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We propose protocols for obliviously evaluating finite-state machines, i.e.,the evaluation is shared between the provider of the finite-state machine andthe provider of the input string in such a manner that neither party learns theother's input, and the states being visited are hidden from both. For alphabetsize $|\Sigma|$, number of states $|Q|$, and input length $n$, previoussolutions have either required a number of rounds linear in $n$ orcommunication $\Omega(n|\Sigma||Q|\log|Q|)$. Our solutions require 2 roundswith communication $O(n(|\Sigma|+|Q|\log|Q|))$. We present two differentsolutions to this problem, a two-party one and a setting with an untrusted butnon-colluding helper.

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