Doing-it-All with Bounded Work and Communication

We consider the Do-All problem, where $p$ cooperating processors need to complete $t$ similar and independent tasks in an adversarial setting. Here we deal with a synchronous message passing system with processors that are subject to crash failures. Efficiency of algorithms in this setting is measured in terms of work complexity (also known as total available processor steps) and communication complexity (total number of point-to-point messages). When work and communication are considered to be comparable resources, then the overall efficiency is meaningfully expressed in terms of effort defined as work + communication. We develop and analyze a constructive algorithm that has work $O( t + p \log p\, (\sqrt{p\log p}+\sqrt{t\log t}\, ) )$ and a nonconstructive algorithm that has work $O(t +p \log^2 p)$. The latter result is close to the lower bound $\Omega(t + p \log p/ \log \log p)$ on work. The effort of each of these algorithms is proportional to its work when the number of crashes is bounded above by $c\,p$, for some positive constant $c < 1$. We also present a nonconstructive algorithm that has effort $O(t + p ^{1.77})$

Simple and Optimal Randomized Fault-Tolerant Rumor Spreading

We revisit the classic problem of spreading a piece of information in a group of $n$ fully connected processors. By suitably adding a small dose of randomness to the protocol of Gasienic and Pelc (1996), we derive for the first time protocols that (i) use a linear number of messages, (ii) are correct even when an arbitrary number of adversarially chosen processors does not participate in the process, and (iii) with high probability have the asymptotically optimal runtime of $O(\log n)$ when at least an arbitrarily small constant fraction of the processors are working. In addition, our protocols do not require that the system is synchronized nor that all processors are simultaneously woken up at time zero, they are fully based on push-operations, and they do not need an a priori estimate on the number of failed nodes. Our protocols thus overcome the typical disadvantages of the two known approaches, algorithms based on random gossip (typically needing a large number of messages due to their unorganized nature) and algorithms based on fair workload splitting (which are either not {time-efficient} or require intricate preprocessing steps plus synchronization).Comment: This is the author-generated version of a paper which is to appear in Distributed Computing, Springer, DOI: 10.1007/s00446-014-0238-z It is available online from http://link.springer.com/article/10.1007/s00446-014-0238-z This version contains some new results (Section 6

EFFICIENT COUNTING WITH OPTIMAL RESILIENCE

Consider a complete communication network of n nodes, where the nodes receive a common clock pulse. We study the synchronous c-counting problem: given any starting state and up to f faulty nodes with arbitrary behavior, the task is to eventually have all correct nodes labeling the pulses with increasing values modulo c in agreement. Thus, we are considering algorithms that are self-stabilizing despite Byzantine failures. In this work, we give new algorithms for the synchronous counting problem that (1) are deterministic, (2) have optimal resilience, (3) have a linear stabilization time in f (asymptotically optimal), (4) use a small number of states, and, consequently, (5) communicate a small number of bits per round. Prior algorithms either resort to randomization, use a large number of states and need high communication bandwidth, or have suboptimal resilience. In particular, we achieve an exponential improvement in both state complexity and message size for deterministic algorithms. Moreover, we present two complementary approaches for reducing the number of bits communicated during and after stabilization.Peer reviewe

Almost-Everywhere Secure Computation with Edge Corruptions

We consider secure multi-party computation (MPC) in a setting where the adversary can separately corrupt not only the parties (nodes) but also the communication channels (edges), and can furthermore choose selectively and adaptively which edges or nodes to corrupt. Note that if an adversary corrupts an edge, even if the two nodes that share that edge are honest, the adversary can control the link and thus deliver wrong messages to both players. We consider this question in the information-theoretic setting, and require security against a computationally unbounded adversary. In a fully connected network the above question is simple (and we also provide an answer that is optimal up to a constant factor). What makes the problem more challenging is to consider the case of sparse networks. Partially connected networks are far more realistic than fully connected networks, which led Garay and Ostrovsky [Eurocrypt\u2708] to formulate the notion of (unconditional) \emph{almost everywhere (a.e.) secure computation} in the node-corruption model, i.e., a model in which not all pairs of nodes are connected by secure channels and the adversary can corrupt some of the nodes (but not the edges). In such a setting, MPC amongst all honest nodes cannot be guaranteed due to the possible poor connectivity of some honest nodes with other honest nodes, and hence some of them must be ``given up\u27\u27 and left out of the computation. The number of such nodes is a function of the underlying communication graph and the adversarial set of nodes. In this work we introduce the notion of \emph{almost-everywhere secure computation with edge corruptions}, which is exactly the same problem as described above, except that we additionally allow the adversary to completely control some of the communication channels between two correct nodes---i.e., to ``corrupt\u27\u27 edges in the network. While it is easy to see that an a.e. secure computation protocol for the original node-corruption model is also an a.e. secure computation protocol tolerating edge corruptions (albeit for a reduced fraction of edge corruptions with respect to the bound for node corruptions), no polynomial-time protocol is known in the case where a {\bf constant fraction} of the edges can be corrupted (i.e., the maximum that can be tolerated) and the degree of the network is sub-linear. We make progress on this front, by constructing graphs of degree $O(n^\epsilon)$ (for arbitrary constant $0<\epsilon<1$) on which we can run a.e. secure computation protocols tolerating a constant fraction of adversarial edges. The number of given-up nodes in our construction is $\mu n$ (for some constant $0<\mu<1$ that depends on the fraction of corrupted edges), which is also asymptotically optimal

Simple and Optimal Fault-tolerant Rumor Spreading

We present rumor spreading protocols for the complete graph topology that are robust against an arbitrary number of adversarial initial node failures. Our protocols are the first rumor spreading protocols combining the following three properties: they can tolerate any number of failures, they distribute the rumor to all nodes using linear number of messages (actually they use strictly minimal $n-1$ messages), and if an arbitrarily small constant fraction $p$ of nodes (including the initiator of the rumor) are working correctly, our protocols communicate the rumor to all members in the network in $O(\log(n))$ rounds. Our protocols are simpler than previous fault-tolerant rumor spreading protocols in this model, they do not require synchronization (i.e., their correctness is independent on the relative speeds of the nodes), and they do not require a simultaneous wakeup of all nodes at time 0

On strong fault tolerance (or strong Menger-connectivity) of multicomputer networks

As the size of networks increases continuously, dealing with networks with faulty nodes becomes unavoidable. In this dissertation, we introduce a new measure for network fault tolerance, the strong fault tolerance (or strong Menger-connectivity)in multicomputer networks, and study the strong fault tolerance for popular multicomputer network structures. Let G be a network in which all nodes have degree d. We say that G is strongly fault tolerant if it has the following property: Let Gf be a copy of G with at most d - 2 faulty nodes. Then for any pair of non-faulty nodes u and v in Gf , there are min{degf (u), degf (v)} node-disjoint paths in Gf from u to v, where degf (u) and degf (v) are the degrees of the nodes u and v in Gf, respectively. First we study the strong fault tolerance for the popular network structures such as star networks and hypercube networks. We show that the star networks and the hypercube networks are strongly fault tolerant and develop efficient algorithms that construct the maximum number of node-disjoint paths of nearly optimal or optimal length in these networks when they contain faulty nodes. Our algorithms are optimal in terms of their time complexity. In addition to studying the strong fault tolerance, we also investigate a more realistic concept to describe the ability of networks for tolerating faults. The traditional definition of fault tolerance, sustaining at most d - 1faulty nodes for a regular graph G of degree d, reflects a very rare situation. In many cases, there is a chance that a routing path between two given nodes can be constructed though the network may have more faulty nodes than its degree. In this dissertation, we study the fault tolerance of hypercube networks under a probability model. When each node of the n-dimensional hypercube network has an independent failure probability p, we develop algorithms that, with very high probability, can construct a fault-free path when the hypercube network can sustain up to 2np faulty nodes

Resource-Efficient Communication in the Presence of Adversaries

This dissertation presents algorithms for achieving communication in the presence of adversarial attacks in large, decentralized, resource-constrained networks. We consider abstract single-hop communication settings where a set of senders wishes to directly communicate with a set of receivers . These results are then extended to provide resource-efficient, multi-hop communication in wireless sensor networks (WSNs), where energy is critically scarce, and peer-to-peer (P2P) networks, where bandwidth and computational power are limited. Our algorithms are provably correct in the face of attacks by a computationally bounded adversary who seeks to disrupt communication between correct participants. The first major result in this dissertation addresses a general scenario involving single-hop communication in a time-slotted network where a single sender in wishes to transmit a message to a single receiver in . The two players share a communication channel; however, there exists an adversary who aims to prevent the transmission of by periodically blocking this channel. There are costs to send, receive or block on the channel, and we ask: How much do the two players need to spend relative to the adversary in order to guarantee transmission of the message? This problem abstracts many types of conflict in information networks, and the associated costs represent an expenditure of network resources. We show that it is significantly more costly for the adversary to block than for the two players to achieve communication. Specifically, if the cost to send, receive and block in a slot are fixed constants, and the adversary spends a total of slots to try to block the message, then both the sender and receiver must be active in only O(ᵠ⁻¹ + 1) slots in expectation to transmit , where φ = (1+ √5)/2 is the golden ratio. Surprisingly, this result holds even if (1) the value of is unknown to either player; (2) the adversary knows the algorithms of both players, but not their random bits; and (3) the adversary is able to launch attacks using total knowledge of past actions of both players. Finally, these results are applied to two concrete problems. First, we consider jamming attacks in WSNs and address the fundamental task of propagating from a single device to all others in a WSN in the presence of faults; this is the problem of reliable broadcast. Second, we examine how our algorithms can mitigate application-level distributed denial-of-service attacks in wired client-server scenarios. The second major result deals with a single-hop communication problem where now consists of multiple senders and there is still a single receiver who wishes to obtain a message . However, many of the senders (strictly less than half) can be faulty, failing to send or sending incorrect messages. While the majority of the senders possess , rather than listening to all of and majority filtering on the received data, we desire an algorithm that allows the single receiver to decide on in a more efficient manner. To investigate this scenario, we define and devise algorithms for a new data streaming problem called the Bad Santa problem which models the selection dilemma faced by the receiver. With our results for the Bad Santa problem, we consider the problem of energy-efficient reliable broadcast. All previous results on reliable broadcast require devices to spend significant time in the energy-expensive receiving state which is a critical problem in WSNs where devices are typically battery powered. In a popular WSN model, we give a reliable broadcast protocol that achieves optimal fault tolerance (i.e., tolerates the maximum number of faults in this WSN model) and improves over previous results by achieving an expected quadratic decrease in the cost to each device. For the case where the number of faults is within a (1-∊)-factor of the optimal fault tolerance, for any constant ∊>0, we give a reliable broadcast protocol that improves further by achieving an expected (roughly) exponential decrease in the cost to each device. The third and final major result of this dissertation addresses single-hop communication where and both consist of multiple peers that need to communicate in an attack-resistant P2P network. There are several analytical results on P2P networks that can tolerate an adversary who controls a large number of peers and uses them to disrupt network functionality. Unfortunately, in such systems, operations such as data retrieval and message sending incur significant communication costs. Here, we employ cryptographic techniques to define two protocols both of which are more efficient than existing solutions. For a network of peers, our first protocol is deterministic with O(log²) message complexity and our second protocol is randomized with expected O(log ) message complexity; both improve over all previous results. The hidden constants and setup costs for our protocols are small and no trusted third party is required. Finally, we present an analysis showing that our protocols are practical for deployment under significant churn and adversarial behaviour