145,406 research outputs found
Landscape Boolean Functions
In this paper we define a class of Boolean and generalized Boolean functions
defined on with values in (mostly, we consider
), which we call landscape functions (whose class containing generalized
bent, semibent, and plateaued) and find their complete characterization in
terms of their components. In particular, we show that the previously published
characterizations of generalized bent and plateaued Boolean functions are in
fact particular cases of this more general setting. Furthermore, we provide an
inductive construction of landscape functions, having any number of nonzero
Walsh-Hadamard coefficients. We also completely characterize generalized
plateaued functions in terms of the second derivatives and fourth moments.Comment: 19 page
Fast Algebraic Attacks and Decomposition of Symmetric Boolean Functions
Algebraic and fast algebraic attacks are power tools to analyze stream
ciphers. A class of symmetric Boolean functions with maximum algebraic immunity
were found vulnerable to fast algebraic attacks at EUROCRYPT'06. Recently, the
notion of AAR (algebraic attack resistant) functions was introduced as a
unified measure of protection against both classical algebraic and fast
algebraic attacks. In this correspondence, we first give a decomposition of
symmetric Boolean functions, then we show that almost all symmetric Boolean
functions, including these functions with good algebraic immunity, behave badly
against fast algebraic attacks, and we also prove that no symmetric Boolean
functions are AAR functions. Besides, we improve the relations between
algebraic degree and algebraic immunity of symmetric Boolean functions.Comment: 13 pages, submitted to IEEE Transactions on Information Theor
Stratification and enumeration of Boolean functions by canalizing depth
Boolean network models have gained popularity in computational systems
biology over the last dozen years. Many of these networks use canalizing
Boolean functions, which has led to increased interest in the study of these
functions. The canalizing depth of a function describes how many canalizing
variables can be recursively picked off, until a non-canalizing function
remains. In this paper, we show how every Boolean function has a unique
algebraic form involving extended monomial layers and a well-defined core
polynomial. This generalizes recent work on the algebraic structure of nested
canalizing functions, and it yields a stratification of all Boolean functions
by their canalizing depth. As a result, we obtain closed formulas for the
number of n-variable Boolean functions with depth k, which simultaneously
generalizes enumeration formulas for canalizing, and nested canalizing
functions
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