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Finite Fields and Their Applications [chapter]

Rudolf Lidl, Harald Niederreiter
1996 Handbook of Algebra  
Acknowledgments We would like to thank the anonymous referee for the very useful suggestions 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 .  ... 
doi:10.1016/s1570-7954(96)80013-1 fatcat:noprxntirvchplkwjbue6amsma

A decade of Finite Fields and Their Applications

Gary L. Mullen
2005 Finite Fields and Their Applications  
The journal 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  ... 
doi:10.1016/j.ffa.2005.05.009 fatcat:rgii7n2uanc27ohm3tse4tcbhi

Binary Representations of Finite Fields and Their Application to Complexity Theory

Jürg Ganz
1996 Finite Fields and Their Applications  
Binary representations of 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.  ... 
doi:10.1006/ffta.1996.0022 fatcat:pevl73mnfndhrbugi2urgia6s4

Finite Upper Half Planes over Finite Fields

Jeff Angel
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.  ... 
doi:10.1006/ffta.1996.0005 fatcat:xgorbcx345eehaore4zamex2fm

On group automata over finite fields

A.V. Kelarev
2004 Finite Fields and Their Applications  
We investigate the structure of incidence rings of group automata over 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  ... 
doi:10.1016/s1071-5797(03)00043-1 fatcat:vjmc4zi6avelrkt5m7mr3qfn5i

On Diagonal Equations over Finite Fields

Sun Qi
1997 Finite Fields and Their Applications  
We get an explicit formula for the number of solutions of a diagonal equation over 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.  ... 
doi:10.1006/ffta.1996.0173 fatcat:3yyi5dlkezhlfcka2oeliv3q7m

Power sums of polynomials over finite fields and applications: A survey

Dinesh S. Thakur
2015 Finite Fields and Their Applications  
We mention several 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.  ... 
doi:10.1016/j.ffa.2014.08.004 fatcat:e7wpvlv3afcxri2he3yadyfvhm

Dickson polynomials over finite fields

Qiang Wang, Joseph L. Yucas
2012 Finite Fields and Their Applications  
In this paper we introduce the notion of Dickson polynomials of the (k + 1)-th kind over 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.  ... 
doi:10.1016/j.ffa.2012.02.001 fatcat:rfqhezt4orh45olpkchahw6lo4

Specific permutation polynomials over finite fields

José E. Marcos
2011 Finite Fields and Their Applications  
We fix a 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 .  ... 
doi:10.1016/j.ffa.2009.02.004 fatcat:rdvd577havg5rmx6rwkrccuwjy

Normal bases and primitive elements over finite fields

Giorgos Kapetanakis
2014 Finite Fields and Their Applications  
Let q be a prime power, m ≥ 2 an integer 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.  ... 
doi:10.1016/j.ffa.2013.12.002 fatcat:fpkf5jcvmracneyrp53ziu6xam

Finite Modular Forms

Ernst-Ulrich Gekeler
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.  ... 
doi:10.1006/ffta.2000.0314 fatcat:v255cjo66fghvjaagyamz2cs4a

Rings Which Are Sums of Finite Fields

A.V Kelarev
1999 Finite Fields and Their Applications  
We describe all semigroup rings 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.  ... 
doi:10.1006/ffta.1998.0240 fatcat:b5n2pjeqhfga5b7a22zgxz2uoa

Finite field arithmetic using quasi-normal bases

Christophe Negre
2007 Finite Fields and Their Applications  
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.  ...  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 .  ... 
doi:10.1016/j.ffa.2006.09.008 fatcat:6pm54fwepfbv5ku52j3q53jsc4

On Diagonal Equations over Finite Fields

JiaGui Luo, Qi Sun
2001 Finite Fields and Their Applications  
Ed. 26 (1989), 55}59) greatly 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.  ... 
doi:10.1006/ffta.2000.0316 fatcat:etqz75qnt5hlbp55w2hacbmaqu

Explicit Computation of Isomorphisms between Finite Fields

Bill Allombert
2002 Finite Fields and Their Applications  
We give algorithms to solve this problem efficiently in practice, 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.  ... 
doi:10.1006/ffta.2001.0344 fatcat:q2a3kekryza27alzhzjab2tvkm
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