ϫ 1. We define that ind g a ϭ h, or simply ind a ϭ h. ...##
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Power sums of polynomials over finite fields and applications: A survey

2015
*
Finite Fields and Their Applications
*

We mention several

doi:10.1016/j.ffa.2014.08.004
fatcat:e7wpvlv3afcxri2he3yadyfvhm
*applications*to the function*field*arithmetic. ... In this brief expository survey, we explain some results*and*conjectures on various aspects of the study of the sums of integral powers of monic polynomials of a given degree over a*finite**field*. ... Acknowledgements: I thank the reviewers for catching typos*and*for helpful comments to improve the exposition. ...##
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Dickson polynomials over finite fields

2012
*
Finite Fields and Their Applications
*

In this paper we introduce the notion of Dickson polynomials of the (k + 1)-th kind over

doi:10.1016/j.ffa.2012.02.001
fatcat:rfqhezt4orh45olpkchahw6lo4
*finite**fields*F p m*and*study basic properties of this family of polynomials. ... In particular, we study the factorization*and*the permutation behavior of Dickson polynomials of the third kind. ... Cohen*and*Gary L. Mullen for spending some time on reading a draft of this paper*and*suggesting some future directions. Finally we thank the anonymous referees for*their*helpful suggestions. ...##
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Specific permutation polynomials over finite fields

2011
*
Finite Fields and Their Applications
*

We fix a

doi:10.1016/j.ffa.2009.02.004
fatcat:rdvd577havg5rmx6rwkrccuwjy
*finite**field*F q*and*a number d 3 such that d|(q − 1). ... Let F q be a*finite**field*such that 5|(q − 1)*and*4 = 0 in F q . ...##
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Normal bases and primitive elements over finite fields

2014
*
Finite Fields and Their Applications
*

Let q be a prime power, m ≥ 2 an integer

doi:10.1016/j.ffa.2013.12.002
fatcat:fpkf5jcvmracneyrp53ziu6xam
*and*A = a b c d ∈ GL 2 (F q ), where A = ( 1 1 0 1 ) if q = 2*and*m is odd. ... q m over F q . 1. q = 2, m = 3*and*A = ( 0 1 1 0 ) or A = ( 1 0 1 1 ), 2. q = 3, m = 4*and*A is anti-diagonal or 3. ... Theodoulos Garefalakis for his encouragment*and*support*and*the anonymous referee for her/his useful corrections*and*improvements to the original manuscript. ...##
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Finite Modular Forms

2001
*
Finite Fields and Their Applications
*

*FINITE*MODULAR FORMS 1.8. PROPOSITION. ... @ A B )3 C we let¸I be the space of rational functions on / 1 (K M ) that have

*their*poles at K6/ 1 (K), all of order less than or equal to k,

*and*¸ I the subspace of functions that vanish at R

*and*... 1), both M IK

*and*S K IO are zero. For k,2m, items (i)

*and*(ii) follow from (7.5), (7.7),

*and*(7.8), as is easily seen. ...

##
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Rings Which Are Sums of Finite Fields

1999
*
Finite Fields and Their Applications
*

We describe all semigroup rings

doi:10.1006/ffta.1998.0240
fatcat:b5n2pjeqhfga5b7a22zgxz2uoa
*and*band-graded rings which are direct sums of*finite**fields*. ... We show that the class of rings which are direct sums of*finite**fields*is closed for taking sums of two subrings. 1999 Academic Press 89 ... In a commutative ring the product of two*finite**fields*is either zero or a*finite*direct sum of*finite**fields*. Proof. Let R be a commutative ring,*and*let A*and*B be*finite**fields*contained in R. ...##
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Finite field arithmetic using quasi-normal bases

2007
*
Finite Fields and Their Applications
*

Efficient multiplication in

doi:10.1016/j.ffa.2006.09.008
fatcat:6pm54fwepfbv5ku52j3q53jsc4
*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. ... These bases generalize the notion of normal bases*and*provide simple exponentiation to the power q in F q n . For some extension*fields*F q n over F q , we construct quasi-normal bases of low density. ... Let F q n be a*finite**field*. (1) Let H be a submodule of F q n*and*ζ ∈ H . ...##
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On Diagonal Equations over Finite Fields

2001
*
Finite Fields and Their Applications
*

Ed. 26 (1989), 55}59) greatly

doi:10.1006/ffta.2000.0316
fatcat:etqz75qnt5hlbp55w2hacbmaqu
*and*proves that the conjecture posed by Powell (J. Number ¹heory 18 (1984), 34}40) holds for general n3N as well. ... ACKNOWLEDGMENT The authors thank the referee for his valuable suggestions, especially, for pointing out references [1]*and*[2] . ... Let ON DIAGONAL EQUATIONS OVER*FINITE**FIELDS*u H "k H !k L '0, j"1, 2, 2 , n!1. We get from (8) that L\ H a H SH K #a L "0. ...##
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Explicit Computation of Isomorphisms between Finite Fields

2002
*
Finite Fields and Their Applications
*

We give algorithms to solve this problem efficiently in practice,

doi:10.1006/ffta.2001.0344
fatcat:q2a3kekryza27alzhzjab2tvkm
*and*as an*application*, we also give an algorithm for factoring a polynomial P 2 F p ½X over a*finite*extension of F p . # 2002 Elsevier ... Although it is easy to prove that two*finite**fields*having the same cardinality are isomorphic, the proof uses embeddings into an algebraic closure (or at least into a common overfield), hence is not constructive ... The two*finite**fields*K 1*and*K 2 both have p n elements, hence are isomorphic. ...
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