Perennial secure multi-party computation of universal Turing machine

Consider a user who needs to perform computation over an initially unbounded stream of input. The user would like to compute functions of the input that change dynamically due to external events. The focus of this work is outsourcing such computation to a set of agents. The outsourcing must meet several constraints. Any large enough subset of agents must correctly emulate the user's computation on the unbounded stream of input. Any small subset of agents must obtain as little information as possible on the user's data, including the computed functions and any initial input. This privacy assurance must be maintained in an information-theoretic sense. Finally, the set of agents is dynamic with agents joining and leaving the set and different sets of agents being merged, cloned or split. In this work, we show how to securely outsource such perennial computation. The user's required computation is modeled as programs for a universal Turing machine. The only information that the agents obtain on the user's secrets is an upper bound on the space complexity required to perform the computation. Each state transition of the user's Turing machine requires computation and communication that are linear in the Turing machine's space complexity and polynomial in the number of agents performing the computation for every round of computation. The communication and computational complexity for an agent joining or leaving the set of computing agents in a transition round are linear in the space complexity of the Turing machine and polynomial in the number of agents. Some of the tools we develop may be of independent interest. We construct a strongly oblivious Turing machine, in which the tape head moves only as a function of its current location. We also show how to securely share the description of a Turing machine among several agents and how to securely compute each Turing machine's transition in a constant number of communication rounds.