Related papers: Equivalences of biprojective almost perfect nonlin…
Establishing the CCZ-equivalence of a pair of APN functions is generally quite difficult. In some cases, when seeking to show that a putative new infinite family of APN functions is CCZ inequivalent to an already known family, we rely on…
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…
We construct infinite classes of almost bent and almost perfect nonlinear polynomials, which are affinely inequivalent to any sum of a power function and an affine function.
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…
An almost perfect nonlinear (APN) function (necessarily a polynomial function) on a finite field $\mathbb{F}$ is called exceptional APN, if it is also APN on infinitely many extensions of $\mathbb{F}$. In this article we consider the most…
In a prior paper [14], along with P. Ellingsen, P. Felke and A. Tkachenko, we defined a new (output) multiplicative differential, and the corresponding c-differential uniformity, which has the potential of extending differential…
In this paper, we establish a lower bound on the total number of inequivalent APN functions on the finite field with $2^{2m}$ elements, where $m$ is even. We obtain this result by proving that the APN functions introduced by Pott and the…
The investigation of partially APN functions has attracted a lot of research interest recently. In this paper, we present several new infinite classes of 0-APN power functions over $\mathbb{F}_{2^n}$ by using the multivariate method and…
A new family of commutative semifields with two parameters is presented. Its left and middle nucleus are both determined. Furthermore, we prove that for any different pairs of parameters, these semifields are not isotopic. It is also shown…
Bent functions as optimal combinatorial objects are difficult to characterize and construct. In the literature, bent idempotents are a special class of bent functions and few constructions have been presented, which are restricted by the…
All almost perfect nonlinear (APN) permutations that we know to date admit a special kind of linear self-equivalence, i.e., there exists a permutation $G$ in their CCZ-equivalence class and two linear permutations $A$ and $B$, such that $G…
We investigate the concept of Best Approximation for Feedforward Neural Networks (FNN) and explore their convergence properties through the lens of Random Projection (RPNNs). RPNNs have predetermined and fixed, once and for all, internal…
The problem of finding APN permutations of ${\mathbb F}_{2^n}$ where $n$ is even and $n>6$ has been called the Big APN Problem. Li, Li, Helleseth and Qu recently characterized APN functions defined on ${\mathbb F}_{q^2}$ of the form…
We present an infinite family of quadratic APN functions on a finite field of dimension over GF(2) divisible by 3.
In a recent work, Beierle, Brinkmann and Leander presented a recursive tree search for finding APN permutations with linear self-equivalences in small dimensions. In this paper, we describe how this search can be adapted to find many new…
Let $F$ be a finite field, let $f$ be a function from $F$ to $F$, and let $a$ be a nonzero element of $F$. The discrete derivative of $f$ in direction $a$ is $\Delta_a f \colon F \to F$ with $(\Delta_a f)(x)=f(x+a)-f(x)$. The differential…
Strong convergence of a new iterative process based on the Shrinking projection method to a common element of the set of common fixed points of an infinite family of relatively quasi-nonexpansive multivalued mappings and the solution set of…
Generalized almost perfect nonlinear (GAPN) functions were defined to satisfy some generalizations of basic properties of almost perfect nonlinear (APN) functions for even characteristic. In this paper, we study monomial GAPN functions for…
A new method, called the method of self-similar approximants, and its recent developments are described. The method is based on the ideas of renormalization group theory and optimal control theory. It allows for the effective extrapolation…
We introduce a new concept, the APN-defect, which can be thought of as measuring the distance of a given function $G:\mathbb{F}_{2^n} \rightarrow \mathbb{F}_{2^n}$ to the set of almost perfect nonlinear (APN) functions. This concept is…