In this paper, we give novel certificates for triangular equivalence and rank profiles. These certificates enable to verify the row or column rank profiles or the whole rank profile matrix faster than recomputing them, with a negligible overall overhead. We first provide quadratic time and space non-interactive certificates saving the logarithmic factors of previously known ones. Then we propose interactive certificates for the same problems whose Monte Carlo verification complexity requires a small constant number of matrixvector multiplications, a linear space, and a linear number of extra field operations. As an application we also give an interactive protocol, certifying the determinant of dense matrices, faster than the best previously known one.

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.

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