Breaking Goppa-based McEliece with hints

We consider the McEliece cryptosystem with a binary Goppa code C⊂F2n specified by an irreducible Goppa polynomial g(x)∈F2m[X] and Goppa points (α1,…,αn)∈F2mn. Since g(x) together with the αi's allow for efficient decoding, these parameters form McEliece secret keys. Such a Goppa code C is an (n−tm)-dimensional subspace of F2n, and therefore C has co-dimension tm. For typical McEliece instantiations we have tm≈n4.