Further results on permutation polynomials of the form (xpm − x + δ)s + L(x) over 𝔽p2m

Several classes of permutation polynomials with given form over [Formula: see text] were recently proposed by Tu, Zeng, Li and Helleseth. In this paper, continuing their work, we present more permutation polynomials of the form [Formula: see text] over the finite field [Formula: see text] where [Formula: see text] is a linearized polynomial with coefficients in [Formula: see text].

Bent and Permutational Properties of Budaghyan-Carlet Hexanomials

The set related to the existence of Budaghyan-Carlet hexanomials is characterized. By investigating the component functions, it is also proved that none of Budaghyan-Carlet hexanomials cannot be turned into a permutation by adding any linearized polynomial. As a byproduct, a class of quadratic bent functions is obtained.

The dynamics of linearized polynomials

Let F = GF(q). To any polynomial G ∈ F[x] there is associated a mapping Ĝ on the set IF of monic irreducible polynomials over F. We present a natural and effective theory of the dynamics of Ĝ for the case in which G is a monic q-linearized polynomial. The main outcome is the following theorem.Assume that G is not of the form , where l ≥ 0 (in which event the dynamics is trivial). Then, for every integer n ≥ 1 and for every integer k ≥ 0, there exist infinitely many μ ∈ IF. having preperiod k and primitive period n with respect to Ĝ.Previously, Morton, by somewhat different means, had studied the primitive periods of Ĝ when G = xq – ax, α a non-zero element of F. Our theorem extends and generalizes Morton's result. Moreover, it establishes a conjecture of Morton for the class of q-linearized polynomials.

The factorable core of polynomials over finite fields

For a polynomial f(x) over a finite field Fq, denote the polynomial f(y)−f(x) by ϕf(x, y). The polynomial ϕf has frequently been used in questions on the values of f. The existence is proved here of a polynomial F over Fq of the form F = Lr, where L is an affine linearized polynomial over Fq, such that f = g(F) for some polynomial g and the part of ϕf which splits completely into linear factors over the algebraic closure of Fq is exactly φF. This illuminates an aspect of work of D. R. Hayes and Daqing Wan on the existence of permutation polynomials of even degree. Related results on value sets, including the exhibition of a class of permutation polynomials, are also mentioned.