Related papers: Progress Towards the Conjecture on APN Functions a…
By replacing standard non-linearities with polynomial activations, Polynomial Neural Networks (PNNs) are pivotal for applications such as privacy-preserving inference via Homomorphic Encryption (HE). However, training PNNs effectively…
APN functions play a big role as primitives in symmetric cryptography as building blocks that yield optimal resistance to differential attacks. In this note, we consider a recent extension of a biprojective APN family by G\"olo\u{g}lu…
We discuss several enumerative results for irreducible polynomials of a given degree and pairs of relatively prime polynomials of given degrees in several variables over finite fields. Two notions of degree, the {\em total degree} and the…
In this paper we determined explicitly the multiplicative inverses of the Dobbertin and Welch APN exponents in Z_{2^n-1}, and we described the binary weights of the inverses of the Gold and Kasami exponents. We studied the function \de(n),…
Let $G$ be a topological Abelian semigroup with unit, let $E$ be a Banach space, and let $C(G,E)$ denote the set of continuous functions $f\colon G\to E$. A function $f\in C(G,E)$ is a generalized polynomial, if there is an $n\ge 0$ such…
We obtain a polynomial upper bound in the finite-field version of the multidimensional polynomial Szemer\'{e}di theorem for distinct-degree polynomials. That is, if $P_1, ..., P_t$ are nonconstant integer polynomials of distinct degrees and…
We study a class of 2-variable polynomials called exact polynomials which contains $A$-polynomials of knot complements. The Mahler measure of these polynomials can be computed in terms of a volume function defined on the vanishing set of…
The vector partition function $p_A$ associated to a $d \times n$ matrix $A$ with integer entries is the function $\mathbb{Z}^d \to \mathbb{N}$ defined by $\mathbf{b} \to \#\{\mathbf{x} \in \mathbb{N}^n : A\mathbf{x} = \mathbf{b}\}$. It is…
In this paper, we investigate the power functions $F(x)=x^d$ over the finite field $\mathbb{F}_{2^{4n}}$, where $n$ is a positive integer and $d=2^{3n}+2^{2n}+2^{n}-1$. It is proved that $F(x)=x^d$ is APcN at certain $c$'s in…
Planar functions are of great importance in the constructions of DES-like iterated ciphers, error-correcting codes, signal sets and the area of mathematics. They are defined over finite fields of odd characteristic originally and…
Recent studies on binomials of the form $F_r(x) = x^r(1 + \chi(x))$ over $\mathbb{F}_{p^n}$ have shown that these functions can exhibit very low boomerang uniformity. In this paper, we focus on the specific behavior of such binomials in…
In this work, a functional variant of the polynomial analogue of the classical Gandy's fixed point theorem is obtained. Sufficient conditions have been found to ensure that the complexity of the recursive function does not go beyond the…
The generalised Gegenbauer functions of fractional degree (GGF-Fs), denoted by ${}^{r\!}G^{(\lambda)}_\nu(x)$ (right GGF-Fs) and ${}^{l}G^{(\lambda)}_\nu(x)$ (left GGF-Fs) with $x\in (-1,1),$ $\lambda>-1/2$ and real $\nu\ge 0,$ are special…
Let F = F_p for any fixed prime p >= 2. An affine-invariant property is a property of functions on F^n that is closed under taking affine transformations of the domain. We prove that all affine-invariant property having local…
Polynomial representations of Boolean functions over various rings such as $\mathbb{Z}$ and $\mathbb{Z}_m$ have been studied since Minsky and Papert (1969). From then on, they have been employed in a large variety of fields including…
We consider the problem of testing if a given function f : F_2^n -> F_2 is close to any degree d polynomial in n variables, also known as the Reed-Muller testing problem. The Gowers norm is based on a natural 2^{d+1}-query test for this…
By introducing the generalized master function of order up to four together with corresponding weight function, we have obtained all quasi-exactly solvable second order differential equations. It is shown that these differntial equations…
In this article, we consider polynomials of the form $f(x)=a_0+a_{n_1}x^{n_1}+a_{n_2}x^{n_2}+\dots+a_{n_r}x^{n_r}\in \mathbb{Z}[x],$ where $|a_0|\ge |a_{n_1}|+\dots+|a_{n_r}|,$ $|a_0|$ is a prime power and $|a_0|\nmid |a_{n_1}a_{n_r}|$. We…
Cilleruelo conjectured that for an irreducible polynomial $f \in \mathbb{Z}[X]$ of degree $d \geq 2$ one has $$\log\left[\mathrm{lcm}(f(1),f(2),\ldots f(N))\right]\sim(d-1)N\log N$$ as $N \to \infty$. He proved it in the case $d=2$ but it…
For each odd prime power q, we construct an infinite sequence of rational functions f(X) in F_q(X), each of which is exceptional, which means that for infinitely many n the map c-->f(c) induces a bijection of P^1(F_{q^n}). Moreover, each of…