Visual properties of generalized Kloosterman sums
نویسندگان
چکیده
For a positive integer m and a subgroup Λ of the unit group (Z/mZ)×, the corresponding generalized Kloosterman sum is the function K(a, b,m,Λ) = ∑ u∈Λ e( au+bu−1 m ). Unlike classical Kloosterman sums, which are real valued, generalized Kloosterman sums display a surprising array of visual features when their values are plotted in the complex plane. In a variety of instances, we identify the precise number-theoretic conditions that give rise to particular phenomena.
منابع مشابه
A transform property of Kloosterman sums
An expression for the number of times a certain trace function associated with a Kloosterman sum on an extension field assumes a given value in the base field is given and its properties explored. The relationship of this result to the enumeration of certain types of irreducible polynomials over fields of characteristic two or three and to the weights in the dual of a Melas code is considered. ...
متن کاملRecursive formulas generating power moments of multi-dimensional Kloosterman sums and m-multiple power moments of Kloosterman sums
Abstract. In this paper, we construct two binary linear codes associated with multi-dimensional and m−multiple power Kloosterman sums (for any fixed m) over the finite field Fq. Here q is a power of two. The former codes are dual to a subcode of the binary hyper-Kloosterman code. Then we obtain two recursive formulas for the power moments of multi-dimensional Kloosterman sums and for the m-mult...
متن کاملExplicit values of multi-dimensional Kloosterman sums for prime powers, II
For any integer m > 1 fix ζm = exp(2πi/m), and let Z ∗ m denote the group of reduced residues modulo m. Let q = pα, a power of a prime p. The hyper-Kloosterman sums of dimension n > 0 are defined for q by R(d, q) = ∑ x1,...,xn∈Z∗ q ζ x1+···+xn+d(x1···xn) q (d ∈ Zq), where x−1 denotes the multiplicative inverse of x modulo q. Salie evaluated R(d, q) in the classical setting n = 1 for even q, and...
متن کاملA new class of hyper-bent functions and Kloosterman sums
This paper is devoted to the characterization of hyper-bent functions. Several classes of hyper-bent functions have been studied, such as Charpin and Gong’s ∑ r∈R Tr1 (arx r(2m−1)) and Mesnager’s ∑ r∈R Tr1 (arx r(2m−1)) + Tr1(bx 2n−1 3 ), where R is a set of representations of the cyclotomic cosets modulo 2 + 1 of full size n and ar ∈ F2m . In this paper, we generalize their results and conside...
متن کاملThe Value 4 of Binary Kloosterman Sums
Kloosterman sums have recently become the focus of much research, most notably due to their applications in cryptography and their relations to coding theory. Very recently Mesnager has showed that the value 4 of binary Kloosterman sums gives rise to several infinite classes of bent functions, hyper-bent functions and semi-bent functions in even dimension. In this paper we analyze the different...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2017