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In this paper, we give novel certificates for triangular equivalence and rankprofiles. These certificates enable to verify the row or column rank profilesor the whole rank profile matrix faster than recomputing them, with anegligible overall overhead. We first provide quadratic time and spacenon-interactive certificates saving the logarithmic factors of previously knownones. Then we propose interactive certificates for the same problems whoseMonte Carlo verification complexity requires a small constant number ofmatrix-vector multiplications, a linear space, and a linear number of extrafield operations. As an application we also give an interactive protocol,certifying the determinant of dense matrices, faster than the best previouslyknown one.

Certificates for triangular equivalence and rank profiles