Permutation polynomials from trace functions over finite fields

In this paper, we propose several classes of permutation polynomials based on trace functions over finite fields of characteristic 2. The main result of this paper is obtained by determining the number of solutions of certain equations over finite fields.     

A note on permutation polynomials over finite fields

Permutation polynomials over finite fields constitute an active research area and have applications in many areas of science and engineering. In this paper, two conjectures on permutation polynomials proposed recently by Wu and Li [19] are settled. Moreover, a new class of permutation trinomials of the form $x+\gamma {\mathrm{Tr}}_{q^n/q}(x^k)$ is also presented, which generalizes two examples of [10].     

Self-dual normal integral bases for infinite unramified extensions

We prove a generalization to infinite Galois extensions of local fields, of a classical result by Noether on the existence of normal integral bases for finite tamely ramified Galois extensions. We also prove a self-dual normal integral basis theorem for infinite unramified Galois field extensions of local fields with finite residue fields of characteristic different from 2. This generalizes a result by Fainsilber for the finite case. To do this, we obtain an injectivity result concerning pointed cohomology sets, defined by inverse limits of norm-one groups of free finite-dimensional algebras with involution over complete discrete valuation rings.     

Existence of primitive normal pairs with one prescribed trace over finite fields

Designs, Codes and Cryptography - Given $$m, n, qin mathbb {N}$$ such that q is a prime power and $$mge 3$$ , $$ain mathbb {F}_q$$ , we establish a sufficient condition for the existence of a...     

A new characterization of dual bases in finite fields and its applications

We propose a new characterization of dual bases in finite fields. Let A=(α₁,…,α_n) be a basis of F over F_q and its dual basis B=(β₁,…,β_n) with the transition matrix C∈GL_n(F_q) such that (β₁,…,β_n)=(α₁,…,α_n)C. We show that holds for all 1?k?n, where T_k∈M_n(F_q) satisfies α_k(α₁,…,α_n)=(α₁,…,α_n)T_k. Conversely, suppose F=F_q(α_k^′) and for some 1?k^′?n and G∈GL_n(F_q), then B is equivalent to (α₁,…,α_n)G. As applications, we can construct the dual basis of a given basis A or determine whether the dual basis of A satisfies the desired conditions from T_k. This generalizes the results obtained by Liao and Sun for normal bases. Furthermore, we give a simple proof of the theorem of Gollmann, Wang and Blake for polynomial bases.     

The number of irreducible polynomials over finite fields of characteristic 2 with given trace and subtrace

In this paper we obtained the formula for the number of irreducible polynomials with degree n over finite fields of characteristic two with given trace and subtrace. This formula is a generalization of the result of Cattell et al. (2003) [2].     

Primitive normal polynomials with a prescribed coefficient

In this paper, we established the existence of a primitive normal polynomial over any finite field with any specified coefficient arbitrarily prescribed. Let n15 be a positive integer and q a prime power. We prove that for any aF_q and any 1m<n, there exists a primitive normal polynomial f(x)=xⁿ−σ₁xⁿ⁻¹++(−1)ⁿ⁻¹σ_n−1x+(−1)ⁿσ_n such that σ_m=a, with the only exceptions σ₁≠0. The theory can be extended to polynomials of smaller degree too.     

Explicit construction of self-dual integral normal bases for the square-root of the inverse different

Let K be a finite extension of Qp, let L/K be a finite abelian Galois extension of odd degree and let OL be the valuation ring of L. We define AL/K to be the unique fractional OL-ideal with square equal to the inverse different of L/K. For p an odd prime and L/Qp contained in certain cyclotomic extensions, Erez has described integral normal bases for AL/Qp that are self-dual with respect to the trace form. Assuming K/Qp to be unramified we generate odd abelian weakly ramified extensions of K using Lubin-Tate formal groups. We then use Dwork's exponential power series to explicitly construct self-dual integral normal bases for the square-root of the inverse different in these extensions.     

Trace formulae for irreducible polynomials over with minimal order roots in

The extension field where q is a prime divisor of (P−1), has a unique structure. This paper describes this unique structure and uses it to derive formulas relating the trace values for elements in . These formulas can be refined for certain elements to produce a formula for the trace.