Distributed Algorithms for Boolean Equations over Networks

In this article, we study systems of Boolean equations over a network, where each node in the network possesses only one Boolean equation from the system. The Boolean equation assigned at any particular node is a private equation known to this node only, and the aim of this article is to develop distributed algorithms that allow all the nodes to obtain solutions to the network Boolean equations without exchanging their local Boolean equations. First, we observe that the Boolean equations can be locally lifted to a system of linear algebraic equations under a basis of Boolean vectors, which is distributedly solvable using existing distributed linear equation algorithms as a subroutine. Next, we construct a distributed Boolean equation solver by the nodes solving the lifted linear network equation for a number of randomly selected initial values, and then converting the algebraic solutions into solutions to the original Boolean equations by a novel Boolean vector search algorithm. We prove that for solvable Boolean equations, when the initial values of the nodes for the distributed linear equation solving step are independent identically distributed selected according to a uniform distribution in a high-dimensional cube, such an algorithm returns the exact solution set of the Boolean equations at each node with high probability. We also present an algorithm for distributed verification of the satisfiability of Boolean equations when the solvability is not known beforehand, and prove its correctness. Finally, we show that by utilizing linear equation solvers with differential privacy to replace the in-network computing routines, the distributed Boolean equation algorithms can be made differentially private