Dickson Polynomials Research Articles

A Note on “Differential Properties of $x \mapsto x^{2^{t}-1}$”

Blondeau and colleagues recently conjectured a characterization of when x2t-1 is an almost perfect nonlinear function on F2n. In this paper, we will prove the conjecture when n is even. The short proof uses a theorem of Hou and colleagues concerning reversed Dickson polynomials and a theorem of Payne.

Dickson polynomials over finite fields

In this paper we introduce the notion of Dickson polynomials of the (k+1)-th kind over finite fields Fpm and study basic properties of this family of polynomials. In particular, we study the factorization and the permutation behavior of Dickson polynomials of the third kind.

The minimal polynomial of 2 cos(π/q) and Dickson polynomials

The number λq=2cos(π/q),q∈N,q⩾3, appears in the study of Hecke groups which are Fuchsian groups, and in the study of regular polyhedra. There are many partial results about the minimal polynomial of this algebraic number. Here we obtain the general formula and it is Möbius inversion for this minimal polynomial by means of the Dickson polynomials and the Möbius inversion theory. Moreover, we investigate the homogeneous cyclotomic, Chebychev and Dickson polynomials in two variables and we show that our main results in one variable case nicely extend to this situation. In this paper, the deep results concerning these polynomials are proved by elementary arguments.

Semibent Functions From Dillon and Niho Exponents, Kloosterman Sums, and Dickson Polynomials

Kloosterman sums have recently become the focus of much research, most notably due to their applications in cryptography and coding theory. In this paper, we extensively investigate the link between the semibentness property of functions in univariate forms obtained via Dillon and Niho functions and Kloosterman sums. In particular, we show that zeros and the value four of binary Kloosterman sums give rise to semibent functions in even dimension with maximum degree. Moreover, we study the semibentness property of functions in polynomial forms with multiple trace terms and exhibit criteria involving Dickson polynomials.

Bent and Hyper-Bent Functions in Polynomial Form and Their Link With Some Exponential Sums and Dickson Polynomials

Bent functions are maximally nonlinear Boolean functions with an even number of variables. They were introduced by Rothaus in 1976. For their own sake as interesting combinatorial objects, but also because of their relations to coding theory (Reed-Muller codes) and applications in cryptography (design of stream ciphers), they have attracted a lot of research, specially in the last 15 years. The class of bent functions contains a subclass of functions, introduced by Youssef and Gong in 2001, the so-called hyper-bent functions, whose properties are still stronger and whose elements are still rarer than bent functions. Bent and hyper-bent functions are not classified. A complete classification of these functions is elusive and looks hopeless. So, it is important to design constructions in order to know as many of (hyper)-bent functions as possible. This paper is devoted to the constructions of bent and hyper-bent Boolean functions in polynomial forms. We survey and present an overview of the constructions discovered recently. We extensively investigate the link between the bentness property of such functions and some exponential sums (involving Dickson polynomials) and give some conjectures that lead to constructions of new hyper-bent functions.

On the Waring problem with Dickson polynomials in finite fields

We improve recent results of D. Gomez and A. Winterhof on the Waring problem with Dickson polynomials in finite fields. Our approach is based on recent advances in arithmetic combinatorics in arbitrary finite fields due to A. Glibichuk and M. Rudnev.

Key exchange based on Dickson polynomials over finite field with 2m

Give a survey of the important properties of Dickson polynomial Dn(x,1), and prove that Dickson polynomial Dn(x,1) over finite field with 2m is a permutation polynomial if and only if n is odd. Use this fact to construct a new key agreement protocol which is secure, feasible and extensible.

Necessary conditions for reversed Dickson polynomials to be permutational

We give several necessary conditions for the reversed Dickson polynomial D n ( 1 , x ) to be a permutation of F q . In particular, we give explicit evaluation of ∑ a ∈ F q D n ( 1 , a ) i for i = 1 , 2 .

Dembowski–Ostrom polynomials from Dickson polynomials

Motivated by several recent results, we determine precisely when F k ( X d , a ) − F k ( 0 , a ) is a Dembowski–Ostrom polynomial, where F k ( X , a ) is a Dickson polynomial of the first or second kind. As a consequence, we obtain a classification of all such polynomials which are also planar; all examples found are equivalent to previously known examples.

Two classes of permutation polynomials over finite fields

Two new classes of permutation polynomials over finite fields are presented: (i) f ( x ) = ( 1 − x − x 2 ) x 3 e + 1 2 − 1 − x + x 2 over F 3 e where e is a positive even integer; (ii) g n , p ( x ) = ∑ n p ⩽ l ⩽ n p − 1 n l ( l n − l ( p − 1 ) ) × x n − l ( p − 1 ) over F p e where e is a positive integer such that e ≡ 0 ( mod 2 ) if p = 2 , and n = ( p − 1 ) p m + p 0 e + p 1 e + ⋯ + p ( p − 1 ) e , ( m − 1 , e ) = 1 . The permutation polynomial in (i) answers an open question about reversed Dickson polynomials.

Crosscorrelation of m-Sequences, Exponential Sums and Dickson Polynomials

Binary maximal-length sequences (or m-sequences) are sequences of period 2 m -1 generated by a linear recursion of degree m. Decimating an m-sequence {s t } by an integer d relatively prime to 2 m 1 leads to another m-sequence {s dt } of the same period. The crosscorrelation of m-sequences has many applications in communication systems and has been an important and well studied problem during more than 40 years. This paper presents an updated survey on the crosscorrelation between binary m-sequences with at most five-valued crosscorrelation and shows some of the many recent connections of this problem to several areas of mathematics such as exponential sums and Dickson polynomials.

Reversed Dickson polynomials over finite fields

Reversed Dickson polynomials over finite fields are obtained from Dickson polynomials D n ( x , a ) over finite fields by reversing the roles of the indeterminate x and the parameter a. We study reversed Dickson polynomials with emphasis on their permutational properties over finite fields. We show that reversed Dickson permutation polynomials (RDPPs) are closely related to almost perfect nonlinear (APN) functions. We find several families of nontrivial RDPPs over finite fields; some of them arise from known APN functions and others are new. Among RDPPs on F q with q < 200 , with only one exception, all belong to the RDPP families established in this paper.

On Certain Values of Kloosterman Sums

Let K q n(a) be a Kloosterman sum over the finite field F q n of characteristic p . In this note so called subfield conjecture is proved: if a ne 0 belongs to the proper subfield F q of F q n, then K q n(a) ne -1. This completes recent works on the subfield conjecture by Shparlinski, and Moisio and Lisonek. The problem is motivated by some applications to bent functions. Moreover, in the course of the proof a large class of translates of Dickson polynomials are shown to be irreducible.

Hyperbent Functions, Kloosterman Sums, and Dickson Polynomials

This paper is devoted to the study of hyperbent functions in n variables, i.e., bent functions which are bent up to a change of primitive roots in the finite field GF(2n). Our main purpose is to obtain an explicit trace representation for some classes of hyperbent functions. We first exhibit an infinite class of monomial functions which is not hyperbent. This result indicates that Kloosterman sums on F2 m cannot be zero at some points. For functions with multiple trace terms, we express their spectra by means of Dickson polynomials. We then introduce a new tool to describe these hyperbent functions. The effectiveness of this new method can be seen from the characterization of a new class of binomial hyperbent functions.

ON THE NUMBER OF SOLUTIONS OF EQUATIONS OF DICKSON POLYNOMIALS OVER FINITE FIELDS

Let $k, n_1, \dots, n_k$ be fixed positive integers, $c_1, \dots, c_k \in GF(q)^*$, and $a_1, \dots, a_k, c \in GF(q)$. We study the number of solutions in $GF(q)$ of the equation $c_1D_{n_1}(x_1, a_1) + c_2D_{n_2}(x_2, a_2) + \cdots + c_kD_{n_k}(x_k, a_k) = c$, where each $D_{n_i}(x_i, a_i)$, $1 \leq i \leq k$, is the Dickson polynomial of degree $n_i$ with parameter $a_i$. We also employ the results of the $k = 1$ case to recover the cardinality of preimages and images of Dickson polynomials obtained earlier by Chou, Gomez-Calderon and Mullen [1].

On the moments of Kloosterman sums and fibre products of Kloosterman curves

Let q = p r with p = 3 and r ⩾ 2 . We give a recursion formula for the moments of a Kloosterman sum over the finite field F q , which utilizes known weight formulae for the ternary Melas code M of length q − 1 . The method is illustrated by giving explicit formulae for the moments up to the tenth moment. As an application for the formulae, and for their analogues obtained earlier in case p = 2 , we get the exact number of rational points on fibre products of certain Kloosterman curves. As a corollary we obtain identities between Ramanujan's tau-function, Kronecker class numbers, and Dickson polynomials.

Complete decomposition of Dickson-type polynomials and related Diophantine equations

We characterize decomposition over C of polynomials f n ( a , B ) ( x ) defined by the generalized Dickson-type recursive relation ( n ⩾ 1 ) f 0 ( a , B ) ( x ) = B , f 1 ( a , B ) ( x ) = x , f n + 1 ( a , B ) ( x ) = x f n ( a , B ) ( x ) − a f n − 1 ( a , B ) ( x ) , where B , a ∈ Q or R . As a direct application of the uniform decomposition result, we fully settle the finiteness problem for the Diophantine equation f n ( a , B ) ( x ) = f m ( a ˆ , B ˆ ) ( y ) . This extends and completes work of Dujella/Tichy and Dujella/Gusić concerning Dickson polynomials of the second kind. The method of the proof involves a new sufficient criterion for indecomposability of polynomials with fixed degree of the right component.

Generalized reciprocals, factors of Dickson polynomials and generalized cyclotomic polynomials over finite fields

We give new descriptions of the factors of Dickson polynomials over finite fields in terms of cyclotomic factors. To do this generalized reciprocal polynomials are introduced and characterized. We also study the factorization of generalized cyclotomic polynomials and their relationship to the factorization of Dickson polynomials.

On the Linear Complexity Profile of Nonlinear Congruential Pseudorandom Number Generators with Dickson Polynomials

Linear complexity and linear complexity profile are important characteristics of a sequence for applications in cryptography and Monte-Carlo methods. The nonlinear congruential method is an attractive alternative to the classical linear congruential method for pseudorandom number generation. Recently, a weak lower bound on the linear complexity profile of a general nonlinear congruential pseudorandom number generator was proven by Gutierrez, Shparlinski and the first author. For most nonlinear generators a much stronger lower bound is expected. Here, we obtain a much stronger lower bound on the linear complexity profile of nonlinear congruential pseudorandom number generators with Dickson polynomials.

Niho type cross-correlation functions via dickson polynomials and Kloosterman sums

Suppose that n=2k is even. We study the cross-correlation function between two m-sequences for Niho type decimations d=(2/sup k/-1)s+1. We develop a new technique to study the value distribution of these cross-correlation functions, which makes use of Dickson polynomials. As a first application, we derive here the distribution of the six-valued cross-correlation function for s=3 and odd k, up to a term which depends on Kloosterman sums. In addition, applying simpler methods, we prove a theorem providing Niho type decimations with four-valued cross-correlation functions and their distribution. We conjecture that the latter result actually covers all such decimations.