On the exponents of APN power functions and Sidon sets, sum-free sets, and Dickson polynomials

<p style='text-indent:20px;'>We derive necessary conditions related to the notions, in additive combinatorics, of Sidon sets and sum-free sets, on those exponents <inline-formula><tex-math id="M1">\begin{document}$ d\in {\mathbb Z}/(2^n-1){\mathbb Z} $\end{document}</tex-math></inline-formula>, which are such that <inline-formula><tex-math id="M2">\begin{document}$ F(x) = x^d $\end{document}</tex-math></inline-formula> is an APN function over <inline-formula><tex-math id="M3">\begin{document}$ {\mathbb F}_{2^n} $\end{document}</tex-math></inline-formula> (which is an important cryptographic property). We study to what extent these new conditions may speed up the search for new APN exponents <inline-formula><tex-math id="M4">\begin{document}$ d $\end{document}</tex-math></inline-formula>. We summarize all the necessary conditions that an exponent must satisfy for having a chance of being an APN, including the new conditions presented in this work. Next, we give results up to <inline-formula><tex-math id="M5">\begin{document}$ n = 48 $\end{document}</tex-math></inline-formula>, providing the number of exponents satisfying all the conditions for a function to be APN.</p><p style='text-indent:20px;'>We also show a new connection between APN exponents and Dickson polynomials: <inline-formula><tex-math id="M6">\begin{document}$ F(x) = x^d $\end{document}</tex-math></inline-formula> is APN if and only if the reciprocal polynomial of the Dickson polynomial of index <inline-formula><tex-math id="M7">\begin{document}$ d $\end{document}</tex-math></inline-formula> is an injective function from <inline-formula><tex-math id="M8">\begin{document}$ \{y\in {\Bbb F}_{2^n}^*; tr_n(y) = 0\} $\end{document}</tex-math></inline-formula> to <inline-formula><tex-math id="M9">\begin{document}$ {\Bbb F}_{2^n}\setminus \{1\} $\end{document}</tex-math></inline-formula>. This also leads to a new and simple connection between Reversed Dickson polynomials and reciprocals of Dickson polynomials in characteristic 2 (which generalizes to every characteristic thanks to a small modification): the squared Reversed Dickson polynomial of some index and the reciprocal of the Dickson polynomial of the same index are equal.</p>

Several classes of permutation trinomials from Niho exponents over finite fields of characteristic 3

Permutation polynomials have important applications in cryptography, coding theory, combinatorial designs, and other areas of mathematics and engineering. Finding new classes of permutation polynomials is therefore an interesting subject of study. Permutation trinomials attract people’s interest due to their simple algebraic forms and additional extraordinary properties. In this paper, based on a seventh-degree and a fifth-degree Dickson polynomial over the finite field [Formula: see text], two conjectures on permutation trinomials over [Formula: see text] presented recently by Li–Qu–Li–Fu are partially settled, where [Formula: see text] is a positive integer.

On Bipartite Cages of Excess 4

The Moore bound $M(k,g)$ is a lower bound on the order of $k$-regular graphs of girth $g$ (denoted $(k,g)$-graphs). The excess $e$ of a $(k,g)$-graph of order $n$ is the difference $ n-M(k,g) $. In this paper we consider the existence of $(k,g)$-bipartite graphs of excess $4$ by studying spectral properties of their adjacency matrices. For a given graph $G$ and for the integers $i$ with $0\leq i\leq diam(G)$, the $i$-distance matrix $A_i$ of $G$ is an $n\times n$ matrix such that the entry in position $(u,v)$ is $1$ if the distance between the vertices $u$ and $v$ is $i$, and zero otherwise. We prove that the $(k,g)$-bipartite graphs of excess $4$ satisfy the equation $kJ=(A+kI)(H_{d-1}(A)+E)$, where $A=A_{1}$ denotes the adjacency matrix of the graph in question, $J$ the $n \times n$ all-ones matrix, $E=A_{d+1}$ the adjacency matrix of a union of vertex-disjoint cycles, and $H_{d-1}(x)$ is the Dickson polynomial of the second kind with parameter $k-1$ and degree $d-1$. We observe that the eigenvalues other than $\pm k$ of these graphs are roots of the polynomials $H_{d-1}(x)+\lambda$, where $\lambda$ is an eigenvalue of $E$. Based on the irreducibility of $H_{d-1}(x)\pm2$, we give necessary conditions for the existence of these graphs. If $E$ is the adjacency matrix of a cycle of order $n$, we call the corresponding graphs graphs with cyclic excess; if $E$ is the adjacency matrix of a disjoint union of two cycles, we call the corresponding graphs graphs with bicyclic excess. In this paper we prove the non-existence of $(k,g)$-graphs with cyclic excess $4$ if $k\geq6$ and $k \equiv1 \!\! \pmod {3}$, $g=8, 12, 16$ or $k \equiv2 \!\! \pmod {3}$, $g=8;$ and the non-existence of $(k,g)$-graphs with bicyclic excess $4$ if $k\geq7$ is an odd number and $g=2d$ such that $d\geq4$ is even.

Dickson curves

Letkqdenote the finite field of orderqand odd characteristicp. Fora∈kq, letgd(x,a)denote the Dickson polynomial of degreeddefined bygd(x,a)=∑i=0[d/2]d/(d−i)(d−ii)(−a)ixd−2i. Letf(x)denote a monic polynomial with coefficients inkq. Assume thatf2(x)−4is not a perfect square andgcd⁡(p,d)=1. Also assume thatf(x)andg2(f(x),1)are not of the formgd(h(x),c). In this note, we show that the polynomialgd(y,1)−f(x)∈kq[x,y]is absolutely irreducible.

On the decomposition ofxd+aexe+⋯+a1x+a0

LetKdenote a field. A polynomialf(x)∈K[x]is said to be decomposable overKiff(x)=g(h(x))for some polynomialsg(x)andh(x)∈K[x]with1<deg(h)<deg(f). Otherwisef(x)is called indecomposable. Iff(x)=g(xm)withm>1, thenf(x)is said to be trivially decomposable. In this paper, we show thatxd+ax+bis indecomposable and that ifedenotes the largest proper divisor ofd, thenxd+ad−e−1xd−e−1+⋯+a1x+a0is either indecomposable or trivially decomposable. We also show that ifgd(x,a)denotes the Dickson polynomial of degreedand parameteraandgd(x,a)=f(h(x)), thenf(x)=gt(x−c,a)andh(x)=ge(x,a)+c.