289 research outputs found
Univariate Niho Bent Functions from o-Polynomials
In this paper, we discover that any univariate Niho bent function is a sum of
functions having the form of Leander-Kholosha bent functions with extra
coefficients of the power terms. This allows immediately, knowing the terms of
an o-polynomial, to obtain the powers of the additive terms in the polynomial
representing corresponding bent function. However, the coefficients are
calculated ambiguously. The explicit form is given for the bent functions
obtained from quadratic and cubic o-polynomials. We also calculate the
algebraic degree of any bent function in the Leander-Kholosha class
Niho Bent Functions and Subiaco/Adelaide Hyperovals
In this paper, the relation between binomial Niho bent functions discovered
by Dobbertin et al. and o-polynomials that give rise to the Subiaco and
Adelaide classes of hyperovals is found. This allows to expand the class of
bent functions that corresponds to Subiaco hyperovals, in the case when
A new class of hyper-bent Boolean functions in binomial forms
Bent functions, which are maximally nonlinear Boolean functions with even
numbers of variables and whose Hamming distance to the set of all affine
functions equals , were introduced by Rothaus in
1976 when he considered problems in combinatorics. Bent functions have been
extensively studied due to their applications in cryptography, such as S-box,
block cipher and stream cipher. Further, they have been applied to coding
theory, spread spectrum and combinatorial design. Hyper-bent functions, as a
special class of bent functions, were introduced by Youssef and Gong in 2001,
which have stronger properties and rarer elements. Many research focus on the
construction of bent and hyper-bent functions. In this paper, we consider
functions defined over by
,
where , , and
. When and ,
with the help of Kloosterman sums and the factorization of , we
present a characterization of hyper-bentness of . Further, we use
generalized Ramanujan-Nagell equations to characterize hyper-bent functions of
in the case
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}
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 Negabent Functions over Finite Fields
Negabent functions as a class of generalized bent functions have attracted a
lot of attention recently due to their applications in cryptography and coding
theory. In this paper, we consider the constructions of negabent functions over
finite fields. First, by using the compositional inverses of certain binomial
and trinomial permutations, we present several classes of negabent functions of
the form f(x)=\Tr_1^n(\lambda x^{2^k+1})+\Tr_1^n(ux)\Tr_1^n(vx), where
\lambda\in \F_{2^n}, , (u,v)\in \F^*_{2^n}\times
\F^*_{2^n}, and \Tr_1^n(\cdot) is the trace function from \F_{2^n} to
\F_{2}. Second, by using Kloosterman sum, we prove that the condition for the
cubic monomials given by Zhou and Qu (Cryptogr. Commun., to appear, DOI
10.1007/s12095-015-0167-0.) to be negabent is also necessary. In addition, a
conjecture on negabent monomials whose exponents are of Niho type is given
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
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
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
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
- β¦