{"title":"On the structural security of a McEliece-type cryptosystem based on the sum of tensor products of binary Reed - Muller codes","authors":"Y. Kosolapov, E. A. Lelyuk","doi":"10.17223/20710410/57/2","DOIUrl":null,"url":null,"abstract":"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.","PeriodicalId":42607,"journal":{"name":"Prikladnaya Diskretnaya Matematika","volume":"1 1","pages":""},"PeriodicalIF":0.2000,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Prikladnaya Diskretnaya Matematika","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.17223/20710410/57/2","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
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.
期刊介绍：
The scientific journal Prikladnaya Diskretnaya Matematika has been issued since 2008. It was registered by Federal Control Service in the Sphere of Communications and Mass Media (Registration Witness PI № FS 77-33762 in October 16th, in 2008). Prikladnaya Diskretnaya Matematika has been selected for coverage in Clarivate Analytics products and services. It is indexed and abstracted in SCOPUS and WoS Core Collection (Emerging Sources Citation Index). The journal is a quarterly. All the papers to be published in it are obligatorily verified by one or two specialists. The publication in the journal is free of charge and may be in Russian or in English. The topics of the journal are the following: 1.theoretical foundations of applied discrete mathematics – algebraic structures, discrete functions, combinatorial analysis, number theory, mathematical logic, information theory, systems of equations over finite fields and rings; 2.mathematical methods in cryptography – synthesis of cryptosystems, methods for cryptanalysis, pseudorandom generators, appreciation of cryptosystem security, cryptographic protocols, mathematical methods in quantum cryptography; 3.mathematical methods in steganography – synthesis of steganosystems, methods for steganoanalysis, appreciation of steganosystem security; 4.mathematical foundations of computer security – mathematical models for computer system security, mathematical methods for the analysis of the computer system security, mathematical methods for the synthesis of protected computer systems;[...]
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