This paper presents a secure multiparty computation protocol for the Strassen-Winograd matrix multiplication algorithm. We focus on the setting in which any given player knows only one row (or one block of rows) of both input matrices and learns the corresponding row (or block of rows) of the resulting product matrix. Neither the player initial data, nor the intermediate values, even during the recurrence part of the algorithm, are ever revealed to other players. We use a combination of partial homomorphic encryption schemes and additive masking techniques together with a novel schedule for the location and encryption layout of all intermediate computations to preserve privacy. Compared to state of the art protocols, the asymptotic communication volume of our construction is reduced from O(n^3) to O(n^{2.81}). This improvement in terms of communication volume arises with matrices of dimension as small as n=96 which is confirmed by experiments.