Structural attack on McEliece-Sidelnikov type public-key cryptosystem based on a combination of random codes with Reed-Muller codes

Ivan Chizhov, Elizaveta Popova


This paper represents the investigation of McEliece-Sidelnikov cryptosystem, based on combination of random codes with Reed-Muller codes. Different modifications of classical McEliece cryptosystem has been studied. Sidelnikov’s work introduced using the several samples of Reed-Muller code, and Kabatiansky and Tavernier’s work proposed to use the concatenation of Goppa and Reed-Muller codes. The popularity of this cryptosystem explains with the fact that it’s strength is based on the hardness of the decoding general linear code problem, so it will remain unbreakable in postquantum era. This paper investigates one of the modifications of McElice cryptosystem in a model when the attacker knows the public-key matrix and the generator matrix of random linear code. The goal is to reconstruct the permutation matrix from the secret-key. During the investigation of the hull of the code built by combining random codes with Reed-Muller codes the theorem about the number of such codes with fixed size of the hull has been proved. An attack based on signature method has been produced and programmed with the use of C++ programming language. All the results of this program’s work are represented in this paper. With the use of proved theorem the hardness of provided attack has been calculated.

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