Multidimensional agreement in Byzantine systems

Abstract

Consider a network of $n$ processes, where each process inputs a $d$-dimensional vector of reals. All processes can communicate directly with others via reliable FIFO channels. We discuss two problems. The multidimensional Byzantine consensus problem, for synchronous systems, requires processes to decide on a single $d$-dimensional vector $v \in {\mathbb {R}}^d$, inside the convex hull of $d$-dimensional vectors that were input by the non-faulty processes. Also, the multidimensional Byzantine approximate agreement (MBAA) problem, for asynchronous systems, requires processes to decide on multiple $d$-dimensional vectors in ${\mathbb {R}}^d$, all within a fixed Euclidean distance $\epsilon$ of each other, and inside the convex hull of $d$-dimensional vectors that were input by the non-faulty processes. We obtain the following results for the problems above, while tolerating up to $f$ Byzantine failures in systems with complete communication graphs: (1) In synchronous systems, $n > \max \{3f, (d+1)f\}$ is necessary and sufficient to solve the multidimensional consensus problem. (2) In asynchronous systems, $n > (d+2)f$ is necessary and sufficient to solve the multidimensional approximate agreement problem. Our sufficiency proofs are constructive, giving explicit protocols for the problems. In particular, for the MBAA problem, we give two protocols with strictly different properties and applications.

Appendix: Tverberg’s Theorem

Theorem 8

(Tverberg’s Theorem [22, 26]) For any integer $f$ with $f \ge 1$ and any multiset $X$ containing at least $(d+1)f+1$ points in ${\mathbb {R}}^d$, there exists a partition $X_1,\ldots , X_{f+1}$ of $X$ into $f+1$ non-empty multisets such that $\cap _{1 \le x \le f+1} {\mathrm {Poly}}({X_x}) \ne \emptyset$.

The points in the above multiset $X$ are not necessarily distinct [22, 26], so the same point may occur multiple times in $X$. Any partition in Theorem 8 is called a Tverberg partition, and the points in $\cap _{1 \le x \le f+1} {\mathrm {Poly}}({X_x})$ in Theorem 8 are called Tverberg points. It can be shown that any Tverberg point of a particular multiset $X$ is necessarily within the safe area ${\mathrm {Safe}}_{f}({X})$ [27].

Algorithm 5 requires non-faulty processes to choose identically any point in $S$ as the output vector. The deterministic procedure in the algorithm could therefore return a Tverberg point. For arbitrary $d$, no algorithm to compute a Tverberg point of an arbitrary multiset is currently known with polynomial complexity [2, 18, 19]. However, in some restricted cases, efficient algorithms are known (e.g., [15]).