Stochastic Consensus-Testing in Relay Networks

Stochastic network codes for consensus testing (CT) via a relay are proposed, where each of two or more parties knows a message and can find out if all these messages are equal, e.g. as an integrity check in a decentralized storage system or the control of mobile autonomous robots. The proposed codes achieve the CT capacity for memoryless uplinks channels when common randomness (CR) is available and no local randomness is used. With only local randomness at the edge nodes, upper and lower bounds for the capacity are given. The lower bound is achieved by CR generation via decode-and-forward transmission, and then using a common-randomness (CR)-assisted code. The upper bound is imposed by the CT over the uplink, when this consists of independent parallel channels to the relay. A recent derandomization result for encoders shows that, unlike deterministic encoding and CR shared between both encoders, the use of local randomness prevents the relay from successfully testing consensus. Therefore, in the proposed coding scheme, the relay recodes only to transmit random seeds and message hashes generated with these seeds. This scheme relies on an underlying CT code based on almost-universal hashing, where hashing is done with random seeds.