31 research outputs found
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
Secondary constructions of vectorial -ary weakly regular bent functions
In \cite{Bapic, Tang, Zheng} a new method for the secondary construction of
vectorial/Boolean bent functions via the so-called property was
introduced. In 2018, Qi et al. generalized the methods in \cite{Tang} for the
construction of -ary weakly regular bent functions. The objective of this
paper is to further generalize these constructions, following the ideas in
\cite{Bapic, Zheng}, for secondary constructions of vectorial -ary weakly
regular bent and plateaued functions. We also present some infinite families of
such functions via the -ary Maiorana-McFarland class. Additionally, we give
another characterization of the property for the -ary case via
second-order derivatives, as it was done for the Boolean case in \cite{Zheng}
Composition construction of new bent functions from known dually isomorphic bent functions
Bent functions are optimal combinatorial objects and have been studied over the last four decades. Secondary construction plays a central role in constructing bent functions since it may generate bent functions outside the primary classes of bent functions. In this study, we improve a theoretical framework of the secondary construction of bent functions in terms of the composition of Boolean functions. Based on this framework, we propose several constructions of bent functions through the composition of a balanced Boolean function and dually isomorphic (DI) bent functions defined herein. In addition, we present a construction of self-dual bent functions
Balanced Boolean Functions with (Almost) Optimal Algebraic Immunity and Very High Nonlinearity
In this paper, we present a class of -variable balanced Boolean
functions and a class of -variable -resilient Boolean functions for an integer , which both have the maximal algebraic degree and very high nonlinearity. Based on a newly proposed conjecture by Tu and Deng, it is shown that the proposed balanced Boolean functions have optimal algebraic immunity and the -resilient Boolean functions have almost optimal algebraic immunity. Among all the known results of balanced Boolean
functions and -resilient Boolean functions, our new functions possess the highest nonlinearity. Based on the fact that the conjecture has been verified for all by computer,
at least we have constructed a class of balanced Boolean functions and a class of -resilient Boolean functions with the even number of variables , which are cryptographically optimal or almost
optimal in terms of balancedness, algebraic degree, nonlinearity, and algebraic immunity
Constructing new superclasses of bent functions from known ones
Some recent research articles [23, 24] addressed an explicit specification of indicators
that specify bent functions in the so-called and classes, derived from the Maiorana-
McFarland () class by C. Carlet in 1994 [5]. Many of these bent functions that belong
to or are provably outside the completed class. Nevertheless, these modifications
are performed on affine subspaces, whereas modifying bent functions on suitable subsets
may provide us with further classes of bent functions. In this article, we exactly specify
new families of bent functions obtained by adding together indicators typical for the
and class, thus essentially modifying bent functions in on suitable subsets instead
of subspaces. It is shown that the modification of certain bent functions in gives rise
to new bent functions which are provably outside the completed class. Moreover, we
consider the so-called 4-bent concatenation (using four different bent functions on the
same variable space) of the (non)modified bent functions in and show that we can
generate new bent functions in this way which do not belong to the completed class
either. This result is obtained by specifying explicitly the duals of four constituent bent
functions used in the concatenation. The question whether these bent functions are also
excluded from the completed versions of , or remains open and is considered
difficult due to the lack of membership indicators for these classes
Programs as Diagrams: From Categorical Computability to Computable Categories
This is a draft of the textbook/monograph that presents computability theory
using string diagrams. The introductory chapters have been taught as graduate
and undergraduate courses and evolved through 8 years of lecture notes. The
later chapters contain new ideas and results about categorical computability
and some first steps into computable category theory. The underlying
categorical view of computation is based on monoidal categories with program
evaluators, called *monoidal computers*. This categorical structure can be
viewed as a single-instruction diagrammatic programming language called Run,
whose only instruction is called RUN. This version: improved text, moved the
final chapter to the next volume. (The final version will continue lots of
exercises and workouts, but already this version has severely degraded graphics
to meet the size bounds.)Comment: 150 pages, 81 figure