Semifields, relative difference sets, and bent functions

Recently, the interest in semifields has increased due to the discovery of several new families and progress in the classification problem. Commutative semifields play an important role since they are equivalent to certain planar functions (in the case of odd characteristic) and to modified planar functions in even characteristic. Similarly, commutative semifields are equivalent to relative difference sets. The goal of this survey is to describe the connection between these concepts. Moreover, we shall discuss power mappings that are planar and consider component functions of planar mappings, which may be also viewed as projections of relative difference sets. It turns out that the component functions in the even characteristic case are related to negabent functions as well as to $\mathbb{Z}_4$-valued bent functions.Comment: Survey paper for the RICAM workshop "Emerging applications of finite fields", 09-13 December 2013, Linz, Austria. This article will appear in the proceedings volume for this workshop, published as part of the "Radon Series on Computational and Applied Mathematics" by DeGruyte

Planar diagrams from optimization

We propose a new toy model of a heteropolymer chain capable of forming planar secondary structures typical for RNA molecules. In this model the sequential intervals between neighboring monomers along a chain are considered as quenched random variables. Using the optimization procedure for a special class of concave--type potentials, borrowed from optimal transport analysis, we derive the local difference equation for the ground state free energy of the chain with the planar (RNA--like) architecture of paired links. We consider various distribution functions of intervals between neighboring monomers (truncated Gaussian and scale--free) and demonstrate the existence of a topological crossover from sequential to essentially embedded (nested) configurations of paired links.Comment: 10 pages, 10 figures, the proof is added. arXiv admin note: text overlap with arXiv:1102.155

Quadratic Zero-Difference Balanced Functions, APN Functions and Strongly Regular Graphs

Let $F$ be a function from $\mathbb{F}_{p^n}$ to itself and $\delta$ a positive integer. $F$ is called zero-difference $\delta$-balanced if the equation $F(x+a)-F(x)=0$ has exactly $\delta$ solutions for all non-zero $a\in\mathbb{F}_{p^n}$. As a particular case, all known quadratic planar functions are zero-difference 1-balanced; and some quadratic APN functions over $\mathbb{F}_{2^n}$ are zero-difference 2-balanced. In this paper, we study the relationship between this notion and differential uniformity; we show that all quadratic zero-difference $\delta$-balanced functions are differentially $\delta$-uniform and we investigate in particular such functions with the form $F=G(x^d)$, where $\gcd(d,p^n-1)=\delta +1$ and where the restriction of $G$ to the set of all non-zero $(\delta +1)$-th powers in $\mathbb{F}_{p^n}$ is an injection. We introduce new families of zero-difference $p^t$-balanced functions. More interestingly, we show that the image set of such functions is a regular partial difference set, and hence yields strongly regular graphs; this generalizes the constructions of strongly regular graphs using planar functions by Weng et al. Using recently discovered quadratic APN functions on $\mathbb{F}_{2^8}$, we obtain $15$ new $(256, 85, 24, 30)$ negative Latin square type strongly regular graphs

Generalization of the Langmuir–Blodgett laws for a nonzero potential gradient

The Langmuir–Blodgett laws for cylindrical and spherical diodes and the Child–Langmuir law for planar diodes repose on the assumption that the electric field at the emission surface is zero. In the case of ion beam extraction from a plasma, the Langmuir–Blodgett relations are the typical tools of study, however, their use under the above assumption can lead to significant error in the beam distribution functions. This is because the potential gradient at the sheath/beam interface is nonzero and attains, in most practical ion beam extractors, some hundreds of kilovolts per meter. In this paper generalizations to the standard analysis of the spherical and cylindrical diodes to incorporate this difference in boundary condition are presented and the results are compared to the familiar Langmuir–Blodgett relation

Planar functions over fields of characteristic two

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 also call planar functions. They again give rise to finite projective planes, as recently shown by the second author. We give a characterisation of planar functions in characteristic two in terms of codes over $\mathbb{Z}_4$. We then specialise to planar monomial functions $f(x)=cx^t$ and present constructions and partial results towards their classification. In particular, we show that $t=1$ is the only odd exponent for which $f(x)=cx^t$ is planar (for some nonzero $c$) over infinitely many fields. The proof techniques involve methods from algebraic geometry.Comment: 23 pages, minor corrections and simplifications compared to the first versio