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It is well-known that keys (generator matrices) of code-based cryptography are very large for its practical usage. To reduce them scientists propose to use families of quasi-cyclic codes. I remind that a linear code $C \subset \mathbb{F}_q^n$ is called $\textit{quasi-cyclic}$ of index $m \mid n$ if for any codeword $c = (c_1, c_2, \cdots\!, c_n) \in C$ the shift by $m$ positions gives another codeword, that is, $(c_{m+1}, c_{m+2}, \cdots\!, c_{m}) \in C$. It is clear that for cyclic codes (when $m=1$) we obtain the most compact representation of the key. However cyclic codes seem to be attacked, i.e., insecure.

Help, please. What is the minimal value $m$ now considered to be secure?

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