Xiwang Cao

New methods for generating permutation polynomials over finite fields

Abstract We present two methods for generating linearized permutation polynomials over an extension of a finite field F q . These polynomials are parameterized by an element of the extension field and are permutation polynomials for all nonzero values of the element. For the case of the extension degree being odd and the size of the ground field satisfying q ≡ 3 ( mod 4 ) , these parameterized linearized permutation polynomials can be used to derive non-parameterized nonlinear permutation polynomials via a recent result of Ding et al.

Constructing permutation polynomials from piecewise permutations

We present a construction of permutation polynomials over finite fields by using some piecewise permutations. Based on a matrix approach and an interpolation approach, several classes of piecewise permutation polynomials are obtained.

Complete permutation polynomials over finite fields of odd characteristic

In this paper, we present three classes of complete permutation monomials over finite fields of odd characteristic. Meanwhile, the compositional inverses of these polynomials are also investigated.

Constructing permutations and complete permutations over finite fields via subfield-valued polynomials

We describe a recursive construction of permutation and complete permutation polynomials over a finite field F p n by using F p k -valued polynomials for several same or different factors k of n. As a result, we obtain some specific permutation polynomials which unify and generalize several previous constructions.

On the number of solutions of certain diagonal equations over finite fields

We use character sums over finite fields to give formulas for the number of solutions of certain diagonal equations of the form a 1 x 1 m 1 + a 2 x 2 m 2 + ź + a n x n m n = c . We also show that if the value distribution of character sums ź x ź F q ź ( a x m + b x ) , a , b ź F q , is known, then one can obtain the number of solutions of the system of equations { x 1 + x 2 + ź + x n = α x 1 m + x 2 m + ź + x n m = β , for some particular m. We finally apply our results to induce some facts about Warings problems and the covering radius of certain cyclic codes.

New Kloosterman sum identities and equalities over finite fields

We present some general equalities between Kloosterman sums over finite fields of arbitrary characteristics. In particular, we obtain an explicit Kloosterman sum identity over finite fields of characteristic 3.

A NOTE ON SOME CHARACTER SUMS OVER FINITE FIELDS

In this paper, we present a decomposition of the elements of a finite field and illustrate the e ciency of this decomposition in evaluating some specific exponential sums over finite fields. The results can be employed in determining the Walsh spectrum of some Boolean functions. 2010 Mathematics subject classification: primary 11T23; secondary 11T71.