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$(G)$ for a graph $G$), introduced by (Pudl\'ak, R\"odl 1992), who showed that proving lower bounds for pd$(G_f)$ for bipartite graphs $G_f$ associated with a Boolean function $f$ imply size lower bounds for branching programs computing $f$. 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 $f$ (on $n$ bits) for which the gap between the projective dimension and size of the optimal branching program computing $f$ (denoted by bpsize$(f)$), is $2^{\Omega(n)}$. 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$(G)$) and bitwise decomposable projective dimension (denoted by bitpdim$(G)$). As our main result, we show that there is an explicit family of graphs on $N = 2^n$ vertices such that the projective dimension is $O(\sqrt{n})$, the projective dimension with intersection dimension $1$ is $\Omega(n)$ and the bitwise decomposable projective dimension is $\Omega(\frac{n^{1.5}}{\log n})$. We also show that there exist a Boolean function $f$ (on $n$ bits) for which the gap between upd$(G_f)$ and bpsize$(f)$ is $2^{\Omega(n)}$. 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 $c>0$ and for any function $f$, $\textrm{bitpdim}(G_f)/6 \le \textrm{bpsize}(f) \le (\textrm{bitpdim}(G_f))^c$. 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 $G = (V,E)$ can be described by the characteristic function of the edge set $\chi_E$ which maps a pair of binary encoded nodes to 1 iff the nodes are adjacent. Using \emph{Ordered Binary Decision Diagrams} (OBDDs) to store $\chi_E$ 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 $O(\ | V \ | /\log \ | V \ |)$ and the OBDD size of interval graphs is $O(\ | V \ | \log \ | V \ |)$which both improve a known result from Nunkesser and Woelfel (2009). Furthermore, we can show that using our variable order and node labeling for interval graphs the worst-case OBDD size is$\Omega(\ | V \ | \log \ | V \ |)$. We use the structure of the adjacency matrices to prove these bounds. This method may be of independent interest and can be applied to other graph classes. We also develop a maximum matching algorithm on unit interval graphs using$O(\log \ | V \ |)$operations and a coloring algorithm for unit and general intervals graphs using$O(\log^2 \ | V \ |)$ operations and evaluate the algorithms empirically.Comment: 29 pages, accepted for 39th International Workshop on Graph-Theoretic Concepts 201