Elimination-based certificates for triangular equivalence and rank profiles

TL;DR

This paper introduces efficient, verifiable certificates for triangular equivalence, rank profiles, and determinants, significantly reducing verification time and computational overhead compared to previous methods.

Contribution

It presents novel quadratic time non-interactive certificates and interactive protocols with minimal verification complexity for matrix properties.

Findings

Quadratic time, space non-interactive certificates for rank profiles.

Interactive certificates with constant verification complexity.

Faster protocols for certifying determinants and signatures.

Abstract

In this paper, we give novel certificates for triangular equivalence and rank profiles. These certificates enable somebody 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 , with a linear number of interactions. As an application we also give an interactive protocol, certifying the determinant or the signature of dense matrices, faster for the Prover than the best previously known one. Finally we give linear space and constant round certificates for the row or column rank profiles.