In this talk, I will present the recent improvements in
algebraic techniques for solving the MinRank problem, which is
ubiquitous in multivariate and rank metric code based cryptography.
Algebraic attacks now outperform the combinatorial ones that were
considered state of the art up until now. In the particular case of
Fqm-linear codes in rank metric, for solving the Rank Decoding problem,
the attack is even more efficient, and completely break the parameters
of various schemes submitted to the NIST-PQC standardisation process for
quantum-resistant public key cryptography.