  We consider the problem of communication efficient secure distributed matrix
multiplication. The previous literature has focused on reducing the number of
servers as a proxy for minimizing communication costs. The intuition being,
that the more servers used, the higher the communication cost. We show that
this is not the case. Our central technique relies on adapting results from the
literature on repairing Reed-Solomon codes where instead of downloading the
whole of the computing task, a user downloads field traces of these
computations. We present field trace polynomial codes, a family of codes, that
explore this technique and characterize regimes for which our codes outperform
the existing codes in the literature.
