Permutation trinomials over Fq3

We consider four classes of polynomials over the fields ${\mathbb{F}}_{q^3}$, $q=p^h$, $p>3$, $f_1(x)=x^{q^2+q-1}+A{x}^{q^2-q+1}+Bx$, $f_2(x)=x^{q^2+q-1}+A{x}^{q^3-q^2+q}+Bx$, $f_3(x)=x^{q^2+q-1}+A{x}^{q^2}-Bx$, $f_4(x)=x^{q^2+q-1}+A{x}^q-Bx$, where $A,B\in {\mathbb{F}}_q$. We find sufficient conditions on the pairs $(A,B)$ for which these polynomials permute ${\mathbb{F}}_{q^3}$ and we give lower bounds on the number of such pairs.     

On the boomerang uniformity of a class of permutation quadrinomials over finite fields

Let ${\mathbb{F}}_{2^n}$ be a finite field with $2^n$ elements and $f_{\underset{\_}{c}}(x)=c_0{x}^{2^m(2^k+1)}+c_1{x}^{2^{m+k}+1}+c_2{x}^{2^m+2^k}+c_3{x}^{2^k+1}\in {\mathbb{F}}_{2^n}[x]$, where n, m and k are positive integers with $n=2m$ and $\gcd ?(m,k)=e$. In this paper, motivated by a recent work of Li, Xiong and Zeng (Li et al. (2021) [12]), we further study the boomerang uniformity of $f_{\underset{\_}{c}}(x)$ by using similar ideas and carrying out particular techniques in solving equations over finite fields. As a consequence, we generalize Li, Xiong and Zeng's result from the case of m being odd and $e=1$ to that of both $m/e$ and $k/e$ being odd.     

Classification of some quadrinomials over finite fields of odd characteristic

In this paper, we completely determine all necessary and sufficient conditions such that the polynomial $f(x)=x^3+a{x}^{q+2}+b{x}^{2q+1}+c{x}^{3q}$, where $a,b,c\in {\mathbb{F}}_q^{⁎}$, is a permutation quadrinomial of ${\mathbb{F}}_{q^2}$ over any finite field of odd characteristic. This quadrinomial has been studied first in [25] by Tu, Zeng and Helleseth, later in [24] Tu, Liu and Zeng revisited these quadrinomials and they proposed a more comprehensive characterization of the coefficients that results with new permutation quadrinomials, where $char({\mathbb{F}}_q)=2$ and finally, in [16], Li, Qu, Li and Chen proved that the sufficient condition given in [24] is also necessary and thus completed the solution in even characteristic case. In [6] Gupta studied the permutation properties of the polynomial $x^3+a{x}^{q+2}+b{x}^{2q+1}+c{x}^{3q}$, where $char({\mathbb{F}}_q)=3,5$ and $a,b,c\in {\mathbb{F}}_q^{⁎}$ and proposed some new classes of permutation quadrinomials of ${\mathbb{F}}_{q^2}$.In particular, in this paper we classify all permutation polynomials of ${\mathbb{F}}_{q^2}$ of the form $f(x)=x^3+a{x}^{q+2}+b{x}^{2q+1}+c{x}^{3q}$, where $a,b,c\in {\mathbb{F}}_q^{⁎}$, over all finite fields of odd characteristic and obtain several new classes of such permutation quadrinomials.     

Low differentially uniform permutations from the Dobbertin APN function over F2n

Substitution boxes (S-boxes) play a central role in block ciphers. In substitution-permutation networks, the S-boxes should be permutation functions over ${\mathbb{F}}_{2^n}$ to realize the invertibility of the encryption. More importantly, the S-boxes should have low differential uniformity, high nonlinearity, and high algebraic degree in order to resist differential attacks, linear attacks, and higher order differential attacks, respectively. In this paper, we construct new classes of differentially 4 and 6-uniform permutations by modifying the image of the Dobbertin APN function $x^d$ with $d=2^{4k}+2^{3k}+2^{2k}+2^k-1$ over a subfield of ${\mathbb{F}}_{2^n}$. In addition, the algebraic degree and the lower bound of the nonlinearity of the constructed functions are given.     

Dynamical behavior of solutions of a reaction–diffusion model in river network

In this paper, we study the long-time behavior of solutions of a reaction–diffusion model in a one-dimensional river network, where the river network has two branches, and the water flow speeds in each branch are the same constant $\beta$. We show the existence of two critical values $c_0$ and 2 with $0<c_0<2$, and prove that when $-c_0\le \beta <2$, the population density in every branch of the river goes to 1 as time goes to infinity; when $-2<\beta <-c_0$, then, as time goes to infinity, the population density in every river branch converges to a positive steady state strictly below 1; when $\left|\beta \right|\ge 2$, the species will be washed down the stream, and so locally the population density converges to 0. Our result indicates that only if the water-flow speed is suitably small (i.e., $\left|\beta \right|<2$), the species will survive in the long run.     

Intersections between the norm-trace curve and some low degree curves

In this paper we analyze the intersection between the norm-trace curve over ${\mathbb{F}}_{q^3}$ and the curves of the form $y=a{x}^3+b{x}^2+cx+d$, giving a complete characterization of the intersection between the curve and the parabolas (a=0), as well as sharp bounds for the other cases. This information is used for the determination of the weight distribution of some one-point AG codes arising from the curve.