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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…
Almost perfect nonlinear (APN) functions on finite fields of characteristic two have been studied by many researchers. Such functions have useful properties and applications in cryptography, finite geometries and so on. However APN…
The article is devoted to investigation of the classes of functions belonging to the gaps between classes $P_{n+1}(I)$ and $P_{n}(I)$ of matrix monotone functions for full matrix algebras of successive dimensions. In this paper we address…
In this technical report, certain interesting classification of arithmetical functions is proposed. The notion of additively decomposable and multiplicatively decomposable arithmetical functions is proposed. The concepts of arithmetical…
Two important problems on almost perfect nonlinear (APN) functions are the enumeration and equivalence problems. In this paper, we solve these two problems for any biprojective APN function family by introducing a strong group theoretic…
We construct an uncountable family of well-quasi-ordered permutation classes, each with a distinct enumeration sequence. This disproves a conjecture that all well-quasi-ordered permutation classes have algebraic generating functions, and in…
Extended Affine (EA) equivalence is the equivalence relation between two vectorial Boolean functions $F$ and $G$ such that there exist two affine permutations $A$, $B$, and an affine function $C$ satisfying $G = A \circ F \circ B + C$.…
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…
It is known that there are classes of 2-CNFs requiring exponential size non-deterministic read-once branching programs to compute them. However, to the best of our knowledge, there are no superpolynomial lower bounds for branching programs…
In this paper, several new classes of Boolean functions with few Walsh transform values, including bent, semi-bent and five-valued functions, are obtained by adding the product of two or three linear functions to some known bent…
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…
Polynomial functions have plenty of useful analytical properties, but they are rarely used as learning models because their function class is considered to be restricted. This work shows that when trained properly polynomial functions can…
Two classes of ternary bent functions of degree four with two and three terms in the univariate representation that belong to the completed Maiorana-McFarland class are found. Binomials are mappings $\F_{3^{4k}}\mapsto\fthree$ given by…
Let A be a finite set with at least two elements. The composition of two classes I and J of operations on A, is defined as the set of all compositions of functions in I with functions in J. This binary operation gives a monoid structure to…
Two emergent properties in aggregation theory are investigated, namely horizontal maxitivity and comonotonic maxitivity (as well as their dual counterparts) which are commonly defined by means of certain functional equations. We completely…
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…
Maximum nonlinear functions on finite fields are widely used in cryptography because the coordinate functions have large distance to linear functions. More precisely, the Hamming distance to the characteristic functions of hyperplanes is…
We construct polynomial approximations of Dzjadyk type (in terms of the k-th modulus of continuity, $k \ge 1$) for analytic functions defined on a continuum E in the complex plane, which simultaneously interpolate at given points of E.…
We prove that asymptotically almost all vectorial functions over finite fields have trivial extended-affine stabilizers. As a consequence, the number of EA-equivalence classes is asymptotically equal to the naive estimate, namely the total…
APN functions play a central role as building blocks in the design of many block ciphers, serving as optimal functions to resist differential attacks. One of the most important properties of APN functions is their linearity, which is…