# Certificates for triangular equivalence and rank profiles

## TL;DR

This paper introduces new certificates for verifying triangular equivalence and rank profiles that are faster and more efficient than previous methods, with applications to determinant certification.

## Contribution

It presents novel quadratic-time, non-interactive certificates and efficient interactive certificates for rank profiles and matrix determinants.

## Key findings

- Quadratic time and space certificates for rank verification
- Interactive certificates with minimal matrix-vector multiplications
- Faster determinant certification protocol

## Abstract

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 matrix-vector 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.