The number of irreducible polynomials of a given form over a finite field

1987 ◽  
Vol 41 (3) ◽  
pp. 165-169 ◽  
Author(s):  
S. A. Stepanov
Keyword(s):  
Finite Field ◽  
Irreducible Polynomials

scholarly journals Short Kloosterman Sums for Polynomials over Finite Fields

2003 ◽  
Vol 55 (2) ◽  
pp. 225-246 ◽  
Author(s):  
William D. Banks ◽  
Asma Harcharras ◽  
Igor E. Shparlinski
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Small Degree ◽  
Kloosterman Sums ◽  
Degree Relative ◽  
Irreducible Polynomials ◽  
Arbitrary Polynomial ◽  
Polynomials Over Finite Fields ◽  
Titchmarsh Theorem

AbstractWe extend to the setting of polynomials over a finite field certain estimates for short Kloosterman sums originally due to Karatsuba. Our estimates are then used to establish some uniformity of distribution results in the ring [x]/M(x) for collections of polynomials either of the form f−1g−1 or of the form f−1g−1 + afg, where f and g are polynomials coprime to M and of very small degree relative to M, and a is an arbitrary polynomial. We also give estimates for short Kloosterman sums where the summation runs over products of two irreducible polynomials of small degree. It is likely that this result can be used to give an improvement of the Brun-Titchmarsh theorem for polynomials over finite fields.

Irreducible polynomials over a finite field

1990 ◽  
Vol 30 (6) ◽  
pp. 915-925 ◽  
Author(s):  
E. N. Kuz'min
Keyword(s):  
Finite Field ◽  
Irreducible Polynomials

scholarly journals The distribution of irreducible polynomials in several indeterminates over a finite field

1968 ◽  
Vol 16 (1) ◽  
pp. 1-17 ◽  
Author(s):  
Stephen D. Cohen
Keyword(s):  
Positive Integer ◽  
Finite Field ◽  
Equivalence Class ◽  
Ordered Set ◽  
Irreducible Polynomials ◽  
Totally Positive

We consider non-zero polynomials f(x1, …, xk) in k variables x1, …, xk with coefficients in the finite field GF[q] (q = pn for some prime p and positive integer n). We assume that the polynomials have been normalised by selecting one polynomial from each equivalence class with respect to multiplication by non-zero elements of GF[q]. By the degree of a polynomial f(x1, …, xk) will be understood the ordered set (m1, …, mk), where mi is the degree of f(x1 ,…, xk) in x1(i = 1, 2, …, K). The degree (m,…, mk) of a polynomial will be called totally positive if mi>0, i = 1, 2, …, k.

scholarly journals Constructing irreducible polynomials with prescribed level curves over finite fields

2001 ◽  
Vol 27 (4) ◽  
pp. 197-200
Author(s):  
Mihai Caragiu
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Irreducible Polynomial ◽  
Irreducible Polynomials ◽  
Level Curves ◽  
Curves Over Finite Fields ◽  
Irreducibility Criterion ◽  
Absolutely Irreducible Polynomial

We use Eisenstein's irreducibility criterion to prove that there exists an absolutely irreducible polynomialP(X,Y)∈GF(q)[X,Y]with coefficients in the finite fieldGF(q)withqelements, with prescribed level curvesXc:={(x,y)∈GF(q)2|P(x,y)=c}.

Low latency systolic multipliers for finite field GF (2 m ) based on irreducible polynomials

2012 ◽  
Vol 19 (5) ◽  
pp. 1283-1289 ◽  
Author(s):  
Jia-feng Xie ◽  
Jian-jun He ◽  
Wei-hua Gui
Keyword(s):  
Finite Field ◽  
Low Latency ◽  
Irreducible Polynomials

scholarly journals The reducibility theorem for linearised polynomials over finite fields

1989 ◽  
Vol 40 (3) ◽  
pp. 407-412 ◽  
Author(s):  
Stephen D. Cohen
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Irreducible Polynomials ◽  
Polynomials Over Finite Fields ◽  
Wide Readership

A self-contained elementary account is given of the theorem of S. Agou that classifies all composite irreducible polynomials of the form over a finite field of characteristic p. Written to appeal to a wide readership, it is intended to complement the original rather technical proof and other contributions by the author and by Moreno.

scholarly journals Automaticity of primitive words and irreducible polynomials

2013 ◽  
Vol Vol. 15 no. 1 (Automata, Logic and Semantics) ◽  
Author(s):  
Anne Lacroix ◽  
Narad Rampersad
Keyword(s):  
Finite Field ◽  
Finite Automaton ◽  
Prime Numbers ◽  
Deterministic Finite Automaton ◽  
Irreducible Polynomials ◽  
Letter Alphabet ◽  
International Audience ◽  
Primitive Words

Automata, Logic and Semantics International audience If L is a language, the automaticity function A_L(n) (resp. N_L(n)) of L counts the number of states of a smallest deterministic (resp. non-deterministic) finite automaton that accepts a language that agrees with L on all inputs of length at most n. We provide bounds for the automaticity of the language of primitive words and the language of unbordered words over a k-letter alphabet. We also give a bound for the automaticity of the language of base-b representations of the irreducible polynomials over a finite field. This latter result is analogous to a result of Shallit concerning the base-k representations of the set of prime numbers.

Analysis of the Number of Terms in the Generation Polynomial of the Decimation Sequence

2011 ◽  
Vol 18 (01) ◽  
pp. 171-180
Author(s):  
Jianhua Zheng
Keyword(s):  
Finite Field ◽  
Upper Bound ◽  
Irreducible Polynomials ◽  
Linear Sequence ◽  
The Relationship

In this paper, we discuss the relationship between the generation polynomial of a linear sequence and its decimation sequence over the finite field 𝔽2. Then we propose an upper bound of the number of terms in the generation polynomial of a decimation sequence of a linear sequence whose generation polynomial is trinomial. Finally, we suggest a method to calculate the terms of a decimation polynomial and their number directly, which can be used in the construction of irreducible polynomials with a controlled number of terms, when the source trinomial and the decimation distance satisfy a certain condition.

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