Related papers: The number of almost perfect nonlinear functions g…

In this paper, we classify $(q,q)$-biprojective almost perfect nonlinear (APN) functions over $\mathbb{LL} \times \mathbb{LL}$ under the natural left and right action of $\mathrm{GL}(2,\mathbb{LL})$ where $\mathbb{LL}$ is a finite field of…

In this paper, two new classes of perfect nonlinear functions over $\mathbb{F}_{p^{2m}}$ are proposed, where $p$ is an odd prime. Furthermore, we investigate the nucleus of the corresponding semifields of these functions and show that the…

This paper proposes a novel technique for the approximation of strong solutions $u \in C(\overline{\Omega}) \cap W^{2,n}_\mathrm{loc}(\Omega)$ to uniformly elliptic linear PDE of second order in nondivergence form with continuous leading…

Euler's totient function, $\varphi(n)$, which counts how many of $0,1,\dots,n-1$ are coprime to $n$, has an explicit asymptotic lower bound of $n/\log \log n$, modulo some constant. In this note, we generalise $\varphi$; given an…

We show that the there exists an infinite family of APN functions of the form $F(x)=x^{2^{s}+1} + x^{2^{k+s}+2^k} + cx^{2^{k+s}+1} + c^{2^k}x^{2^k + 2^s} + \delta x^{2^{k}+1}$, over $\gf_{2^{2k}}$, where $k$ is an even integer and…

Let $m\ge 2$ be an integer. For any open domain $\Omega\subset\mathbb{R}^{2m}$, non-positive function $\varphi\in C^\infty(\Omega)$ such that $\Delta^m \varphi\equiv 0$, and bounded sequence $(V_k)\subset L^\infty(\Omega)$ we prove the…

We obtain results for the following question where $m\ge 1$ and $n\ge 2$ are integers. {\bf Question.} For which continuous functions $f\colon [0,\infty)\to [0,\infty)$ does there exist a continuous function $\phi\colon (0,1)\to (0,\infty)$…

Let $\phi(\cdot)$ and $\sigma(\cdot)$ denote the Euler function and the sum of divisors function, respectively. In this paper, we give a lower bound for the number of positive integers $m\le x$ for which the equation $m=n-\phi(n)$ has no…

Boolean functions on the space $F_{2}^m$ are not only important in the theory of error-correcting codes, but also in cryptography, where they occur in private key systems. In these two cases, the nonlinearity of these function is a main…

Exhibiting an explicit Boolean function with a large high-order nonlinearity is an important problem in cryptography, coding theory, and computational complexity. We prove lower bounds on the second-order, third-order, and higher-order…

Using the framework first presented by Ruf and Sani, we give a proof of an Adams type inequality which can be applied to a nonlinear functional. Under two kinds of assumptions on the nonlinearity, we estimate the min-max level of the…

In 1887, Minkowski determined the least common multiple of the orders of all finite subgroups of $GL_n(\mathbb{Q})$; we refer to this number as $M(n)$. In (Katznelson, 1994), Katznelson provides the asymptotic behaviour of $M(n)$, with a…

We study the nonlinearity of functions defined on a finite field with 2^m elements which are the trace of a polynomial of degree 7 or more general polynomials. We show that for m odd such functions have rather good nonlinearity properties.…

We study the problem of approximating and learning coverage functions. A function $c: 2^{[n]} \rightarrow \mathbf{R}^{+}$ is a coverage function, if there exists a universe $U$ with non-negative weights $w(u)$ for each $u \in U$ and subsets…

We design a two-scale finite element method (FEM) for linear elliptic PDEs in non-divergence form $A(x) : D^2 u(x) = f(x)$ in a bounded but not necessarily convex domain $\Omega$ and study it in the max norm. The fine scale is given by the…

The study of non-linearity (linearity) of Boolean function was initiated by Rothaus in 1976. The classical non-linearity of a Boolean function is the minimum Hamming distance of its truth table to that of affine functions. In this note we…

The paper studies semi-almost periodic holomorphic functions on a polydisk which have, in a sense, the weakest possible discontinuities on the boundary torus. The basic result used in the proofs is an extension of the classical Bohr…

Block ciphers use S-boxes to create confusion in the cryptosystems. Such S-boxes are functions over $\mathbb{F}_{2^{n}}$. These functions should have low differential uniformity, high nonlinearity, and high algebraic degree in order to…

Let $p>3$ be a prime. We show that, for each integer $d$ with $p \leq d \leq 2(p-1)$, there exists a generalized almost perfect nonlinear (GAPN) binomial or trinomial over $\mathbb{F}_{p^2}$ of algebraic degree $d$. We start by deriving…

This article summarises the theory of several bounded functional calculi for unbounded operators that have recently been discovered. The extend the Hille--Phillips calculus for (negative) generators $A$ of certain bounded $C_0$-semigroups,…