Limitations and tradeoffs in synchronization of large-scale networks with uncertain links

We study synchronization in scalar nonlinear systems connected over a linear network with stochastic uncertainty in their interactions. We provide a sufficient condition for the synchronization of such network systems expressed in terms of the parameters of the nonlinear scalar dynamics, the second and largest eigenvalues of the mean interconnection Laplacian, and the variance of the stochastic uncertainty. The sufficient condition is independent of network size thereby making it attractive for verification of synchronization in a large size network. The main contribution of this paper is to provide analytical characterization for the interplay of roles played by the internal dynamics of the nonlinear system, network topology, and uncertainty statistics in network synchronization. We show there exist important tradeoffs between these various network parameters necessary to achieve synchronization. We show for nearest neighbor networks with stochastic uncertainty in interactions there exists an optimal number of neighbors with maximum margin for synchronization. This proves in the presence of interaction uncertainty, too many connections among network components is just as harmful for synchronization as the lack of connection. We provide an analytical formula for the optimal gain required to achieve maximum synchronization margin thereby allowing us to compare various complex network topology for their synchronization property

Re-visiting the One-Time Pad

In 1949, Shannon proved the perfect secrecy of the Vernam cryptographic system,also popularly known as the One-Time Pad (OTP). Since then, it has been believed that the perfectly random and uncompressible OTP which is transmitted needs to have a length equal to the message length for this result to be true. In this paper, we prove that the length of the transmitted OTP which actually contains useful information need not be compromised and could be less than the message length without sacrificing perfect secrecy. We also provide a new interpretation for the OTP encryption by treating the message bits as making True/False statements about the pad, which we define as a private-object. We introduce the paradigm of private-object cryptography where messages are transmitted by verifying statements about a secret-object. We conclude by suggesting the use of Formal Axiomatic Systems for investing N bits of secret.Comment: 13 pages, 3 figures, submitted for publication to IndoCrypt 2005 conferenc

Matrix Representation of Iterative Approximate Byzantine Consensus in Directed Graphs

This paper presents a proof of correctness of an iterative approximate Byzantine consensus (IABC) algorithm for directed graphs. The iterative algorithm allows fault- free nodes to reach approximate conensus despite the presence of up to f Byzantine faults. Necessary conditions on the underlying network graph for the existence of a correct IABC algorithm were shown in our recent work [15, 16]. [15] also analyzed a specific IABC algorithm and showed that it performs correctly in any network graph that satisfies the necessary condition, proving that the necessary condition is also sufficient. In this paper, we present an alternate proof of correctness of the IABC algorithm, using a familiar technique based on transition matrices [9, 3, 17, 19]. The key contribution of this paper is to exploit the following observation: for a given evolution of the state vector corresponding to the state of the fault-free nodes, many alternate state transition matrices may be chosen to model that evolution cor- rectly. For a given state evolution, we identify one approach to suitably "design" the transition matrices so that the standard tools for proving convergence can be applied to the Byzantine fault-tolerant algorithm as well. In particular, the transition matrix for each iteration is designed such that each row of the matrix contains a large enough number of elements that are bounded away from 0