GPT is a very promising public key cryptosystem published at Eurocrypt 91. GPT stands for Gabidulin, Paramonov and Trietakov. It san be seen as an analogue of McEliece based on Gabidulin's maximum rank-distance error correcting codes.

Unlike McEliece, Gibson has shown at Eurocrypt 1996 that structural attacks on GPT that are better (in many cases) that the general decoding attacks. Other structural attacks have been proposed by Overbeck in 2005.
At WCC 2001 Gabiulin and Ourivski have shown how to construct modified versions of GPT. Modified versions seem to make Gibson's structural attacks work with a probability that can be made exponentially small. These modifications are analogous to the operations such as "-" that are usually done to increase the security of HFE cryptosystem. The public key is however bigger than in the orginal GPT, for example about 4 kbytes, but not as big as in some other multivariate cryptosystems, namely Quartz and McEliece.

The best decoding attacks are the Stern-Chabaud attack published at Asiacrypt 96, and a direct reduction to a known algorithm for the MinRank problem (to be published soon). All these decoding attacks are exponential.

It is very likely that some versions of GPT are very secure, but it is too early to say which. It is also still an open problem to know if decoding of rank-distance codes is an NP-complete problem.