Certificates for Triangular Equivalence and Rank Profiles

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.

[1]  Lap Chi Lau,et al.  Fast matrix rank algorithms and applications , 2012, JACM.

[2]  Clément Pernet,et al.  Faster algorithms for the characteristic polynomial , 2007, ISSAC '07.

[3]  Claude-Pierre Jeannerod,et al.  Rank-profile revealing Gaussian elimination and the CUP matrix decomposition , 2011, J. Symb. Comput..

[4]  Jon Howell,et al.  Geppetto: Versatile Verifiable Computation , 2015, 2015 IEEE Symposium on Security and Privacy.

[5]  Erich Kaltofen,et al.  Linear Time Interactive Certificates for the Minimal Polynomial and the Determinant of a Sparse Matrix , 2016, ISSAC.

[6]  C. L. Dodgson,et al.  IV. Condensation of determinants, being a new and brief method for computing their arithmetical values , 1867, Proceedings of the Royal Society of London.

[7]  Erich Kaltofen,et al.  Essentially optimal interactive certificates in linear algebra , 2014, ISSAC.

[8]  Rusins Freivalds,et al.  Fast Probabilistic Algorithms , 1979, MFCS.

[9]  Amos Fiat,et al.  How to Prove Yourself: Practical Solutions to Identification and Signature Problems , 1986, CRYPTO.

[10]  Yael Tauman Kalai,et al.  Delegating computation: interactive proofs for muggles , 2008, STOC.

[11]  Jean-Guillaume Dumas,et al.  Computing the Rank Profile Matrix , 2015, ISSAC.

[12]  Jean-Guillaume Dumas,et al.  Fast computation of the rank profile matrix and the generalized Bruhat decomposition , 2016, J. Symb. Comput..

[13]  Arne Storjohann,et al.  A Relaxed Algorithm for Online Matrix Inversion , 2015, ISSAC.

[14]  Jean-Guillaume Dumas,et al.  Simultaneous computation of the row and column rank profiles , 2013, ISSAC '13.

[15]  Erich Kaltofen,et al.  Quadratic-time certificates in linear algebra , 2011, ISSAC '11.