We provide a new way of performing an algebraic attack on the McEliece cryptosystem based on binary Goppa codes. It also applies in general to the case where the field over which the Goppa code is defined is of even characteristic. It is based on a new algebraic modeling for finding as in [CMT23,M25,BLT26] matrices of rank 22 in the code of quadratic relations related to the Goppa code that is attacked. Such matrices are then used to recover the secret algebraic structure of the code. This brea