Fountain Codes for Private Distributed Matrix-Matrix Multiplication

We consider the problem of designing codes with flexible rate (referred to as rateless codes) for private distributed matrix-matrix multiplication. A master server owns two private matrices A and B and wants to hire worker nodes to help compute the multiplication. The matrices should remain private from the workers, in an information-theoretic sense. This problem has been considered in the literature and codes with a predesigned threshold are constructed. More precisely, the master assigns tasks to the workers and waits for a predetermined number of workers to finish their assigned tasks. The size of the tasks assigned to the workers depends on the designed threshold. We are interested in settings where the size of the task must be small and independent of the designed threshold. We design a rateless private matrix-matrix multiplication scheme, called RPM3. Our scheme fixes the size of the tasks and allows the master to send multiple tasks to the workers. The master keeps receiving results until it can decode the multiplication; hence the flexibility of the rate. Two main applications require this property: i) leverage the possible heterogeneity in the system and assign more tasks to workers that are faster; and ii) assign tasks adaptively to account for a possibly time-varying system.