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Combinatorial privacy: Packing splinters in polytopes at scale for private bit sums via SecureHull

We present a scheme to obtain counts of 0’s and 1’s at a server based on private bit streams hosted by multiple clients. The goal is to obtain this solution at the server while maintaining privacy of client data. The bit sums need to be obtained with respect to data from all clients; and not at a per client granularity. In our scheme called SecureHull, we hide the private data encoded as permutations amidst publicly shareable permutation matrices and form a secret doubly stochastic matrix via a convex combination with secret coefficients. We exploit the nonuniqueness of the Birkhoff-von Neumann decomposition and use some remnants of the splintering scheme to provide an unconventional secure computation method to this private bitsum problem. This scheme does not require any private datadependent communication with the server as is ideal. We also provide lower bounds to quantify the probability of a successful attack. We show that the lower bound can be quadratically reduced with a linear increase in communication upto a constant. Our solution also involves a cryptographic shuffling routine that scales linearly with number of clients as against to the size of the datasets. The rest of the operations do not require a cryptographic approach and are secured through our scheme thereby benefiting its scalability.

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