# Broadcasting in hypercubes with randomly distributed Byzantine faults

## Abstract

We study all-to-all broadcasting in hypercubes with randomly distributed Byzantine faults. We construct an efficient broadcasting scheme BC1-*n*-cube running on the *n*-dimensional hypercube (*n*-cube for short) in *2n* rounds, where for communication by each node of the *n*-cube, only one of its links is used in each round. The scheme BC1-*n*-cube can tolerate ⌊(*n* −1)/2⌋ Byzantine node and/or link faults in the worst case. If there are at most *f* Byzantine faulty nodes randomly distributed in the *n*-cube, BC1-*n*-cube succeeds with a probability higher than 1-(64*nf*/2^{n})^{⌈n/2⌉}. In other words, if 1/(64nk) of all the nodes (i.e., 2^{n}/(64nk) nodes) fail in Byzantine manner randomly in the *n*-cube, then the scheme succeeds with a probability higher than 1 −k^{−⌈ n/2 ⌉}. We also consider the case where all nodes are faultless but links may fail randomly in the *n*-cube. Scheme BC1-*n*-cube succeeds with a probability higher than 1 −k^{−⌈ n/2 ⌉} provided that not more than l/(64(*n* + 1)k) of all the links in the *n-cube* fail in Byzantine manner randomly. For the case where only links may fail, we give another broadcasting scheme BC2-*n*-cube which runs in 2n^{2} rounds. Broadcasting by BC2-*n*-cube is successful with a high probability if the number of Byzantine faulty links randomly distributed in the *n*-cube is not more than a constant fraction of the total number of links. That is, it succeeds with a probability higher than 1−n·k−⌈ n/2 ⌉ if l/(48k) of all the links in the *n*-cube fail in Byzantine manner randomly.

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