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(k 2 ). We show that oblivious Gaussian elimination is a powerful primitive. In particular, it allows us to securely decide the intersection of linear and affine subspaces, pick a random vector from the intersection, and obliviously solve 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.
Nissim and Weinreb REF also considered the problem of securely solving a set of linear equations in the computational two-party model, focusing on low (nearly optimal) communication complexity.
Communication efficient secure linear algebra
In the Third Theory of Cryptography Conference – TCC 2006
Computer Science