# Niederreiter cryptosystem

I can't understand how Niederreiter cryptosystem works. If $$c=mH^{'T}$$ than why we cannot compute $$m$$ directly by multiplying $$c$$ with the $$(H^{'T})^{-1}$$? Can you give me an example of a "fast decoding algorithm"?

Thank you!

• did you consider the noise removal by $D$? – kelalaka Feb 8 '19 at 11:43
• What noise? That's confuses me. In the McEliece cryptosystem we add some error $e$ but in all Niederreiter documentations I didn't see any error adding to the plaintext message – mip Feb 8 '19 at 11:46

In the Niederreiter system, the plaintext is mapped to some error vector of weight $$t$$, where the code correction capability is $$d=2t+1.$$
• Is $2t+1$ excessive? – kelalaka Feb 9 '19 at 15:33
• @kelalaka If you took $d=t+k+1$ you might be unlucky and the actual codeword might only be Hamming distance $k+1$ away which can be found with $O(2^{k+1})$ effort. – kodlu Feb 10 '19 at 20:46
• $H'$ is rectangular, so has no inverse. – kodlu Feb 10 '19 at 20:48