# 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 $D$? 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 $$d=2t+1.$$

With the trapdoor information (permutation) this can be decoded by the legitimate receiver by syndrome decoding.

Without the trapdoor information, this is equivalent to decoding a random vector, which is hard, as in the McEliece cryptosystem.

• Is $2t+1$ excessive? Feb 9 '19 at 15:33
• It is always guaranteed that the matrix H' doesn't has an inverse?
– mip
Feb 10 '19 at 6:44
• @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. Feb 10 '19 at 20:46
• $H'$ is rectangular, so has no inverse. Feb 10 '19 at 20:48
• Can be uniquely and correctly found? Feb 10 '19 at 20:51