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**Published in:** LIPIcs, Volume 246, 36th International Symposium on Distributed Computing (DISC 2022)

We initiate the study of distributed graph algorithms under the presence of Byzantine nodes. We consider the fundamental problem of testing the connectivity of a graph in the congested clique model in a Byzantine setting. We are given a n-vertex (arbitrary) graph G embedded in a n-node congested clique where an arbitrary subset of B nodes of the clique of size up to (1/3-ε)n (for any arbitrary small constant ε > 0) can be Byzantine. We consider the full information model where Byzantine nodes can behave arbitrarily, collude with each other, and have unlimited computational power and full knowledge of the states and actions of the honest nodes, including random choices made up to the current round.
Our main result is an efficient randomized distributed algorithm that is able to correctly distinguish between two contrasting cases: (1) the graph G⧵ B (i.e., the graph induced by the removal of the vertices assigned to the Byzantine nodes in the clique) is connected or (2) the graph G is far from connected, i.e., it has at least 2|B|+1 connected components. Our algorithm runs in O(polylog n) rounds in the congested clique model and guarantees that all honest nodes will decide on the correct case with high probability. Since Byzantine nodes can lie about the vertices assigned to them, we show that this is essentially the best possible that can be done by any algorithm. Our result can be viewed also in the spirit of property testing, where our algorithm is able to distinguish between two contrasting cases while giving no guarantees if the graph falls in the grey area (i.e., neither of the cases occur).
Our work is a step towards robust and secure distributed graph computation that can output meaningful results even in the presence of a large number of faulty or malicious nodes.

John Augustine, Anisur Rahaman Molla, Gopal Pandurangan, and Yadu Vasudev. Byzantine Connectivity Testing in the Congested Clique. In 36th International Symposium on Distributed Computing (DISC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 246, pp. 7:1-7:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{augustine_et_al:LIPIcs.DISC.2022.7, author = {Augustine, John and Molla, Anisur Rahaman and Pandurangan, Gopal and Vasudev, Yadu}, title = {{Byzantine Connectivity Testing in the Congested Clique}}, booktitle = {36th International Symposium on Distributed Computing (DISC 2022)}, pages = {7:1--7:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-255-6}, ISSN = {1868-8969}, year = {2022}, volume = {246}, editor = {Scheideler, Christian}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2022.7}, URN = {urn:nbn:de:0030-drops-171987}, doi = {10.4230/LIPIcs.DISC.2022.7}, annote = {Keywords: Byzantine protocols, distributed graph algorithms, congested clique, graph connectivity, fault-tolerant computation, randomized algorithms} }

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**Published in:** LIPIcs, Volume 179, 34th International Symposium on Distributed Computing (DISC 2020)

The last decade has seen substantial progress on designing Byzantine agreement algorithms which do not require all-to-all communication. However, these protocols do require each node to play a particular role determined by its ID. Motivated by the rise of permissionless systems such as Bitcoin, where nodes can join and leave at will, we extend this research to a more practical model where initially, each node does not know the identity of its neighbors. In particular, a node can send to new destinations only by sending to random (or arbitrary) nodes, or responding to messages received from those destinations. We assume a synchronous and fully-connected network, with a full-information, but static Byzantine adversary. A major drawback of existing Byzantine protocols in this setting is that they have at least Ω(n²) message complexity, where n is the total number of nodes. In particular, the communication cost incurred by the honest nodes is Ω(n²), even when Byzantine node send no messages. In this paper, we design protocols for fundamental problems which are message-competitive, i.e., the total number of bits sent by honest nodes is not significantly more than the total sent by Byzantine nodes.
We describe a message-competitive algorithm to solve Byzantine agreement, leader election, and committee election. Our algorithm sends an expected O((T+n)log n) bits and has latency O(polylog(n)) (even in the CONGEST model), where T = O(n²) is the number of bits sent by Byzantine nodes. The algorithm is resilient to (1/4-ε)n Byzantine nodes for any fixed ε > 0, and succeeds with high probability. Our message bounds are essentially optimal up to polylagarithmic factors, for algorithms that run in polylogarithmic rounds in the CONGEST model.
We also show lower bounds for message-competitive Byzantine agreement regardless of rounds. We prove that, in general, one cannot hope to design Byzantine protocols that have communication cost that is significantly smaller than the cost of the Byzantine adversary.

John Augustine, Valerie King, Anisur Rahaman Molla, Gopal Pandurangan, and Jared Saia. Scalable and Secure Computation Among Strangers: Message-Competitive Byzantine Protocols. In 34th International Symposium on Distributed Computing (DISC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 179, pp. 31:1-31:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{augustine_et_al:LIPIcs.DISC.2020.31, author = {Augustine, John and King, Valerie and Molla, Anisur Rahaman and Pandurangan, Gopal and Saia, Jared}, title = {{Scalable and Secure Computation Among Strangers: Message-Competitive Byzantine Protocols}}, booktitle = {34th International Symposium on Distributed Computing (DISC 2020)}, pages = {31:1--31:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-168-9}, ISSN = {1868-8969}, year = {2020}, volume = {179}, editor = {Attiya, Hagit}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2020.31}, URN = {urn:nbn:de:0030-drops-131093}, doi = {10.4230/LIPIcs.DISC.2020.31}, annote = {Keywords: Byzantine protocols, Byzantine agreement, Leader election, Committee election, Message-competitive protocol, Randomized protocol} }

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**Published in:** LIPIcs, Volume 46, 19th International Conference on Principles of Distributed Systems (OPODIS 2015)

We study the single-message broadcast problem in dynamic radio networks. We show that the time complexity of the problem depends on the amount of stability and connectivity of the dynamic network topology and on the adaptiveness of the adversary providing the dynamic topology. More formally, we model communication using the standard graph-based radio network model. To model the dynamic network, we use a variant of the synchronous dynamic graph model introduced in [Kuhn et al., STOC 2010]. For integer parameters T >= 1 and k => 1, we call a dynamic graph T-interval k-connected if for every interval of T consecutive rounds, there exists a k-vertex-connected stable subgraph. Further, for an integer parameter tau >= 0, we say that the adversary providing the dynamic network is tau-oblivious if for constructing the graph of some round t, the adversary has access to all the randomness (and states) of the algorithm up to round t-tau.
As our main result, we show that for any T >= 1, any k >= 1, and any tau = 1, for a tau-oblivious adversary, there is a distributed algorithm to broadcast a single message in time O((1+n/(k * min(tau,T)) * n *log^3(n)). We further show that even for large interval k-connectivity, efficient broadcast is not possible for the usual adaptive adversaries. For a 1-oblivious adversary, we show that even for any T <= (n/k)^{1-epsilon} (for any constant epsilon > 0) and for any k >= 1, global broadcast in T-interval k-connected networks requires at least Omega(n^2/k^2*log(n)) time. Further, for a 0-oblivious adversary, broadcast cannot be solved in T-interval k-connected networks as long as T < n-k.

Mohamad Ahmadi, Abdolhamid Ghodselahi, Fabian Kuhn, and Anisur Rahaman Molla. The Cost of Global Broadcast in Dynamic Radio Networks. In 19th International Conference on Principles of Distributed Systems (OPODIS 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 46, pp. 7:1-7:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{ahmadi_et_al:LIPIcs.OPODIS.2015.7, author = {Ahmadi, Mohamad and Ghodselahi, Abdolhamid and Kuhn, Fabian and Molla, Anisur Rahaman}, title = {{The Cost of Global Broadcast in Dynamic Radio Networks}}, booktitle = {19th International Conference on Principles of Distributed Systems (OPODIS 2015)}, pages = {7:1--7:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-98-9}, ISSN = {1868-8969}, year = {2016}, volume = {46}, editor = {Anceaume, Emmanuelle and Cachin, Christian and Potop-Butucaru, Maria}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.OPODIS.2015.7}, URN = {urn:nbn:de:0030-drops-65989}, doi = {10.4230/LIPIcs.OPODIS.2015.7}, annote = {Keywords: radio network, dynamic network, global broadcast, interval connectivity, hitting game} }

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**Published in:** LIPIcs, Volume 46, 19th International Conference on Principles of Distributed Systems (OPODIS 2015)

We study the problem of computing a sparse cut in an undirected network graph G=(V,E). We measure the sparsity of a cut (S,V\S) by its conductance phi(S), i.e., by the ratio of the number of edges crossing the cut and the sum of the degrees on the smaller of the two sides. We present an efficient distributed algorithm to compute a cut of low conductance. Specifically, given two parameters b and phi, if there exists a cut of balance at least b and conductance at most phi, our algorithm outputs a cut of balance at least b/2 and conductance at most ~O(sqrt{phi}), where ~O(.) hides polylogarithmic factors in the number of nodes n. Our distributed algorithm works in the \congest model, i.e., it only requires to send messages of size at most O(log(n)) bits. The time complexity of the algorithm is ~O(D + 1/b*phi), where D is the diameter of G. This is a significant improvement over a result by Das Sarma et al. [ICDCN 2015], where it is shown that a cut of the same quality can be computed in time ~O(n + 1/b*phi). The improved running time is in particular achieved by devising and applying an efficient distributed algorithm for the all-prefix-sums problem in a distributed search tree. This algorithm, which is based on the classic parallel all-prefix-sums algorithm, might be of independent interest.

Fabian Kuhn and Anisur Rahaman Molla. Distributed Sparse Cut Approximation. In 19th International Conference on Principles of Distributed Systems (OPODIS 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 46, pp. 10:1-10:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{kuhn_et_al:LIPIcs.OPODIS.2015.10, author = {Kuhn, Fabian and Molla, Anisur Rahaman}, title = {{Distributed Sparse Cut Approximation}}, booktitle = {19th International Conference on Principles of Distributed Systems (OPODIS 2015)}, pages = {10:1--10:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-98-9}, ISSN = {1868-8969}, year = {2016}, volume = {46}, editor = {Anceaume, Emmanuelle and Cachin, Christian and Potop-Butucaru, Maria}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.OPODIS.2015.10}, URN = {urn:nbn:de:0030-drops-66014}, doi = {10.4230/LIPIcs.OPODIS.2015.10}, annote = {Keywords: sparsest cut, conductance, random walks, all-prefix-sums} }

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