The coefficients of primitive polynomials over finite fields

For , we prove that there always exists a primitive polynomial of degree over a finite field with the first and second coefficients prescribed in advance.

     

A method for constructing a self-dual normal basis in odd characteristic extension fields

This paper proposes a useful method for constructing a self-dual normal basis in an arbitrary extension field F_p^m such that 4p does not divide m(p−1) and m is odd. In detail, when the characteristic p and extension degree m satisfies the following conditions (1) and either (2a) or (2b); (1) 2km+1 is a prime number, (2a) the order of p in F_2km+1 is 2km, (2b) 2km and the order of p in F_2km+1 is km, we can consider a class of Gauss period normal bases. Using this Gauss period normal basis, this paper shows a method to construct a self-dual normal basis in the extension field F_p^m.     

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.     

Several classes of complete permutation polynomials over finite fields of even characteristic

In this paper, we find three classes of complete permutation polynomials over finite fields of even characteristic. The first class of quadrinomials is complete in the sense of addition. The second and third classes of binomials and trinomials are complete in multiplication. Moreover, a result related to the complete property in multiplication of a special class of polynomials is also given.     

Some classes of monomial complete permutation polynomials over finite fields of characteristic two

In this paper, four classes of complete permutation polynomials over finite fields of characteristic two are presented. To consider the permutation property of the first three classes, Dickson polynomials play a key role. The fourth class is a generalization of a known result. In addition, we also calculate the inverses of these bijective monomials.     

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].     

On a conjecture of Fernando,Hou and Lappano concerning permutation polynomials over finite fields

Let ${\mathbb{F}}_q$ be the finite field with q elements and let $p=\text{char}{\mathbb{F}}_q$. It was conjectured that for integers $e\ge 2$ and $1\le a\le pe-2$, the polynomial $X^{q-2}+X^{q^2-2}+\cdots +X^{q^a-2}$ is a permutation polynomial of ${\mathbb{F}}_{q^e}$ if and only if (i) $a=2$ and $q=2$, or (ii) $a=1$ and $\text{gcd}(q-2,q^e-1)=1$. In the present paper we confirm this conjecture.     

Ambiguity,deficiency and differential spectrum of normalized permutation polynomials over finite fields

We obtain exact formulas for the differential spectrum, deficiency and ambiguity of all normalized permutation polynomials of degree up to six over finite fields.     

Weak version of restriction estimates for spheres and paraboloids in finite fields

We study $L^p-L^r$ restriction estimates for algebraic varieties in d-dimensional vector spaces over finite fields. Unlike the Euclidean case, if the dimension d is even, then it is conjectured that the $L^{(2d+2)/(d+3)}-L^2$ Stein–Tomas restriction result can be improved to the $L^{(2d+4)/(d+4)}-L^2$ estimate for both spheres and paraboloids in finite fields. In this paper we show that the conjectured $L^p-L^2$ restriction estimate holds in the specific case when test functions under consideration are restricted to d-coordinate functions or homogeneous functions of degree zero. To deduce our result, we use the connection between the restriction phenomena for our varieties in d dimensions and those for homogeneous varieties in $(d+1)$ dimensions.     

-adic formal series and primitive polynomials over finite fields

In this paper, we investigate the Hansen-Mullen conjecture with the help of some formal series similar to the Artin-Hasse exponential series over -adic number fields and the estimates of character sums over Galois rings. Given we prove, for large enough , the Hansen-Mullen conjecture that there exists a primitive polynomial over of degree with the -th ( coefficient fixed in advance except when if is odd and when if is even.