17 research outputs found
A new class of negabent functions
Negabent functions were introduced as a generalization of bent functions,
which have applications in coding theory and cryptography. In this paper, we
have extended the notion of negabent functions to the functions defined from
to (-negabent), where is a
positive integer and is the ring of integers modulo . For
this, a new unitary transform (the nega-Hadamard transform) is introduced in
the current set up, and some of its properties are discussed. Some results
related to -negabent functions are presented. We present two constructions
of -negabent functions. In the first construction, -negabent functions
on variables are constructed when is an even positive integer. In the
second construction, -negabent functions on two variables are constructed
for arbitrary positive integer . Some examples of -negabent
functions for different values of and are also presented
On Negabent Functions and Nega-Hadamard Transform
The Boolean function which has equal absolute spectral values under the nega-Hadamard transform is called negabent function. In this paper, the special Boolean functions by concatenation are presented. We investigate their nega-Hadamard transforms, nega-autocorrelation coefficients, sum-of-squares indicators, and so on. We establish a new equivalent statement on f1∥f2 which is negabent function. Based on them, the construction for generating the negabent functions by concatenation is given. Finally, the function expressed as f(Ax⊕a)⊕b·x⊕c is discussed. The nega-Hadamard transform and nega-autocorrelation coefficient of this function are derived. By applying these results, some properties are obtained
Systematic Constructions of Bent-Negabent Functions, 2-Rotation Symmetric Bent-Negabent Functions and Their Duals
Bent-negabent functions have many important properties for their application
in cryptography since they have the flat absolute spectrum under the both
Walsh-Hadamard transform and nega-Hadamard transform. In this paper, we present
four new systematic constructions of bent-negabent functions on
and variables, respectively, by modifying the truth tables of two
classes of quadratic bent-negabent functions with simple form. The algebraic
normal forms and duals of these constructed functions are also determined. We
further identify necessary and sufficient conditions for those bent-negabent
functions which have the maximum algebraic degree. At last, by modifying the
truth tables of a class of quadratic 2-rotation symmetric bent-negabent
functions, we present a construction of 2-rotation symmetric bent-negabent
functions with any possible algebraic degrees. Considering that there are
probably no bent-negabent functions in the rotation symmetric class, it is the
first significant attempt to construct bent-negabent functions in the
generalized rotation symmetric class
Root-Hadamard transforms and complementary sequences
In this paper we define a new transform on (generalized) Boolean functions,
which generalizes the Walsh-Hadamard, nega-Hadamard, -Hadamard,
consta-Hadamard and all -transforms. We describe the behavior of what we
call the root- Hadamard transform for a generalized Boolean function in
terms of the binary components of . Further, we define a notion of
complementarity (in the spirit of the Golay sequences) with respect to this
transform and furthermore, we describe the complementarity of a generalized
Boolean set with respect to the binary components of the elements of that set.Comment: 19 page
Decomposing generalized bent and hyperbent functions
In this paper we introduce generalized hyperbent functions from to
, and investigate decompositions of generalized (hyper)bent functions.
We show that generalized (hyper)bent functions from to
consist of components which are generalized (hyper)bent functions from
to for some . For odd , we show
that the Boolean functions associated to a generalized bent function form an
affine space of semibent functions. This complements a recent result for even
, where the associated Boolean functions are bent.Comment: 24 page