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We present a solution to consensus on a torus with Byzantine faults. Anysolution to classic consensus that is tolerant to $f$ Byzantine faults requires$2f+1$ node-disjoint paths. Due to limited torus connectivity, this boundnecessitates spatial separation between faults. Our solution does not requirethis many disjoint paths and tolerates dense faults. Specifically, we considerthe case where all faults are in one column. We address the version ofconsensus where only processes in fault-free columns must agree. We prove thateven this weaker version is not solvable if the column may be completelyfaulty. We then present a solution for the case where at least one row isfault-free. The correct processes share orientation but do not know theidentities of other processes or the torus dimensions. The communication issynchronous. To achieve our solution, we build and prove correct an all-to-allbroadcast algorithm $\mathcal{BAT}$ that guarantees delivery to all processesin fault-free columns. We use this algorithm to solve our weak consensusproblem. Our solution, $\mathcal{CBAT}$, runs in $O(H+W)$ rounds, where $H$ and$W$ are torus height and width respectively. We extend our consensus solutionto the fixed message size model where it runs in $O(H^3W^2)$ rounds. Ourresults are immediately applicable if the faults are located in a single row,rather than a column.

Consensus on an Unknown Torus with Dense Byzantine Faults