  We prove that any variant of the GPT cryptosystem which uses a right column
scrambler over the extension field as advocated by the works of Gabidulin et
al. with the goal to resist to Overbeck's structural attack are actually still
vulnerable to that attack. We show that by applying the Frobenius operator
appropriately on the public key, it is possible to build a Gabidulin code
having the same dimension as the original secret Gabidulin code but with a
lower length. In particular, the code obtained by this way correct less errors
than the secret one but its error correction capabilities are beyond the number
of errors added by a sender, and consequently an attacker is able to decrypt
any ciphertext with this degraded Gabidulin code. We also considered the case
where an isometric transformation is applied in conjunction with a right column
scrambler which has its entries in the extension field. We proved that this
protection is useless both in terms of performance and security. Consequently,
our results show that all the existing techniques aiming to hide the inherent
algebraic structure of Gabidulin codes have failed.
