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In this paper, we propose distributed algorithms that solve a system ofBoolean equations over a network, where each node in the network possesses onlyone Boolean equation from the system. The Boolean equation assigned at anyparticular node is a {\em private} equation known to this node only, and thenodes aim to compute the exact set of solutions to the system withoutexchanging their local equations. We show that each private Boolean equationcan be locally lifted to a linear algebraic equation under a basis of Booleanvectors, leading to a network linear equation that is distributedly solvableusing existing distributed linear equation algorithms as a subroutine. A numberof exact or approximate solutions to the induced linear equation are thencomputed at each node from different initial values. The solutions to theoriginal Boolean equations are eventually computed locally via a Boolean vectorsearch algorithm. We prove that given solvable Boolean equations, when theinitial values of the nodes for the distributed linear equation solving stepare i.i.d selected according to a uniform distribution in a high-dimensionalcube, our algorithms return the exact solution set of the Boolean equations ateach node with high probability. Furthermore, we present an algorithm fordistributed verification of the satisfiability of Boolean equations, and proveits correctness. Finally, we show that by utilizing linear equation solverswith differential privacy to replace the in-network computing routines, theoverall distributed Boolean equation algorithms can be made differentiallyprivate. Under the standard Laplace mechanism, we prove an explicit level ofnoises that can be injected in the linear equation steps for ensuring aprescribed level of differential privacy.

Distributed Algorithms that Solve Boolean Equations with Local and Differential Privacies