On the McEliece public-key cryptosystem based on Reed-Muller binary codes

We study the McEliece cryptosystem with $u$-fold use of binary Reed–Muller codes $\mathit{RM}(r,m)$. This modification of the McEliece cryptosystem was proposed by V. M. Sidelnikov in 1994 and combines high cryptographic security, transmission rate close to one, and moderate complexity of both enciphering and deciphering. For arbitrary values of the parameters $u$, $r$, and $m$ we give an upper bound for the cardinality of the set of public keys of this cryptosystem and calculate its exact value in the case of $u=2$ and $r=1$.