Communication efficient secure linear algebra

We present communication efficient secure protocols for a variety of linear algebra problems. Our main building block is a protocol for computing Gaussian Elimination on encrypted data. As input for this protocol, Bob holds a k × k matrix M, encrypted with Alice's key. At the end of the protocol run, Bob holds an encryption of an upper-triangular matrix M′ such that the number of nonzero elements on the diagonal equals the rank of M. The communication complexity of our protocol is roughly O(k²). Building on Oblivious Gaussian elimination, we present secure protocols for several problems: deciding the intersection of linear and affine subspaces, picking a random vector from the intersection, and obliviously solving a set of linear equations. Our protocols match known (insecure) communication complexity lower bounds, and improve the communication complexity of both Yao's garbled circuits and that of specific previously published protocols.