Secure Private and Adaptive Matrix Multiplication Beyond the Singleton Bound.

Consider the problem of designing secure and private codes for distributed matrix-matrix multiplication. A master server owns two private matrices $\mA$ and $\mB$ and hires worker nodes to help computing their multiplication. The matrices should remain information-theoretically private from the workers. Some of the workers are malicious and return corrupted results to the master. This work is motivated by the literature on myopic adversaries in network coding and distributed storage. Security beyond the Singleton bound is possible when the adversary has limited knowledge about the master's data and probabilistic decoding is acceptable. The key observation in this setting is that the master is the sender and the receiver. Therefore, the master enjoys a plethora of advantages that enable coding for security beyond the Singleton bound. We design a framework for security against malicious adversaries in private matrix-matrix multiplication. Our main goal is to apply this security framework to schemes with adaptive rates previously introduced by a subset of the authors. Adaptive schemes divide the workers into clusters and thus provide flexibility in trading decoding complexity for efficiency. Checking the integrity of the computation per cluster has low complexity but costs deleting the results of a whole cluster with at least one malicious worker. Checking the integrity of the results per worker is more complex but allows an efficient use of the non-malicious workers. Our scheme, called SRPM3, provides a computationally efficient security check that detects malicious workers with high probability and can tolerate the presence of an arbitrary number of malicious workers. We provide simulation results that validate our theoretical findings.