Notes on Communication and Computation in Secure Distributed Matrix Multiplication

We consider the problem of secure distributed matrix multiplication in which a user wishes to compute the product of two matrices with the assistance of honest but curious servers. In this paper, we answer the following question: Is it beneficial to offload the computations if security is a concern? We answer this question in the affirmative by showing that by adjusting the parameters in a polynomial code we can obtain a trade-off between the user's and the servers' computational time. Indeed, we show that if the computational time complexity of an operation in F
_q
is at most Z
_q
and the computational time complexity of multiplying two n × n matrices is O(n
^ω
Z
_q
) then, by optimizing the trade-off, the user together with the servers can compute the multiplication in O(n
⁴
-6/ω+1 Z
_q
) time. We also show that if the user is only concerned in optimizing the download rate, a common assumption in the literature, then the problem can be converted into a simple private information retrieval problem by means of a scheme we call Private Oracle Querying. However, this comes at large upload and computational costs for both the user and the servers.