Secure Private and Adaptive Matrix Multiplication Beyond the Singleton Bound

We consider the problem of designing secure and private codes for distributed matrix-matrix multiplication. A master server owns two private matrices and hires worker nodes to help compute their product. The matrices should remain information-theoretically private from the workers. Some of the workers are malicious and return corrupted results to the master. We design a framework for security against malicious workers in private matrix-matrix multiplication. The main idea is a careful use of Freivalds’ algorithm to detect erroneous matrix multiplications. Our main goal is to apply this security framework to schemes with adaptive rates. Adaptive schemes divide the workers into clusters and thus provide flexibility in trading decoding complexity for efficiency. Our new scheme, SRPM3, provides a computationally efficient security check per cluster that detects the presence of one or more malicious workers with high probability. An additional per worker check is used to identify the malicious nodes. SRPM3 can tolerate the presence of an arbitrary number of malicious workers. We provide theoretical guarantees on the complexity of the security checks and simulation results on both, the missed detection rate as well as on the time needed for the integrity check.

I. Introduction

Motivated by distributed machine learning, we consider matrix-matrix multiplication, which is the core computation of many common machine learning algorithms. Due to the tremendous amount of data being collected and processed, computing matrix multiplications locally is becoming a bottleneck. Distributed computing emerged as a solution to alleviate this computation bottleneck. A master node possessing the input matrices partitions them into smaller matrices and sends the partitions to worker nodes. The workers compute the smaller matrix multiplications in parallel and return the results to the master. The master aggregates the results received from the workers to obtain the desired matrix product. The main challenges of distributed computing, which we focus on, are: stragglers, privacy and security.