Decoding the Tensor Product of MLD Codes and Applications for Code Cryptosystems

• Main Authors: Vladimir Mikhailovich Deundyak, Yury Vladimirovich Kosolapov, Evgeniy Andreevich Leluk
• Format: Article
• Language: English
• Published: Yaroslavl State University 2017-04-01
• Series: Modelirovanie i Analiz Informacionnyh Sistem
• Subjects: majority decoder; reed–muller–berman codes; tensor product codes
• Online Access: https://www.mais-journal.ru/jour/article/view/512
• ISSN: 1818-1015; 2313-5417

For the practical application of code cryptosystems such as McEliece, it is necessary that the code used in the cryptosystem should have a fast decoding algorithm. On the other hand, the code used must be such that finding a secret key from a known public key would be impractical with a relatively small key size. In this connection, in the present paper it is proposed to use the tensor product \( C_1 \otimes C_2 \) of group \(\textrm{MLD}\) codes \( C_1 \) and \( C_2 \) in a McEliece-type cryptosystem. The algebraic structure of the code \( C_1 \otimes C_2 \) in the general case differs from the structure of the codes \( C_1 \) and \( C_2 \), so it is possible to build stable cryptosystems of the McEliece type even on the basis of codes \( C_i \) for which successful attacks on the key are known. However, in this way there is a problem of decoding the code \( C_1 \otimes C_2 \). The main result of this paper is the construction and justification of a set of fast algorithms needed for decoding this code. The process of constructing the decoder relies heavily on the group properties of the code \( C_1 \otimes C_2 \). As an application, the McEliece-type cryptosystem is constructed on the code \( C_1 \otimes C_2 \) and an estimate is given of its resistance to attack on the key under the assumption that for code cryptosystems on codes \( C_i \) an effective attack on the key is possible. The results obtained are numerically illustrated in the case when \( C_1 \), \( C_2 \) are Reed--Muller--Berman codes for which the corresponding code cryptosystem was hacked by L. Minder and A. Shokrollahi (2007).