  In this paper, we propose distributed algorithms that solve a system 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 {\em private} equation known to this node only, and the
nodes aim to compute the exact set of solutions to the system without
exchanging their local equations. We show that each private Boolean equation
can be locally lifted to a linear algebraic equation under a basis of Boolean
vectors, leading to a network linear equation that is distributedly solvable
using existing distributed linear equation algorithms as a subroutine. A number
of exact or approximate solutions to the induced linear equation are then
computed at each node from different initial values. The solutions to the
original Boolean equations are eventually computed locally via a Boolean vector
search algorithm. We prove that given solvable Boolean equations, when the
initial values of the nodes for the distributed linear equation solving step
are i.i.d selected according to a uniform distribution in a high-dimensional
cube, our algorithms return the exact solution set of the Boolean equations at
each node with high probability. Furthermore, we present an algorithm for
distributed verification of the satisfiability of Boolean equations, and prove
its correctness. Finally, we show that by utilizing linear equation solvers
with differential privacy to replace the in-network computing routines, the
overall distributed Boolean equation algorithms can be made differentially
private. Under the standard Laplace mechanism, we prove an explicit level of
noises that can be injected in the linear equation steps for ensuring a
prescribed level of differential privacy.
