  We consider the problem of designing rateless coded private distributed
matrix-matrix multiplication. A master server owns two private matrices
$\mathbf{A}$ and $\mathbf{B}$ 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 multiplications 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. 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.
