## Abstract

We present two distributed algorithms for the Byzantine counting problem, which is concerned with estimating the size of a network in the presence of a large number of Byzantine nodes.In an n-node network (n is unknown), our first algorithm, which is deterministic, finishes in O(log n) rounds and is time-optimal. This algorithm can tolerate up to O(n1-γ) arbitrarily (adversarially) placed Byzantine nodes for any arbitrarily small (but fixed) positive constant γ. It outputs a (fixed) constant factor estimate of log n that would be known to all but o(1) fraction of the good nodes. This algorithm works for any bounded degree expander network. However, this algorithms assumes that good nodes can send arbitrarily large-sized messages in a round.Our second algorithm is randomized and most good nodes send only small-sized messages.1 This algorithm works in almost all d-regular graphs. It tolerates up to B(n) = {n1/2- ξ (note that n and B(n) are unknown to the algorithm) arbitrarily (adversarially) placed Byzantine nodes, where ξ is any arbitrarily small (but fixed) positive constant. This algorithm takes O(B(n) log2 n) rounds and outputs a constant factor estimate of log n with probability at least 1-o(1). The said estimate is known to most nodes, i.e., ≥ (1 - β)n nodes for any arbitrarily small (but fixed) positive constant β.To complement our algorithms, we also present an impossibility result that shows that it is impossible to estimate the network size with any reasonable approximation with any non-trivial probability of success if the network does not have sufficient vertex expansion.Both algorithms are the first such algorithms that solve Byzantine counting in sparse, bounded degree networks under very general assumptions. Both algorithms are fully local and need no global knowledge.Our algorithms can be used for the design of efficient distributed algorithms resilient against Byzantine failures, where the knowledge of the network size - a global parameter - may not be known a priori.

Original language | English (US) |
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Title of host publication | Proceedings - 2022 IEEE 42nd International Conference on Distributed Computing Systems, ICDCS 2022 |

Publisher | Institute of Electrical and Electronics Engineers Inc. |

Pages | 12-22 |

Number of pages | 11 |

ISBN (Electronic) | 9781665471770 |

DOIs | |

State | Published - 2022 |

Event | 42nd IEEE International Conference on Distributed Computing Systems, ICDCS 2022 - Bologna, Italy Duration: Jul 10 2022 → Jul 13 2022 |

### Publication series

Name | Proceedings - International Conference on Distributed Computing Systems |
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Volume | 2022-July |

### Conference

Conference | 42nd IEEE International Conference on Distributed Computing Systems, ICDCS 2022 |
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Country/Territory | Italy |

City | Bologna |

Period | 7/10/22 → 7/13/22 |

## Keywords

- Byzantine counting
- Byzantine faults
- expander graphs
- network size estimation
- randomization

## ASJC Scopus subject areas

- Software
- Hardware and Architecture
- Computer Networks and Communications