# Underlying codes for Niederreiter cryptosystems

Niederreiter cryptosystem is usually described by a parity check matrix $$H$$ over $$\mathbb{F}_{2^n}$$.

The minimum distance $$d$$ is given by

$$d := min\lbrace k \text{ such that there are k linearly dependant columns in H}\rbrace$$

Encryption in this case is done by encoding messages $$m$$ of weight less than $$d/2$$.

Now keeping Niederreiter systems in mind, my question is, since $$m$$ is over $$\mathbb{F}_2$$, it makes sense to consider linearly dependence over $$\mathbb{F}_2$$ rather than $$\mathbb{F}_{2^n}$$. In other words, is it okay to consider code as a code over $$F_{2}$$ with alphabet extension or do we have to consider it as codes over $$\mathbb{F}_{2^n}$$.

• In recent years there have been many attempts with quasi cyclic codes particularly LDPC and MDPC(although as you pointed out most of them turned out to be vulnerable) so there is nothing principally wrong or demotivating for someone to look into them. Anyway, my question is not about security but rather about understanding algebraic structure. Would the decoding algorithm work if I use codes with either of the above structures namely codes over $\mathbb{F}_{2^l}$ or over $\mathbb{F}_2$ extension symbols. – Root Jul 20 at 2:46