On the structural security of a McEliece-type cryptosystem based on the sum of tensor products of binary Reed — Muller codes

The current task of cryptography is the development of cryptosystems resistant to attacks using quantum computing. One of the promising encryption schemes is the McEliece system based on Goppa codes. However, this system has a number of disadvantages due to the structure of Goppa codes, which makes it relevant to search for other codes for the McEliece scheme. Important requirements for these codes are the presence of a fast decoder and ensuring the resistance of the corresponding cryptosystem to known attacks, including attacks with the Schur — Hadamard product. Many attempts to replace Goppa codes have failed because the corresponding cryptosystems have proven to be unstable against structural attacks. In this paper, it is proposed to use the $D$-construction ($D$-code) on binary Reed — Muller codes in the McEliece cryptosystem. This construction is a sum of a special kind of tensor products of binary Reed — Muller codes. There is a fast decoding algorithm for it. To analyze the security of the McEliece scheme on $D$-codes, we have constructed a structural attack that uses the Schur — Hadamard product of a $D$-code. To select the parameters that ensure the resistance of the cryptosystem to the constructed attack, we investigate the decomposition of the degree of the $D$-code into the direct sum of Reed — Muller codes and conclude about the set of strong keys of the cryptosystem.