C. Pandu Rangan

Progress in Cryptology – INDOCRYPT 2007

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Linear algorithm for optimal path cover problem on interval graphs

Abstract A path cover of a graph G is a set of vertex-disjoint paths that cover all the vertices of G. An optimal path cover of G is a path cover of minimum cardinality. This problem is known to be NP-complete for arbitrary graphs. We present a linear algorithm for this problem on interval graphs. Given the adjacency lists of an interval graph with n vertices and m edges, our algorithm runs in O(m+n) time.

On perfectly secure communication over arbitrary networks

We study the interplay of network connectivity and perfectly secure message transmission under the corrupting influence of generalized Byzantine adversaries. It is known that in the threshold adversary model, where the Byzantine adversary can corrupt upto any t among the n players (nodes), perfectly secure communication among any pair of players is possible if and only if the underlying synchronous network is (2t + 1)-connected. Strictly generalizing these results to the non-threshold setting, we show that perfectly secure communication among any pair of players is possible if and only if the union of no two sets in the adversary structure is a vertex cutset of the synchronous network. The computation and communication complexities of the transmission protocol are polynomial in the size of the network and the maximal basis of the adversary structure.

Optimal Perfectly Secure Message Transmission

In the perfectly secure message transmission (PSMT) problem, two synchronized non-faulty players (or processors), the Sender S and the Receiver R are connected by n wires (each of which facilitates 2-way communication); S has an l-bit message that he wishes to send to R; after exchanging messages in phases R should correctly obtain S’s message, while an adversary listening on and actively controlling any set of t (or less) wires should have no information about S’s message.

Algorithmic aspects of clique-transversal and clique-independent sets

Abstract A minimum clique-transversal set MCT ( G ) of a graph G =( V , E ) is a set S ⊆ V of minimum cardinality that meets all maximal cliques in G . A maximum clique-independent set MCI ( G ) of G is a set of maximum number of pairwise vertex-disjoint maximal cliques. We prove that the problem of finding an MCT ( G ) and an MCI ( G ) is NP -hard for cocomparability, planar, line and total graphs. As an interesting corollary we obtain that the problem of finding a minimum number of elements of a poset to meet all maximal antichains of the poset remains NP -hard even if the poset has height two, thereby generalizing a result of Duffas et al. (J. Combin. Theory Ser. A 58 (1991) 158–164). We present a polynomial algorithm for the above problems on Helly circular-arc graphs which is the first such algorithm for a class of graphs that is not clique-perfect . We also present polynomial algorithms for the weighted version of the clique-transversal problem on strongly chordal graphs, chordal graphs of bounded clique size, and cographs. The algorithms presented run in linear time for strongly chordal graphs and cographs. These mark the first attempts at the weighted version of the problem.

Restrictions of minimum spanner problems

Abstract A t -spanner of a graph G is a spanning subgraph H such that the distance between any two vertices in H is at most t times their distance in G . Spanners arise in the context of approximating the original graph with a sparse subgraph (Peleg, D., and Schaffer, A. A. (1989), J. Graph. Theory 13 (1), 99–116). The MINIMUM t -SPANNER problem seeks to find a t -spanner with the minimum number of edges for the given graph. In this paper, we completely settle the complexity status of this problem for various values of t , on chordal graphs, split graphs, bipartite graphs and convex bipartite graphs. Our results settle an open question raised by L. Cai (1994, Discrete Appl. Math. 48 , 187–194) and also greatly simplify some of the proofs presented by Cai and by L. Cai and M. Keil (1994, Networks 24 , 233–249). We also give a factor 2 approximation algorithm for the MINIMUM 2-SPANNER problem on interval graphs. Finally, we provide approximation algorithms for the bandwidth minimization problem on convex bipartite graphs and split graphs using the notion of tree spanners.

Tree 3-spanners on interval, permutation and regular bipartite graphs

The tree 3-spanner problem consists of finding a tree 3-spanner of a given graph G(V, E). We study the tree 3-spanner problem for some special classes of graphs namely interval graphs, permutation graphs and regular bipartite graphs.

Round-Optimal and efficient verifiable secret sharing

We consider perfect verifiable secret sharing (VSS) in a synchronous network of n processors (players) where a designated player called the dealer wishes to distribute a secret s among the players in a way that no t of them obtain any information, but any t + 1 players obtain full information about the secret. The round complexity of a VSS protocol is defined as the number of rounds performed in the sharing phase. Gennaro, Ishai, Kushilevitz and Rabin showed that three rounds are necessary and sufficient when n > 3t. Sufficiency, however, was only demonstrated by means of an inefficient (i.e., exponential-time) protocol, and the construction of an efficient three-round protocol was left as an open problem. In this paper, we present an efficient three-round protocol for VSS. The solution is based on a three-round solution of so-called weak verifiable secret sharing (WSS), for which we also prove that three rounds is a lower bound. Furthermore, we also demonstrate that one round is sufficient for WSS when n > 4t, and that VSS can be achieved in 1 + e amortized rounds (for any e > 0 ) when n>3t.

Clique transversal and clique independence on comparability graphs

Abstract We present O ( m √ n + M ( n )) algorithms for finding the clique transversal number and the clique independence number for a comparability graph of n nodes, where M ( n ) is the complexity of multiplying two n × n matrices.

Progress in Cryptology — INDOCRYPT 2001

Function Secret Sharing (FSS) and Homomorphic Secret Sharing (HSS) are two extensions of standard secret sharing, which support rich forms of homomorphism on secret shared values. – An m-party FSS scheme for a given function family F enables splitting a function f : {0, 1}n → G from F (for Abelian group G) into m succinctly described functions f1, . . . , fm such that strict subsets of the fi hide f , and f(x) = f1(x) + · · · + fm(x) for every input x. – An m-party HSS is a dual notion, where an input x is split into shares x, . . . , x, such that strict subsets of x hide x, and one can recover the evaluation P (x) of a program P on x given homomorphically evaluated share values Eval(x, P ), . . . ,Eval(x, P ). In the last few years, many new constructions and applications of FSS and HSS have been discovered, yielding implications ranging from efficient private database manipulation and secure computation protocols, to worst-case to average-case reductions. In this treatise, we introduce the reader to the background required to understand these developments, and give a roadmap of recent advances (up to October 2017).