Round and Communication Balanced Protocols for Oblivious Evaluation of Finite State Machines

We propose protocols for obliviously evaluating finite-state machines, i.e., the evaluation is shared between the provider of the finite-state machine and the provider of the input string in such a manner that neither party learns the other’s input, and the states being visited are hidden from both. For alphabet size $\mathrm{\mid }\mathrm{\Sigma }\mathrm{\mid }$, number of states $\mathrm{\mid }Q\mathrm{\mid }$, and input length $n$, previous solutions have either required a number of rounds linear in $n$ or communication $\mathrm{\Omega }(n\mathrm{\mid }\mathrm{\Sigma }\mathrm{\mid }\mathrm{\mid }Q\mathrm{\mid }\log \mathrm{\mid }Q\mathrm{\mid })$. Our solutions require 2 rounds with communication $O(n(\mathrm{\mid }\mathrm{\Sigma }\mathrm{\mid }+\mathrm{\mid }Q\mathrm{\mid }\log \mathrm{\mid }Q\mathrm{\mid }))$. We present two different solutions to this problem, a two-party one and a setting with an untrusted but non-colluding helper.