Editing McEliece cryptosystem (section)

== Proof of message decryption ==
Note that <math>{\hat c} = cP^{-1} = m{\hat G}P^{-1} + zP^{-1} = mSG + zP^{-1}</math>,
and that <math>P</math> is a permutation matrix, thus <math>zP^{-1}</math> has weight <math>t</math>.

The Goppa code <math>G</math> can correct up to <math>t</math> errors, and the word <math>mSG</math> is at distance at most <math>t</math> from <math>cP^{-1}</math>. Therefore, the correct code word <math>{\hat m} = mS</math> is obtained.

Multiplying with the inverse of '''<math>S</math>''' gives <math>m = {\hat m}S^{-1}= mSS^{-1}</math>, which is the plain text message.