13 research outputs found
On isotopisms and strong isotopisms of commutative presemifields
In this paper we prove that the ( odd prime power and
odd) commutative semifields constructed by Bierbrauer in \cite{BierbrauerSub}
are isotopic to some commutative presemifields constructed by Budaghyan and
Helleseth in \cite{BuHe2008}. Also, we show that they are strongly isotopic if
and only if . Consequently, for each
there exist isotopic commutative presemifields of order (
odd) defining CCZ--inequivalent planar DO polynomials.Comment: References updated, pag. 5 corrected Multiplication of commutative
LMPTB semifield
Non-Boolean almost perfect nonlinear functions on non-Abelian groups
The purpose of this paper is to present the extended definitions and
characterizations of the classical notions of APN and maximum nonlinear Boolean
functions to deal with the case of mappings from a finite group K to another
one N with the possibility that one or both groups are non-Abelian.Comment: 17 page
On isotopisms of commutative presemifields and CCZ-equivalence of functions
A function from \textbf{F} to itself is planar if for any \textbf{F} the function is a permutation. CCZ-equivalence is the most general known equivalence relation of functions preserving planar property. This paper considers two possible extensions of CCZ-equivalence for functions over fields of odd characteristics, one proposed by Coulter and Henderson and the other by Budaghyan and Carlet. We show that the second one in fact coincides with CCZ-equivalence, while using the first one we generalize one of the known families of PN functions. In particular, we prove that, for any odd prime and any positive integers and , the indicators of the graphs of functions and from \textbf{F} to \textbf{F} are CCZ-equivalent if and only if and are CCZ-equivalent.
We also prove that, for any odd prime , CCZ-equivalence of functions from \textbf{F} to \textbf{F}, is strictly more general than EA-equivalence when and is greater or equal to the smallest positive divisor of different from 1
Finite semifields and nonsingular tensors
In this article, we give an overview of the classification results in the theory of finite semifields (note that this is not intended as a survey of finite semifields including a complete state of the art (see also Remark 1.10)) and elaborate on the approach using nonsingular tensors based on Liebler (Geom Dedicata 11(4):455-464, 1981)
A New Family of Perfect Nonlinear Binomials
We prove that the binomials
define perfect nonlinear mappings in for an appropriate choice of the integer and . We show that these binomials are inequivalent to known perfect nonlinear monomials. As a consequence we obtain new commutative semifields for and odd