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.