513 research outputs found
Constructing bent functions and bent idempotents of any possible algebraic degrees
Bent functions as optimal combinatorial objects are difficult to characterize
and construct. In the literature, bent idempotents are a special class of bent
functions and few constructions have been presented, which are restricted by
the degree of finite fields and have algebraic degree no more than 4. In this
paper, several new infinite families of bent functions are obtained by adding
the the algebraic combination of linear functions to some known bent functions
and their duals are calculated. These bent functions contain some previous work
on infinite families of bent functions by Mesnager \cite{M2014} and Xu et al.
\cite{XCX2015}. Further, infinite families of bent idempotents of any possible
algebraic degree are constructed from any quadratic bent idempotent. To our
knowledge, it is the first univariate representation construction of infinite
families of bent idempotents over of algebraic degree
between 2 and , which solves the open problem on bent idempotents proposed
by Carlet \cite{C2014}. And an infinite family of anti-self-dual bent functions
are obtained. The sum of three anti-self-dual bent functions in such a family
is also anti-self-dual bent and belongs to this family. This solves the open
problem proposed by Mesnager \cite{M2014}
New infinite families of p-ary weakly regular bent functions
The characterization and construction of bent functions are challenging
problems. The paper generalizes the constructions of Boolean bent functions by
Mesnager \cite{M2014}, Xu et al. \cite{XCX2015} and -ary bent functions by
Xu et al. \cite{XC2015} to the construction of -ary weakly regular bent
functions and presents new infinite families of -ary weakly regular bent
functions from some known weakly regular bent functions (square functions,
Kasami functions, and the Maiorana-McFarland class of bent functions). Further,
new infinite families of -ary bent idempotents are obtained
A general framework for secondary constructions of bent and plateaued functions
In this work, we employ the concept of {\em composite representation} of
Boolean functions, which represents an arbitrary Boolean function as a
composition of one Boolean function and one vectorial function, for the purpose
of specifying new secondary constructions of bent/plateaued functions. This
representation gives a better understanding of the existing secondary
constructions and it also allows us to provide a general construction framework
of these objects. This framework essentially gives rise to an {\em infinite
number} of possibilities to specify such secondary construction methods (with
some induced sufficient conditions imposed on initial functions) and in
particular we solve several open problems in this context. We provide several
explicit methods for specifying new classes of bent/plateaued functions and
demonstrate through examples that the imposed initial conditions can be easily
satisfied. Our approach is especially efficient when defining new
bent/plateaued functions on larger variable spaces than initial functions. For
instance, it is shown that the indirect sum methods and Rothaus' construction
are just special cases of this general framework and some explicit extensions
of these methods are given. In particular, similarly to the basic indirect sum
method of Carlet, we show that it is possible to derive (many) secondary
constructions of bent functions without any additional condition on initial
functions apart from the requirement that these are bent functions. In another
direction, a few construction methods that generalize the secondary
constructions which do not extend the variable space of the employed initial
functions are also proposed.Comment: 30 page
Bent functions from triples of permutation polynomials
We provide constructions of bent functions using triples of permutations.
This approach is due to Mesnager. In general, involutions have been mostly
considered in such a machinery; we provide some other suitable triples of
permutations, using monomials, binomials, trinomials, and quadrinomials
Frobenius linear translators giving rise to new infinite classes of permutations and bent functions
We show the existence of many infinite classes of permutations over finite
fields and bent functions by extending the notion of linear translators,
introduced by Kyureghyan [12]. We call these translators Frobenius translators
since the derivatives of , where , are
of the form , for a fixed and
all , rather than considering the standard case corresponding to
. This considerably extends a rather rare family {f} admitting linear
translators of the above form. Furthermore, we solve a few open problems in the
recent article [4] concerning the existence and an exact specification of
admitting classical linear translators, and an open problem introduced in [9]
of finding a triple of bent functions such that their sum
is bent and that the sum of their duals . Finally,
we also specify two huge families of permutations over related to the
condition that permutes the set
, where and .
Finally, we offer generalizations of constructions of bent functions from [16]
and described some new bent families using the permutations found in [4]
Several new classes of Boolean functions with few Walsh transform values
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 functions.Numerical results show that the proposed class contains cubic
bent functions that are affinely inequivalent to all known quadratic ones.
Meanwhile, we determine the distribution of the Walsh spectrum of five-valued
functions constructed in this paper
Two infinite classes of rotation symmetric bent functions with simple representation
In the literature, few -variable rotation symmetric bent functions have
been constructed. In this paper, we present two infinite classes of rotation
symmetric bent functions on of the two forms:
{\rm (i)} ,
{\rm (ii)} ,
\noindent where , is any rotation
symmetric polynomial, and is odd. The class (i) of rotation
symmetric bent functions has algebraic degree ranging from 2 to and the
other class (ii) has algebraic degree ranging from 3 to
Constructing vectorial bent functions via second-order derivatives
Let be an even positive integer, and be one of its positive
divisors. In this paper, inspired by a nice work of Tang et al. on constructing
large classes of bent functions from known bent functions [27, IEEE TIT,
63(10): 6149-6157, 2017], we consider the construction of vectorial bent and
vectorial plateaued -functions of the form , where
is a vectorial bent -function, and is a Boolean function
over . We find an efficient generic method to construct
vectorial bent and vectorial plateaued functions of this form by establishing a
link between the condition on the second-order derivatives and the key
condition given by [27]. This allows us to provide (at least) three new
infinite families of vectorial bent functions with high algebraic degrees. New
vectorial plateaued -functions are also obtained ( depending
on can be taken as a very large number), two classes of which have the
maximal number of bent components
A Construction of Binary Linear Codes from Boolean Functions
Boolean functions have important applications in cryptography and coding
theory. Two famous classes of binary codes derived from Boolean functions are
the Reed-Muller codes and Kerdock codes. In the past two decades, a lot of
progress on the study of applications of Boolean functions in coding theory has
been made. Two generic constructions of binary linear codes with Boolean
functions have been well investigated in the literature. The objective of this
paper is twofold. The first is to provide a survey on recent results, and the
other is to propose open problems on one of the two generic constructions of
binary linear codes with Boolean functions. These open problems are expected to
stimulate further research on binary linear codes from Boolean functions.Comment: arXiv admin note: text overlap with arXiv:1503.06511; text overlap
with arXiv:1505.07726 by other author
Several classes of bent, near-bent and 2-plateaued functions over finite fields of odd characteristic
Inspired by a recent work of Mesnager, we present several new infinite
families of quadratic ternary bent, near-bent and 2-plateaued functions from
some known quadratic ternary bent functions. Meanwhile, the distribution of the
Walsh spectrum of two class of 2-plateaued functions obtained in this paper is
completely determined. Additionally, we construct the first class of -ary
bent functions of algebraic degree over the fields of an arbitrary odd
characteristic. The proposed class contains non-quadratic -ary bent
functions that are affinely inequivalent to known monomial and binomial ones
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