On a conjecture about a class of permutation quadrinomials

Very recently, Tu et al. presented a sufficient condition on $(a_1,a_2,a_3)$, see Theorem 1.1, such that $f(x)=x^{3\cdot 2^m}+a_1{x}^{2^{m+1}+1}+a_2{x}^{2^m+2}+a_3{x}^3$ is a class of permutation polynomials over ${\mathbb{F}}_{2^n}$ with $n=2m$ and m odd. In this present paper, we prove that the sufficient condition is also necessary.     

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.     

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.     

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.     

Remarks on the distribution of colors in Gallai colorings

A Gallai coloring of a complete graph $K_n$ is an edge coloring without triangles colored with three different colors. A sequence $e_1\ge \cdots \ge e_k$ of positive integers is an $(n,k)$-sequence if ${\sum }_{i=1}^k{e}_i=\left(\genfrac{}{}{0ex}{}{n}{2}\right)$. An $(n,k)$-sequence is a G-sequence if there is a Gallai coloring of $K_n$ with $k$ colors such that there are $e_i$ edges of color $i$ for all $i,1\le i\le k$. Gyárfás, Pálvölgyi, Patkós and Wales proved that for any integer $k\ge 3$ there exists an integer $g\left(k\right)$ such that every $(n,k)$-sequence is a G-sequence if and only if $n\ge g\left(k\right)$. They showed that $g\left(3\right)=5,g\left(4\right)=8$ and $2k-2\le g\left(k\right)\le 8k^2+1$.We show that $g\left(5\right)=10$ and give almost matching lower and upper bounds for $g\left(k\right)$ by showing that with suitable constants $\alpha ,\beta >0$, $\frac{\alpha k^{1.5}}{lnk}\le g\left(k\right)\le \beta k^{1.5}$ for all sufficiently large $k$.     

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.     

Maximal curves and Tate-Shafarevich results for quartic and sextic twists

We study elliptic surfaces corresponding to an equation of the specific type $y^2=x^3+f(t)x$, defined over the finite field ${\mathbb{F}}_q$ for a prime power $q\equiv 3\mathrm{mod}4$. It is shown that if $s^4=f(t)$ defines a curve that is maximal over ${\mathbb{F}}_{q^2}$ then the rank of the group of sections defined over ${\mathbb{F}}_q$ on the elliptic surface is determined in terms of elementary properties of the rational function $f(t)$. Similar results are shown for elliptic surfaces given by $y^2=x^3+g(t)$ using prime powers $q\equiv 5\mathrm{mod}6$ and curves $s^6=g(t)$. Finally, for each of the forms used here, existence of curves with the property that they are maximal over ${\mathbb{F}}_{q^2}$ is discussed, as well as various examples.     

On simultaneous conjugation of permutations

Let α and β be two permutations in $S_n$. We prove that if the commutator $[\alpha ,\beta ]$ has at least $n?4$ fixed points then there exists a permutation $\gamma \in S_n$ such that ${\alpha }^{\gamma }={\alpha }^{?1}$ and ${\beta }^{\gamma }={\beta }^{?1}$. This gives an affirmative answer to a conjecture proposed by Danny Neftin, which leads to the completion of the classification of monodromy groups for sufficiently large degree.