17 research outputs found
Bent functions of maximal degree
In this article a technique for constructing p-ary bent functions
from plateaued functions is presented. This generalizes earlier techniques
of constructing bent from near-bent functions. The Fourier spectrum of quadratic
monomials is analysed, examples of quadratic functions with highest possible
absolute values in their Fourier spectrum are given. Applying the construction of
bent functions to the latter class of functions yields bent functions attaining
upper bounds for the algebraic degree when . Until now no construction
of bent functions attaining these bounds was known
A construction of bent functions from plateaued functions
In this presentation, a technique for constructing bent functions from plateaued functions is introduced and analysed. This generalizes earlier techniques for constructing bent from near-bent functions. Using this construction, we obtain a big variety of inequivalent bent functions, some weakly regular and some non-weakly regular. Classes of bent function with some additional properties that enable the construction of strongly regular graphs are constructed, and explicit expressions for bent functions with maximal degree are presented
On the normality of -ary bent functions
Depending on the parity of and the regularity of a bent function from
to , can be affine on a subspace of dimension
at most , or . We point out that many -ary bent
functions take on this bound, and it seems not easy to find examples for which
one can show a different behaviour. This resembles the situation for Boolean
bent functions of which many are (weakly) -normal, i.e. affine on a
-dimensional subspace. However applying an algorithm by Canteaut et.al.,
some Boolean bent functions were shown to be not - normal. We develop an
algorithm for testing normality for functions from to . Applying the algorithm, for some bent functions in small dimension we
show that they do not take on the bound on normality. Applying direct sum of
functions this yields bent functions with this property in infinitely many
dimensions.Comment: 13 page
Secondary constructions of (non)weakly regular plateaued functions over finite fields
Plateaued (vectorial) functions over finite fields have diverse applications in symmetric cryptography, coding theory, and sequence theory. Constructing these functions is an attractive research topic in the literature. We can distinguish two kinds of constructions of plateaued functions: secondary constructions and primary constructions. The first method uses already known functions to obtain new functions while the latter do not need to use previously constructed functions to obtain new functions. In this work, the first secondary constructions of (non)weakly regular plateaued (vectorial) functions are presented over the finite fields of odd characteristics. We also introduce some recursive constructions of (non)weakly regular plateaued p-ary functions by using already known such functions. We obtain nontrivial plateaued functions from the previously known trivial plateaued (partially bent) functions in the proposed construction methods