Shreyas Sundaram

Distributed Function Calculation via Linear Iterative Strategies in the Presence of Malicious Agents

Given a network of interconnected nodes, each with its own value (such as a measurement, position, vote, or other data), we develop a distributed strategy that enables some or all of the nodes to calculate any arbitrary function of the node values, despite the actions of malicious nodes in the network. Our scheme assumes a broadcast model of communication (where all nodes transmit the same value to all of their neighbors) and utilizes a linear iteration where, at each time-step, each node updates its value to be a weighted average of its own previous value and those of its neighbors. We consider a node to be malicious or faulty if, instead of following the predefined linear strategy, it updates its value arbitrarily at each time-step (perhaps conspiring with other malicious nodes in the process). We show that the topology of the network completely characterizes the resilience of linear iterative strategies to this kind of malicious behavior. First, when the network contains 2f or fewer vertex-disjoint paths from some node xj to another node xi , we provide an explicit strategy for f malicious nodes to follow in order to prevent node xi from receiving any information about xjs value. Next, if node xi has at least 2f+1 vertex-disjoint paths from every other (non-neighboring) node, we show that xi is guaranteed to be able to calculate any arbitrary function of all node values when the number of malicious nodes is f or less. Furthermore, we show that this function can be calculated after running the linear iteration for a finite number of time-steps (upper bounded by the number of nodes in the network) with almost any set of weights (i.e., for all weights except for a set of measure zero).

Resilient Asymptotic Consensus in Robust Networks

This paper addresses the problem of resilient in-network consensus in the presence of misbehaving nodes. Secure and fault-tolerant consensus algorithms typically assume knowledge of nonlocal information; however, this assumption is not suitable for large-scale dynamic networks. To remedy this, we focus on local strategies that provide resilience to faults and compromised nodes. We design a consensus protocol based on local information that is resilient to worst-case security breaches, assuming the compromised nodes have full knowledge of the network and the intentions of the other nodes. We provide necessary and sufficient conditions for the normal nodes to reach asymptotic consensus despite the influence of the misbehaving nodes under different threat assumptions. We show that traditional metrics such as connectivity are not adequate to characterize the behavior of such algorithms, and develop a novel graph-theoretic property referred to as network robustness. Network robustness formalizes the notion of redundancy of direct information exchange between subsets of nodes in the network, and is a fundamental property for analyzing the behavior of certain distributed algorithms that use only local information.

The Wireless Control Network: A New Approach for Control Over Networks

We present a method to stabilize a plant with a network of resource constrained wireless nodes. As opposed to traditional networked control schemes where the nodes simply route information to and from a dedicated controller (perhaps performing some encoding along the way), our approach treats the network itself as the controller. Specifically, we formulate a strategy for each node in the network to follow, where at each time-step, each node updates its internal state to be a linear combination of the states of the nodes in its neighborhood. We show that this causes the entire network to behave as a linear dynamical system, with sparsity constraints imposed by the network topology. We provide a numerical design procedure to determine appropriate linear combinations to be applied by each node so that the transmissions of the nodes closest to the actuators will stabilize the plant. We also show how our design procedure can be modified to maintain mean square stability under packet drops in the network, and present a distributed scheme that can handle node failures while preserving stability. We call this architecture a Wireless Control Network, and show that it introduces very low computational and communication overhead to the nodes in the network, allows the use of simple transmission scheduling algorithms, and enables compositional design (where the existing wireless control infrastructure can be easily extended to handle new plants that are brought online in the vicinity of the network).

Distributed function calculation and consensus using linear iterative strategies

Given an arbitrary network of interconnected nodes, we develop and analyze a distributed strategy that enables a subset of the nodes to calculate any given function of the node values. Our scheme utilizes a linear iteration where, at each time-step, each node updates its value to be a weighted average of its own previous value and those of its neighbors. We show that this approach can be viewed as a linear dynamical system, with dynamics that are given by the weight matrix of the linear iteration, and with outputs for each node that are captured by the set of values that are available to that node at each time-step. In connected networks with time-invariant topologies, we use observability theory to show that after running the linear iteration for a finite number of time-steps with almost any choice of weight matrix, each node obtains enough information to calculate any arbitrary function of the initial node values. The problem of distributed consensus via linear iterations, where all nodes in the network calculate the same function, is treated as a special case of our approach. In particular, our scheme allows nodes in connected networks with time-invariant topologies to reach consensus on any arbitrary function of the initial node values in a finite number of steps for almost any choice of weight matrix.

Finite-Time Distributed Consensus in Graphs with Time-Invariant Topologies

We present a method for achieving consensus in distributed systems in a finite number of time-steps. Our scheme involves a linear iteration where, at each time-step, each node updates its value to be a weighted average of its own previous value and those of its neighbors. If D denotes the degree of the minimal polynomial of the weight matrix associated with the linear iteration, we show that each node can immediately calculate the consensus value as a linear combination of its own past values over at most D time-steps. We also show that each node can determine the coefficients for this linear combination in a decentralized manner. The proposed scheme has the potential to significantly reduce the time and communication required to reach consensus in distributed systems.

Delayed Observers for Linear Systems With Unknown Inputs

We present a method for constructing reduced-order state observers for linear systems with unknown inputs. Our design provides a characterization of observers with delay, which eases the established necessary conditions for existence of unknown input observers with zero-delay. In order to obtain the observer parameters, we develop a systematic design procedure that is quite general in that it encompasses the design of full-order observers via appropriate choices of design matrices.

Distributed Autonomous Online Learning: Regrets and Intrinsic Privacy-Preserving Properties

Online learning has become increasingly popular on handling massive data. The sequential nature of online learning, however, requires a centralized learner to store data and update parameters. In this paper, we consider online learning with distributed data sources. The autonomous learners update local parameters based on local data sources and periodically exchange information with a small subset of neighbors in a communication network. We derive the regret bound for strongly convex functions that generalizes the work by Ram et al. for convex functions. More importantly, we show that our algorithm has intrinsic privacy-preserving properties, and we prove the sufficient and necessary conditions for privacy preservation in the network. These conditions imply that for networks with greater-than-one connectivity, a malicious learner cannot reconstruct the subgradients (and sensitive raw data) of other learners, which makes our algorithm appealing in privacy-sensitive applications.

A public-key cryptographic processor for RSA and ECC

We describe a general-purpose processor architecture for accelerating public-key computations on server systems that demand high performance and flexibility to accommodate large numbers of secure connections with heterogeneous clients that are likely to be limited in the set of cryptographic algorithms supported. Flexibility is achieved in that the processor supports multiple public-key cryptosystems, namely RSA, DSA, DH, and ECC, arbitrary key sizes and, in the case of ECC, arbitrary curves over fields GF(p) and GF(2/sup m/). At the core of the processor is a novel dual-field multiplier based on a modified carry-save adder (CSA) tree that supports both GF(p) and GF(2/sup m/). In the case of a 64-bit integer multiplier, the necessary modifications increase its size by a mere 5%. To efficiently schedule the multiplier, we implemented a multiply-accumulate instruction that combines several steps of a multiple-precision multiplication in a single operation: multiplication, carry propagation, and partial product accumulation. We have developed a hardware prototype of the cryptographic processor in FPGA technology. If implemented in current 1.5 GHz processor technology, the processor executes 5,265 RSA-1024 op/s and 25,756 ECC-163 op/s - the given key sizes offer comparable security strength. Looking at future security levels, performance is 786 op/s for RSA-2048 and 9,576 op/s for ECC-233.

Consensus of multi-agent networks in the presence of adversaries using only local information

This paper addresses the problem of resilient consensus in the presence of misbehaving nodes. Although it is typical to assume knowledge of at least some nonlocal information when studying secure and fault-tolerant consensus algorithms, this assumption is not suitable for large-scale dynamic networks. To remedy this, we emphasize the use of local strategies to deal with resilience to security breaches. We study a consensus protocol that uses only local information and we consider worst-case security breaches, where the compromised nodes have full knowledge of the network and the intentions of the other nodes. We provide necessary and sufficient conditions for the normal nodes to reach consensus despite the influence of the malicious nodes under different threat assumptions. These conditions are stated in terms of a novel graph-theoretic property referred to as network robustness.

The wireless control network: Monitoring for malicious behavior

We consider the problem of stabilizing a plant with a network of resource constrained wireless nodes. In a companion paper, we developed a protocol where each node repeatedly transmits an appropriate (stabilizing) linear combination of the values in its neighborhood. In this paper, we design an Intrusion Detection System (IDS) for this control scheme, which observes the transmissions of certain nodes and uses that information to (a) recover the plant outputs (for data-logging and diagnostic purposes) and (b) identify malicious behavior by any of the wireless nodes in the network. We show that if the connectivity of the network is sufficiently high, the IDS only needs to observe a subset of the nodes in the network in order to achieve this objective. Our approach provides a characterization of the set of nodes that should be observed, a systematic procedure for the IDS to use to identify the malicious nodes and recover the outputs of the plant, and an upper bound on the delay required to obtain the necessary information.