On the equivalence of different variants of the GPT cryptosystem

We consider different variants of the modified GPT cryptosystem. The main approach for increasing the security of GPT is randomizing the public key. Randomization is done through addition or concatenation of a distortion matrix to the generator matrix of the rank code or to the column scrambler.We show that all techniques are somewhat equivalent. In particular, public keys of different variants can be transformed to one another in some superextension field. This allows applying known attacks to the variants of GPT and analyzing their security.