5,501 research outputs found
Improved lower bound on the number of balanced symmetric functions over GF(p)
The lower bound on the number of n-variable balanced symmetric
functions over finite fields GF(p) presented in
{\cite{Cusick}} is improved in this paper
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
Characterization and Lower Bounds for Branching Program Size using Projective Dimension
We study projective dimension, a graph parameter (denoted by pd for a
graph ), introduced by (Pudl\'ak, R\"odl 1992), who showed that proving
lower bounds for pd for bipartite graphs associated with a Boolean
function imply size lower bounds for branching programs computing .
Despite several attempts (Pudl\'ak, R\"odl 1992 ; Babai, R\'{o}nyai, Ganapathy
2000), proving super-linear lower bounds for projective dimension of explicit
families of graphs has remained elusive.
We show that there exist a Boolean function (on bits) for which the
gap between the projective dimension and size of the optimal branching program
computing (denoted by bpsize), is . Motivated by the
argument in (Pudl\'ak, R\"odl 1992), we define two variants of projective
dimension - projective dimension with intersection dimension 1 (denoted by
upd) and bitwise decomposable projective dimension (denoted by
bitpdim).
As our main result, we show that there is an explicit family of graphs on vertices such that the projective dimension is , the
projective dimension with intersection dimension is and the
bitwise decomposable projective dimension is .
We also show that there exist a Boolean function (on bits) for which
the gap between upd and bpsize is . In contrast, we
also show that the bitwise decomposable projective dimension characterizes size
of the branching program up to a polynomial factor. That is, there exists a
constant and for any function , . We also study two other
variants of projective dimension and show that they are exactly equal to
well-studied graph parameters - bipartite clique cover number and bipartite
partition number respectively.Comment: 24 pages, 3 figure
Distance-regular graphs
This is a survey of distance-regular graphs. We present an introduction to
distance-regular graphs for the reader who is unfamiliar with the subject, and
then give an overview of some developments in the area of distance-regular
graphs since the monograph 'BCN' [Brouwer, A.E., Cohen, A.M., Neumaier, A.,
Distance-Regular Graphs, Springer-Verlag, Berlin, 1989] was written.Comment: 156 page
OBDD-Based Representation of Interval Graphs
A graph can be described by the characteristic function of the
edge set which maps a pair of binary encoded nodes to 1 iff the nodes
are adjacent. Using \emph{Ordered Binary Decision Diagrams} (OBDDs) to store
can lead to a (hopefully) compact representation. Given the OBDD as an
input, symbolic/implicit OBDD-based graph algorithms can solve optimization
problems by mainly using functional operations, e.g. quantification or binary
synthesis. While the OBDD representation size can not be small in general, it
can be provable small for special graph classes and then also lead to fast
algorithms. In this paper, we show that the OBDD size of unit interval graphs
is and the OBDD size of interval graphs is $O(\
| V \ | \log \ | V \ |)\Omega(\ | V \ | \log
\ | V \ |)O(\log \ | V \ |)O(\log^2 \ | V \ |)$ operations and
evaluate the algorithms empirically.Comment: 29 pages, accepted for 39th International Workshop on Graph-Theoretic
Concepts 201
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