Irrationality of the Angles of Kloosterman Sums over Finite Field

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.

Download Full-text

On the curve Yn = Xℓ(Xm + 1) over finite fields

Advances in Geometry ◽

10.1515/advgeom-2017-0041 ◽

2019 ◽

Vol 19 (2) ◽

pp. 263-268 ◽

Cited By ~ 1

Author(s):

Saeed Tafazolian ◽

Fernando Torres

Keyword(s):

Finite Field ◽

Finite Fields ◽

Plane Curve ◽

Rational Points ◽

Weil Bound

Abstract Let 𝓧 be the nonsingular model of a plane curve of type yn = f(x) over the finite field F of order q2, where f(x) is a separable polynomial of degree coprime to n. If the number of F-rational points of 𝓧 attains the Hasse–Weil bound, then the condition that n divides q+1 is equivalent to the solubility of f(x) in F; see [20]. In this paper, we investigate this condition for f(x) = xℓ(xm+1).

Download Full-text

The analog of the Erdös distance problem in finite fields

International Journal of Number Theory ◽

10.1142/s1793042117501275 ◽

2017 ◽

Vol 13 (09) ◽

pp. 2319-2333

Author(s):

S. D. Adhikari ◽

Anirban Mukhopadhyay ◽

M. Ram Murty

Keyword(s):

Finite Field ◽

Finite Fields ◽

Kloosterman Sums ◽

Vector Spaces ◽

Characteristic 2 ◽

Distance Problem ◽

Erdős Distance Problem ◽

Erdos Distance Problem

In this paper, we give a proof of the result of Iosevich and Rudnev [Erdös distance problem in vector spaces over finite fields, Trans. Amer. Math. Soc. 359 (2007) 6127–6142] on the analog of the Erdös–Falconer distance problem in the case of a finite field of characteristic [Formula: see text], where [Formula: see text] is an odd prime, without using estimates for Kloosterman sums. We also address the case of characteristic 2.

Download Full-text

Maximal Sets of Pairwise Orthogonal Vectors in Finite Fields

Canadian Mathematical Bulletin ◽

10.4153/cmb-2011-160-x ◽

2012 ◽

Vol 55 (2) ◽

pp. 418-423 ◽

Cited By ~ 3

Author(s):

Le Anh Vinh

Keyword(s):

Positive Integer ◽

Finite Field ◽

Finite Fields ◽

Bilinear Form ◽

Symmetric Bilinear Form

AbstractGiven a positive integern, a finite fieldofqelements (qodd), and a non-degenerate symmetric bilinear formBon, we determine the largest possible cardinality of pairwiseB-orthogonal subsets, that is, for any two vectorsx,y∈ Ε, one hasB(x,y) = 0.

Download Full-text

Monomial Generalized Almost Perfect Nonlinear Functions

International Journal of Foundations of Computer Science ◽

10.1142/s0129054120500161 ◽

2020 ◽

Vol 31 (03) ◽

pp. 411-419

Author(s):

Masamichi Kuroda

Keyword(s):

Finite Field ◽

Finite Fields ◽

Algebraic Degree ◽

Apn Functions ◽

Nonlinear Functions ◽

Almost Perfect Nonlinear ◽

Basic Properties ◽

Almost Perfect Nonlinear Functions

Generalized almost perfect nonlinear (GAPN) functions were defined to satisfy some generalizations of basic properties of almost perfect nonlinear (APN) functions for even characteristic. In particular, on finite fields of even characteristic, GAPN functions coincide with APN functions. In this paper, we study monomial GAPN functions for odd characteristic. We give monomial GAPN functions whose algebraic degree are maximum or minimum on a finite field of odd characteristic. Moreover, we define a generalization of exceptional APN functions and give typical examples.

Download Full-text

Restriction Operators Acting on Radial Functions on Vector Spaces over Finite Fields

Canadian Mathematical Bulletin ◽

10.4153/cmb-2014-016-2 ◽

2014 ◽

Vol 57 (4) ◽

pp. 834-844

Author(s):

Doowon Koh

Keyword(s):

Finite Field ◽

Finite Fields ◽

Vector Spaces ◽

Dimensional Vector ◽

Algebraic Varieties ◽

Test Functions ◽

Radial Functions ◽

Zero Radius ◽

Intersection Points ◽

Field Setting

AbstractWe study Lp → Lr restriction estimates for algebraic varieties V in the case when restriction operators act on radial functions in the finite field setting. We show that if the varieties V lie in odd dimensional vector spaces over finite fields, then the conjectured restriction estimates are possible for all radial test functions. In addition, assuming that the varieties V are defined in even dimensional spaces and have few intersection points with the sphere of zero radius, we also obtain the conjectured exponents for all radial test functions.

Download Full-text

On the number of permutation polynomials over a finite field

International Journal of Number Theory ◽

10.1142/s1793042116500949 ◽

2016 ◽

Vol 12 (06) ◽

pp. 1519-1528

Author(s):

Kwang Yon Kim ◽

Ryul Kim ◽

Jin Song Kim

Keyword(s):

Finite Field ◽

Finite Fields ◽

Roots Of Unity ◽

Permutation Polynomials

In order to extend the results of [Formula: see text] in [P. Das, The number of permutation polynomials of a given degree over a finite field, Finite Fields Appl. 8(4) (2002) 478–490], where [Formula: see text] is a prime, to arbitrary finite fields [Formula: see text], we find a formula for the number of permutation polynomials of degree [Formula: see text] over a finite field [Formula: see text], which has [Formula: see text] elements, in terms of the permanent of a matrix. We write down an expression for the number of permutation polynomials of degree [Formula: see text] over a finite field [Formula: see text], using the permanent of a matrix whose entries are [Formula: see text]th roots of unity and using this we obtain a nontrivial bound for the number. Finally, we provide a formula for the number of permutation polynomials of degree [Formula: see text] less than [Formula: see text].

Download Full-text

Quadratic form Approach for the Number of Zeros of Homogeneous Linear Recurring Sequences over Finite Fields

Journal of Mathematics Research ◽

10.5539/jmr.v9n3p8 ◽

2017 ◽

Vol 9 (3) ◽

pp. 8

Author(s):

Yasanthi Kottegoda

Keyword(s):

Quadratic Form ◽

Finite Field ◽

Characteristic Polynomial ◽

Finite Fields ◽

Quadratic Forms ◽

Linear Recurring Sequences ◽

Exact Number ◽

Number Of Zeros ◽

Homogeneous Linear ◽

Form Approach

We consider homogeneous linear recurring sequences over a finite field $\mathbb{F}_{q}$, based on an irreducible characteristic polynomial of degree $n$ and order $m$. Let $t=(q^{n}-1)/ m$. We use quadratic forms over finite fields to give the exact number of occurrences of zeros of the sequence within its least period when $t$ has q-adic weight 2. Consequently we prove that the cardinality of the set of zeros for sequences from this category is equal to two.

Download Full-text

THE AVERAGE NUMBER OF SUBGROUPS OF ELLIPTIC CURVES OVER FINITE FIELDS

The Quarterly Journal of Mathematics ◽

10.1093/qmathj/haaa002 ◽

2020 ◽

Vol 71 (3) ◽

pp. 781-822

Author(s):

Corentin Perret-Gentil

Keyword(s):

Finite Field ◽

Elliptic Curves ◽

Finite Fields ◽

Short Interval ◽

Divisor Function ◽

Finite Abelian Groups ◽

Cyclic Subgroups ◽

Number Of Divisors ◽

Curves Over Finite Fields ◽

Euler Products

Abstract By adapting the technique of David, Koukoulopoulos and Smith for computing sums of Euler products, and using their interpretation of results of Schoof à la Gekeler, we determine the average number of subgroups (or cyclic subgroups) of an elliptic curve over a fixed finite field of prime size. This is in line with previous works computing the average number of (cyclic) subgroups of finite abelian groups of rank at most $2$. A required input is a good estimate for the divisor function in both short interval and arithmetic progressions, that we obtain by combining ideas of Ivić–Zhai and Blomer. With the same tools, an asymptotic for the average of the number of divisors of the number of rational points could also be given.

Download Full-text

On the degrees of polynomial divisors over finite fields

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s030500411600044x ◽

2016 ◽

Vol 161 (3) ◽

pp. 469-487 ◽

Cited By ~ 1

Author(s):

ANDREAS WEINGARTNER

Keyword(s):

Asymptotic Formula ◽

Finite Field ◽

Finite Fields ◽

Cycle Structure

AbstractWe show that the proportion of polynomials of degree n over the finite field with q elements, which have a divisor of every degree below n, is given by cqn−1 + O(n−2). More generally, we give an asymptotic formula for the proportion of polynomials, whose set of degrees of divisors has no gaps of size greater than m. To that end, we first derive an improved estimate for the proportion of polynomials of degree n, all of whose non-constant divisors have degree greater than m. In the limit as q → ∞, these results coincide with corresponding estimates related to the cycle structure of permutations.

Download Full-text