On the Berger–Loidreau cryptosystem on the tensor product of codes

In the post-quantum era, asymmetric cryptosystems based on linear codes (code cryptosystems) are considered as an alternative to modern asymmetric cryptosystems. However, the research of the strength of code McEliece-type cryptosystems shows that algebraically structured codes do not provide sufficient strength of these cryptosystems. On the other hand, the use of random codes in such cryptosystems is impossible because of the high complexity of its decoding. Strengthening of code cryptosystems is currently conducted, usually, either by using codes for which no attacks are known, or by modifying the cryptographic protocol. In this paper both of these approaches are used. On the one hand, it is proposed to use the tensor product $C^1\otimes C^2$ of the known codes $C^1$ and $C^2$, since for $C^1\otimes C^2$ in some cases it is possible to construct an effective decoding algorithm. On the other hand, instead of a McEliece-type cryptosystem, it is proposed to use its modification, a Berger–Loidreau cryptosystem. The paper proves a high strength of the constructed code cryptosystem to attacks on the key even in the case when code cryptosystems on codes $C^1$ and $C^2$ are cracked.