scholarly journals Fast construction of irreducible polynomials over finite fields

2012 ◽  
Vol 194 (1) ◽  
pp. 77-105 ◽  
Author(s):  
Jean-Marc Couveignes ◽  
Reynald Lercier
Keyword(s):  
Finite Fields ◽  
Irreducible Polynomials ◽  
Polynomials Over Finite Fields

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.

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 Counting irreducible polynomials over finite fields

2010 ◽  
Vol 60 (3) ◽  
pp. 881-886
Author(s):  
Qichun Wang ◽  
Haibin Kan
Keyword(s):  
Finite Fields ◽  
Irreducible Polynomials ◽  
Polynomials Over Finite Fields
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