Codes in Dihedral Group Algebra
Robert McEliece developed an asymmetric encryption algorithm based on the use of binary Goppa codes in 1978 and no effective key attacks has been described yet. Variants of this cryptosystem are known due to the use of different codes types, but most of them were proven to be less secure. Code crypt...
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Yaroslavl State University
2018-04-01
|
| Series: | Моделирование и анализ информационных систем |
| Subjects: | |
| Online Access: | https://www.mais-journal.ru/jour/article/view/682 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1839573285853986816 |
|---|---|
| author | Kirill V. Vedenev Vladimir M. Deundyak |
| author_facet | Kirill V. Vedenev Vladimir M. Deundyak |
| author_sort | Kirill V. Vedenev |
| collection | DOAJ |
| description | Robert McEliece developed an asymmetric encryption algorithm based on the use of binary Goppa codes in 1978 and no effective key attacks has been described yet. Variants of this cryptosystem are known due to the use of different codes types, but most of them were proven to be less secure. Code cryptosystems are considered an alternate to number-theoretical ones in connection with the development of quantum computing. So, the new classes of error-correcting codes are required for building new resistant code cryptosystems. Non-commutative codes, which simply are ideals of finite non-commutative group algebras, are an option. The Artin–Wedderburn theorem implies that a group algebra is isomorphic to a finite direct sum of matrix algebras, when the order of the group and the field characteristics are relatively prime. This theorem is important to study the structure of a non-commutative code, but it gives no information about summands and the isomorphism. In case of a dihedral group these summands and the isomorphism were found by F. E. Brochero Martinez. The purpose of the paper is to study codes in dihedral group algebras as and when the order of a group and a field characteristics are relatively prime. Using the result of F. E. Brochero Martinez, we consider a structure of all dihedral codes in this case and the codes induced by cyclic subgroup codes. |
| format | Article |
| id | doaj-art-db111ba2f2d9425a9fc1d45c8ae5c08f |
| institution | Matheson Library |
| issn | 1818-1015 2313-5417 |
| language | English |
| publishDate | 2018-04-01 |
| publisher | Yaroslavl State University |
| record_format | Article |
| series | Моделирование и анализ информационных систем |
| spelling | doaj-art-db111ba2f2d9425a9fc1d45c8ae5c08f2025-08-04T14:06:38ZengYaroslavl State UniversityМоделирование и анализ информационных систем1818-10152313-54172018-04-0125223224510.18255/1818-1015-2018-2-232-245502Codes in Dihedral Group AlgebraKirill V. Vedenev0Vladimir M. Deundyak1Southern Federal UniversitySouthern Federal University FGNU NII "Specvuzavtomatika"Robert McEliece developed an asymmetric encryption algorithm based on the use of binary Goppa codes in 1978 and no effective key attacks has been described yet. Variants of this cryptosystem are known due to the use of different codes types, but most of them were proven to be less secure. Code cryptosystems are considered an alternate to number-theoretical ones in connection with the development of quantum computing. So, the new classes of error-correcting codes are required for building new resistant code cryptosystems. Non-commutative codes, which simply are ideals of finite non-commutative group algebras, are an option. The Artin–Wedderburn theorem implies that a group algebra is isomorphic to a finite direct sum of matrix algebras, when the order of the group and the field characteristics are relatively prime. This theorem is important to study the structure of a non-commutative code, but it gives no information about summands and the isomorphism. In case of a dihedral group these summands and the isomorphism were found by F. E. Brochero Martinez. The purpose of the paper is to study codes in dihedral group algebras as and when the order of a group and a field characteristics are relatively prime. Using the result of F. E. Brochero Martinez, we consider a structure of all dihedral codes in this case and the codes induced by cyclic subgroup codes.https://www.mais-journal.ru/jour/article/view/682non-commutative groupsgroup algebranon-commutative codescode cryptosystems |
| spellingShingle | Kirill V. Vedenev Vladimir M. Deundyak Codes in Dihedral Group Algebra Моделирование и анализ информационных систем non-commutative groups group algebra non-commutative codes code cryptosystems |
| title | Codes in Dihedral Group Algebra |
| title_full | Codes in Dihedral Group Algebra |
| title_fullStr | Codes in Dihedral Group Algebra |
| title_full_unstemmed | Codes in Dihedral Group Algebra |
| title_short | Codes in Dihedral Group Algebra |
| title_sort | codes in dihedral group algebra |
| topic | non-commutative groups group algebra non-commutative codes code cryptosystems |
| url | https://www.mais-journal.ru/jour/article/view/682 |
| work_keys_str_mv | AT kirillvvedenev codesindihedralgroupalgebra AT vladimirmdeundyak codesindihedralgroupalgebra |