Dynamical Privacy in Distributed Computing -- Part II: PPSC Gossip Algorithms

In the first part of the paper, we have studied the computational privacy risks in distributed computing protocols against local or global dynamics eavesdroppers, and proposed a Privacy-Preserving-Summation-Consistent (PPSC) mechanism as a generic privacy encryption subroutine for consensus-based distributed computations. In this part of this paper, we show that the conventional deterministic and random gossip algorithms can be used to realize the PPSC mechanism over a given network. At each time step, a node is selected to interact with one of its neighbors via deterministic or random gossiping. Such node generates a random number as its new state, and sends the subtraction between its current state and that random number to the neighbor; then the neighbor updates its state by adding the received value to its current state. We establish concrete privacy-preservation conditions by proving the impossibility for the reconstruction of the network input from the output of the gossip-based PPSC mechanism against eavesdroppers with full network knowledge, and by showing that the PPSC mechanism can achieve differential privacy at arbitrary privacy levels. The convergence is characterized explicitly and analytically for both deterministic and randomized gossiping, which is essentially achieved in a finite number of steps. Additionally, we illustrate that the proposed algorithms can be easily made adaptive in real-world applications by making realtime trade-offs between resilience against node dropout or communication failure and privacy preservation capabilities.