Dickson Polynomials Research Articles

Residue sums of Dickson polynomials over finite fields

Given a polynomial with integral coefficients, one can inquire about the possible residues it can take in its image modulo a prime p. The sum over the distinct residues can sometimes be computed independent of the prime p; for example, Gauss showed that the sum over quadratic residues vanishes modulo a prime. In this paper we provide a closed form for the sum over distinct residues in the image of Dickson polynomials of arbitrary degree over finite fields of odd characteristic, and prove a complete characterization of the size of the value set. Our result provides the first non-trivial classification of such a sum for a family of polynomials of unbounded degree.

Dickson polynomial-based secure group authentication scheme for Internet of Things

Internet of Things (IoT) paves the way for the modern smart industrial applications and cities. Trusted Authority acts as a sole control in monitoring and maintaining the communications between the IoT devices and the infrastructure. The communication between the IoT devices happens from one trusted entity of an area to the other by way of generating security certificates. Establishing trust by way of generating security certificates for the IoT devices in a smart city application can be of high cost and expensive. In order to facilitate this, a secure group authentication scheme that creates trust amongst a group of IoT devices owned by several entities has been proposed. The majority of proposed authentication techniques are made for individual device authentication and are also utilized for group authentication; nevertheless, a unique solution for group authentication is the Dickson polynomial based secure group authentication scheme. The secret keys used in our proposed authentication technique are generated using the Dickson polynomial, which enables the group to authenticate without generating an excessive amount of network traffic overhead. IoT devices' group authentication has made use of the Dickson polynomial. Blockchain technology is employed to enable secure, efficient, and fast data transfer among the unique IoT devices of each group deployed at different places. Also, the proposed secure group authentication scheme developed based on Dickson polynomials is resistant to replay, man-in-the-middle, tampering, side channel and signature forgeries, impersonation, and ephemeral key secret leakage attacks. In order to accomplish this, we have implemented a hardware-based physically unclonable function. Implementation has been carried using python language and deployed and tested on Blockchain using Ethereum Goerli’s Testnet framework. Performance analysis has been carried out by choosing various benchmarks and found that the proposed framework outperforms its counterparts through various metrics. Different parameters are also utilized to assess the performance of the proposed blockchain framework and shows that it has better performance in terms of computation, communication, storage and latency.

Dembowski-Ostrom polynomials and Dickson polynomials

<p style='text-indent:20px;'>We give a complete classification of Dembowski-Ostrom polynomials from the composition of Dickson polynomials of arbitrary kind and monomials over finite fields. Moreover, by using a variant of the Weil bound for the number of points of affine algebraic curves over finite fields, we discuss the planarity of the obtained Dembowski-Ostrom polynomials.</p>

PERMUTATION POLYNOMIALS IN ONE INDETERMINATE OVER MODULAR ALGEBRAS

Known results about permutation polynomials in connection with coefficients in a finite field extend to algebras of the form Lv = K[X]/(p(X)v)\), where K is a finite field,(p(X) ∈ ‍ K[X] is an irreducible polynomial, and v = 1,2,..., and to the algebra of power series L[[Z]], where L = K[X]/(p(X)). Analogues of Dickson polynomials are also studied in this context.

On values of the Bessel function for generic representations of finite general linear groups

We find a recursive expression for the Bessel function of S. I. Gelfand for irreducible generic representations of GLn(Fq). We show that special values of the Bessel function can be realized as the coefficients of L-functions associated with exotic Kloosterman sums, and as traces of exterior powers of Katz's exotic Kloosterman sheaves. As an application, we show that certain polynomials, having special values of the Bessel function as their coefficients, have all of their roots lying on the unit circle. As another application, we show that special values of the Bessel function of the Shintani base change of an irreducible generic representation are related to special values of the Bessel function of the representation through Dickson polynomials.

HERMITE INTERPOLATION WITH DICKSON POLYNOMIALS AND BERNSTEIN BASIS POLYNOMIALS

In this manuscript we introduce three new algorithms: (1) An algorithm to recover an unknown polynomial in terms of Dickson polynomials of the first kind, (2) an algorithm to recover an unknown polynomial in terms Dickson polynomials of the second kind, (3) an algorithm to recover an unknown polynomial in terms of Bernstein basis polynomials, from given black boxes for the polynomial itself and its first derivative. In each algorithm, we assume that the unknown polynomial has a sparse representation in the corresponding basis. The methods presented use transformations from Dickson polynomials to Laurent polynomials, a transformation from Bernstein basis polynomials to Laurent polynomials, and a recently developed algorithm as a middle step.

On the exponents of APN power functions and Sidon sets, sum-free sets, and Dickson polynomials

<p style='text-indent:20px;'>We derive necessary conditions related to the notions, in additive combinatorics, of Sidon sets and sum-free sets, on those exponents <inline-formula><tex-math id="M1">\begin{document}$ d\in {\mathbb Z}/(2^n-1){\mathbb Z} $\end{document}</tex-math></inline-formula>, which are such that <inline-formula><tex-math id="M2">\begin{document}$ F(x) = x^d $\end{document}</tex-math></inline-formula> is an APN function over <inline-formula><tex-math id="M3">\begin{document}$ {\mathbb F}_{2^n} $\end{document}</tex-math></inline-formula> (which is an important cryptographic property). We study to what extent these new conditions may speed up the search for new APN exponents <inline-formula><tex-math id="M4">\begin{document}$ d $\end{document}</tex-math></inline-formula>. We summarize all the necessary conditions that an exponent must satisfy for having a chance of being an APN, including the new conditions presented in this work. Next, we give results up to <inline-formula><tex-math id="M5">\begin{document}$ n = 48 $\end{document}</tex-math></inline-formula>, providing the number of exponents satisfying all the conditions for a function to be APN.</p><p style='text-indent:20px;'>We also show a new connection between APN exponents and Dickson polynomials: <inline-formula><tex-math id="M6">\begin{document}$ F(x) = x^d $\end{document}</tex-math></inline-formula> is APN if and only if the reciprocal polynomial of the Dickson polynomial of index <inline-formula><tex-math id="M7">\begin{document}$ d $\end{document}</tex-math></inline-formula> is an injective function from <inline-formula><tex-math id="M8">\begin{document}$ \{y\in {\Bbb F}_{2^n}^*; tr_n(y) = 0\} $\end{document}</tex-math></inline-formula> to <inline-formula><tex-math id="M9">\begin{document}$ {\Bbb F}_{2^n}\setminus \{1\} $\end{document}</tex-math></inline-formula>. This also leads to a new and simple connection between Reversed Dickson polynomials and reciprocals of Dickson polynomials in characteristic 2 (which generalizes to every characteristic thanks to a small modification): the squared Reversed Dickson polynomial of some index and the reciprocal of the Dickson polynomial of the same index are equal.</p>

An explanation for the similar appearance of two identities involving Dickson polynomials

This article explains the similar appearance of two polynomial identities involving Dickson polynomials in char. 2, one found by Abhyankar, Cohen and Zieve, and the other found by the author.

Quantum supergroup Ur,s(osp(1,2)), Scasimir operators and Dickson polynomials

We study the center of two-parameter quantum supergroup [Formula: see text] using the Dickson polynomial. We show that the Scasimir operator is completely determined by the [Formula: see text]-deformed Chebychev polynomial, generalizing an earlier work of Arnaudon and Bauer.

Cyclic polynomials arising from the functional equation for Dickson polynomials

In this paper, we study algebraic properties of a family of certain polynomials arising from the functional equation for Dickson polynomials. We see that the roots and discriminants of those polynomials have very simple expressions, and each polynomial is cyclic. Further, we provide an irreducibility criterion analogous to the well-known criterion of Vahlen-Capelli. We finish the paper by showing that any cyclic extension of a certain field comes from a member of the family.

New Permutation Reversed Dickson Polynomials over Finite Fields

Let [Formula: see text] be an odd prime, and [Formula: see text], [Formula: see text] be nonnegative integers. Let [Formula: see text] be the reversed Dickson polynomial of the [Formula: see text]-th kind. In this paper, by using Hermite's criterion, we study the permutational properties of the reversed Dickson polynomials [Formula: see text] over finite fields in the case of [Formula: see text] with [Formula: see text]. In particular, we provide some precise characterizations for [Formula: see text] being permutation polynomials over finite fields with characteristic [Formula: see text] when [Formula: see text], or [Formula: see text], or [Formula: see text].

A new identity of Dickson polynomials

A new polynomial identity is found for Dickson polynomials in characteristic 2. The identity is used to prove that the two polynomials xq+1+x+1/a and C(x)+a have the same splitting fields over F, where F is a field of characteristic 2, 0≠a∈F, q=2n>2, and C(x) is a Müller–Cohen–Matthews polynomial of degree (q2−q)/2. In addition, a new proof is obtained of the known result that C(x) induces a permutation on F2m if and only if 2m and n are relatively prime.

A class of computational approaches for simulating fractional functional differential equations via Dickson polynomials

In this paper, a new class of computational techniques for the numerical solution of fractional functional differential equations is discussed. The proposed technique is based on Dickson polynomials with which well-known polynomials such as Fibonacci, Lucas and Chebyshev polynomials are related with some parameters. In general, the proposed combined scheme is improved by the fractional Dickson-Tau collocation technique in which a Dickson operation matrix is constructed for fractional differentiation. Then, a genetic algorithm is used to tune the unknown parameters of the proposed methods. Moreover, the error estimates and convergence of the proposed scheme are analysed. The significance of the accuracy and low computational time of the proposed scheme is verified in several numerical examples.

AN ALGORITHM TO COMPUTE THE DEGREE OF A DICKSON POLYNOMIAL

In this study, we describe an algorithm that computes the degree of a Dickson Polynomial of the First Kind from its known value at a point. Our algorithm is based on a mathematical relation between Dickson Polynomials of the First Kind and Chebyshev Polynomials of the First Kind.

LCD codes from tridiagonal Toeplitz matrices

Double Toeplitz (DT) codes are codes with a generator matrix of the form (I,T) with T a Toeplitz matrix, that is to say constant on the diagonals parallel to the main. When T is tridiagonal and symmetric we determine its spectrum explicitly by using Dickson polynomials, and deduce from there conditions for the code to be LCD. Using a special concatenation process, we construct optimal or quasi-optimal examples of binary and ternary LCD codes from DT codes over extension fields.

Factorization of Dickson polynomials over finite fields

Factorization of Dickson polynomials over finite fields

OPTIMAL CONTROL OF NONLINEAR TIME-DELAY FRACTIONAL DIFFERENTIAL EQUATIONS WITH DICKSON POLYNOMIALS

In this paper, a novel direct scheme to solve a set of time-delay fractional optimal control problems is introduced. The method firstly uses a set of Dickson polynomials as basis functions to approximate the states and control variables of the system. Next, the context of these basis functions and the use of a collocation method allow to transform the problem into a system of nonlinear algebraic equations. Finally, the unknown coefficients and control parameters in the original problem can be easily estimated by resolving the new system of equations. Given the high efficiency of the Dickson polynomials to deal with time-delay fractional systems, the proposed strategy involves a very tunable framework for direct trajectory optimization, according to the discretization procedure and the range of arbitrary nodes. The convergence analysis of the proposed method is presented, along with some illustrative examples which demonstrate its most relevant features.

Dembowski–Ostrom polynomials and reversed Dickson polynomials

We discuss the problem of classifying Dembowski–Ostrom polynomials from the composition of reversed Dickson polynomials of arbitrary kind and monomials over finite fields of odd characteristic. Moreover, by using a variant of the Weil bound for the number of points of affine algebraic curves over finite fields, we discuss the planarity of all such Dembowski–Ostrom polynomials. Planar Dembowski–Ostrom polynomials have applications in many areas including cryptography and coding theory.

Numerical solution of the fractional‐order logistic equation via the first‐kind Dickson polynomials and spectral tau method

In this article, a numerical technique for solving numerically the fractional‐order logistic equation (FLE) is presented. The fractional‐order derivative of the studied problem is given through the Caputo operator of fractional derivatives. The first kind of Dickson polynomials is used as a basis for the desirable approximate solution. The presented technique is based on the operational matrices of these polynomials in both cases of integer and fractional‐order derivatives. The handled problem will be transformed into a system of matrices that will be solved via an appropriate numerical technique. In detail, the convergence and error estimate of the suggested technique is discussed. Many numerical applications in several fields will be given to demonstrate the applicability, accuracy, and efficiency of the introduced technique. These numerical results will be compared to support our obtained theoretical results from one side and from another side to show the advantage of our method over some of the previous methods used in solving such applications.

Orthogonality of the Dickson polynomials of the $(k+1)$-th kind

We study the Dickson polynomials of the (k + 1)-th kind over the field of complex numbers. We show that they are a family of co-recursive orthogonal polynomials with respect to a quasi-definite moment functional Lk. We find an integral representation for Lk and compute explicit expressions for all of its moments.