Isomorphism classes of Doche–Icart–Kohel curves over finite fields

We give explicit formulas for the number of distinct elliptic curves over a finite field, up to isomorphism, in two families of curves introduced by C. Doche, T. Icart and D.R. Kohel.     

Mean value of mixed exponential sums

For integers , , , with , and Dirichlet character , we define a mixed exponential sum

where , and denotes the summation over all with . The main purpose of this paper is to study the mean value of

and to give a related identity on the mean value of the general Kloosterman sum

where .

     

Estimate for exponential sums and its applications

In this paper, we establish a new estimate on exponential sums by using the Bombieri-type theorem and the modified Huxley-Hooley contour. We also generalize the famous Goldbach-Vinogradov theorem, via different argument from that of Vinogradov. In particular, our major arcs are quite large and these enlarged major arcs are treated by the estimate we have established.     

Degree matrices and divisibility of exponential sums over finite fields

By using the degree matrix, we provide an elementary and algorithmic approach to estimating the divisibility of exponential sums over prime fields, which improves the Adolphson–Sperber theorem obtained by using the Newton polyhedron. Our result also improves the Ax–Katz theorem on estimating the number of rational points on hypersurfaces over prime fields.     

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.     

On the isomorphism classes of Legendre elliptic curves over finite fields

In this paper,the number of isomorphism classes of Legendre elliptic curves over finite field is enumerated.     

Several classes of complete permutation polynomials

In this paper, three classes of monomials and one class of trinomials over finite fields of even characteristic are proposed. They are proved to be complete permutation polynomials.     

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.     

Characterizations and constructions of n-to-1 mappings over finite fields

n-to-1 mappings have wide applications in many areas, especially in cryptography, finite geometry, coding theory and combinatorial design. In this paper, many classes of n-to-1 mappings over finite fields are studied. First, we provide a characterization of general n-to-1 mappings over ${\mathbb{F}}_{p^m}$ by means of the Walsh transform. Then, we completely determine 3-to-1 polynomials with degree no more than 4 over ${\mathbb{F}}_{p^m}$. Furthermore, we obtain an AGW-like criterion for characterizing some close relationship between the n-to-1 property of a mapping over finite set A and that of another mapping over a subset of A. Finally, we apply the AGW-like criterion into several forms of polynomials and obtain some explicit n-to-1 mappings. Especially, three explicit constructions of the form $x^{r}h\left(x^s\right)$ from the cyclotomic perspective, and several classes of n-to-1 mappings of the form $g(x^{q^k}-x+\delta )+cx$ are provided.     

有限域上多项式的指数和及其L-函数

L-函数蕴藏着深刻的算术信息,是数论中重要的研究对象.有限域上多项式的指数和及其L-函数在一般情形下难以计算.通过利用高斯和及多项式的次数矩阵的Smith标准形,得到了在特定情形下有限域上一类多项式的指数和及其L-函数的具体公式.     

The depth spectrums of constacyclic codes over finite chain rings

In light of the generator polynomials of constacyclic codes over finite chain rings, the depth spectrum of constacyclic codes can be determined if (n,p)=1$(n,p)=1$.     

Maximality of Ciani curves over finite fields

In this paper, we will study Ciani curves in characteristic $p\ge 3$, in particular their standard forms $C:x^4+y^4+z^4+r{x}^2{y}^2+s{y}^2{z}^2+t{z}^2{x}^2=0$. It is well-known that any Ciani curve is a non-hyperelliptic curve of genus 3, and its Jacobian variety is isogenous to the product of three elliptic curves. As a main result, we will show that if C is superspecial, then $r,s,t$ belong to ${\mathbb{F}}_{p^2}$ and C is maximal or minimal over ${\mathbb{F}}_{p^2}$. Moreover, in this case we will provide a simple criterion in terms of $r,s,t,p$ that tells whether C is maximal (resp. minimal) over ${\mathbb{F}}_{p^2}$.