McEliece cryptosystem is an public-key cryptosystem; its security is based on the complexity of decoding problem for an arbitrary error-correcting code. V. M. Sidel’nikov in 1994 suggested to construct the cryptosystem on the base of binary Reed-Muller code. In 2007 L. Minder and A. Shokrollahi had designed a structural attack on theMcEliece cryptosystem based on the Reed-Muller codes.Herewe improve their attack and suggest a polynomial attack on the McEliece cryptosystem based on Reed-Muller codes RM(r, m) such that GCD(r,m − 1) = 1.

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