Short Kloosterman Sums for Polynomials over Finite Fields

Abstract

Abstract. We 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 F q [x]/M(x) for collections of polynomials either of the form f −1 g −1 or of the form f −1 g −1 + a f g, 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.

Energy bounds, bilinear forms and their applications in function fields

“…This proof is quite similar to the proof of Theorem 2.2, and just requires slightly more attention to detail to obtain more explicit bounds. This expands upon some ideas from [6,14]. Again, recall the notation a 0 , r 0 and F 0 from (5.1).…”

“…As mentioned previously, bounds on sums of this form are used as tools to establish some of the main results in [25]. Here we take a different approach to bounding these sums which can hold for arbitrary F , based on the ideas of Bourgain and Garaev [8], Garaev [14], Fouvry and Shparlinski [12], Banks, Harcharras and Shparlinski [6] and Irving [17].…”

A Variant of NTRU with Non-invertible Polynomials

“…It has been shown in [1] that using the method of [7,8] (see also [4]), one can obtain an analogue of Theorem 1 for polynomials G of the form G = G 1 G 2 where G 1 , G 2 are irreducible polynomials of very small degree; see also [2].…”