Related papers: Planar Difference Functions
Sequences with optimal correlation properties are much sought after for applications in communication systems. In 1980, Alltop (IEEE Trans. Inf. Theory 26(3):350-354, 1980) described a set of sequences based on a cubic function and showed…
Planar functions are functions over a finite field that have optimal combinatorial properties and they have applications in several branches of mathematics, including algebra, projective geometry and cryptography. There are two relevant…
Planar functions, introduced by Dembowski and Ostrom, have attracted much attention in the last decade. As shown in this paper, we present a new class of planar functions of the form $\operatorname{Tr}(ax^{q+1})+\ell(x^2)$ on an extension…
Classical planar functions are functions from a finite field to itself and give rise to finite projective planes. They exist however only for fields of odd characteristic. We study their natural counterparts in characteristic two, which we…
This paper presents an algorithm for generating all imaginary and unusual discriminants up to a fixed degree bound that define a quadratic function field of positive 3-rank. Our method makes use of function field adaptations of a method due…
In a previous article the authors determined the best-known upper bound for the cardinality of the image set for several classes of functions, including planar functions. Here, we show that the upper bound cannot be tight for planar…
Planar functions are mappings from a finite field $\mathbb{F}_q$ to itself with an extremal differential property. Such functions give rise to finite projective planes and other combinatorial objects. There is a subtle difference between…
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…
Many quantum field theories in one, two and four dimensions possess remarkable limits in which the instantons are present, the anti-instantons are absent, and the perturbative corrections are reduced to one-loop. We analyze the…
Almost perfect nonlinear (APN) functions play an important role in the design of block ciphers as they offer the strongest resistance against differential cryptanalysis. Despite more than 25 years of research, only a limited number of APN…
In a recent paper, it is shown that functions of the form $L_1(x^3)+L_2(x^9)$, where $L_1$ and $L_2$ are linear, are a good source for construction of new infinite families of APN functions. In the present work we study necessary and…
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…
A new almost perfect nonlinear function (APN) on the finite field GF(2^10) which is not equivalent to any of the previously known APN mappings is constructed. This is the first example of an APN mapping which is not equivalent to a power…
The classification of maximal function fields over a finite field is a difficult open problem, and even determining isomorphism classes among known function fields is challenging in general. We study a particular family of maximal function…
We introduce a method based on Bezout's theorem on intersection points of two projective plane curves, for determining the nonlinearity of some classes of quadratic functions on $\mathbb{F}_{2^{2m}}$. Among those are the functions of…
Let $p$ and $q$ be anisotropic quadratic forms of dimension $\geq 2$ over a field $F$. In a recent article, we formulated a conjecture describing the general constraints which the dimensions of $p$ and $q$ impose on the isotropy index of…
The theory of quantum symmetric pairs is applied to $q$-special functions. Previous work shows the existence of a family $\chi$-spherical functions indexed by the integers for each Hermitian quantum symmetric pair. A distinguished family of…
This is a brief summary of our studies of quantum field theories in a special limit in which the instantons are present, the anti-instantons are absent, and the perturbative corrections are reduced to one-loop. We analyze the corresponding…
For each non-constant $q$ in the set of $n$-variable Boolean functions, the {\em $q$-transform} of a Boolean function $f$ is related to the Hamming distances from $f$ to the functions obtainable from $q$ by nonsingular linear change of…
Planar functions over finite fields give rise to finite projective planes and other combinatorial objects. They exist only in odd characteristic, but recently Zhou introduced an even characteristic analogue which has similar applications.…