A geometrical approach for consensus security

Guaranteeing security against eavesdroppers while allowing agents to reach an agreement on some shared variables is an essential feature to foster the adoption of distributed protocols. In the literature, a wide variety of methodologies have been developed in the discrete-time case, and most of them rely on encryption and undirected graphs, often exhibiting quantization or accuracy issues. In this paper, we develop a continuous-time secure consensus algorithm for agents that communicate via a broadcast-only protocol over directed and strongly connected graphs. In detail, the legitimate agents interact by exchanging an extended state, which can be converted into the hidden consensus state via a pre-deployed shared vector, thus preventing eavesdropping. In this paper, we take a geometrical perspective, and we implement our encryption via coordinate shifts. This choice guarantees, on one side, the existence of an uncountable infinity of possible vectors and, on the other side, provides a control-theoretical viewpoint on encryption that naturally blends with a dynamical system. We complement our framework by providing an additional layer of protection against honest but curious legitimate nodes, preventing them from gaining insights into the initial condition of their peers.