Secure Computation, I/O-Efficient Algorithms and Distributed Signatures

We consider a setting where a set of n players use a set of m servers to store a large, private data set. Later the players decide on functions they want to compute on the data without the servers needing to know which computation is done, while the computation should be secure against a malicious adversary corrupting a constant fraction of the players and servers. Using packed secret sharing, the data can be stored in a compact way but will only be accessible in a block-wise fashion. We explore the possibility of using I/O-efficient algorithms to nevertheless compute on the data as efficiently as if random access was possible. We show that for sorting, priority queues and data mining, this can indeed be done. We show actively secure protocols of complexity within a constant factor of the passively secure solution. As a technical contribution towards this goal, we develop techniques for generating values of form r, gr for random secret-shared r∈ℤq and gr in a group of order q. This costs a constant number of exponentiation per player per value generated, even if less than n/3 players are malicious. This can be used for efficient distributed computing of Schnorr signatures. We further develop the technique so we can sign secret data in a distributed fashion at essentially the same cost.