Abelian sharing, common informations, and linear rank inequalities
Dougherty et al. introduced the common information (CI) method as a method to produce non-Shannon inequalities satisfied by linear random variables, which are called linear rank inequalities. This method is based on the fact that linear random variables have CI. Dougerthy et al. asked whether this m...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
De Gruyter
2022-08-01
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| Series: | Journal of Mathematical Cryptology |
| Subjects: | |
| Online Access: | https://doi.org/10.1515/jmc-2022-0020 |
| _version_ | 1811280855613571072 |
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| author | Mejia Carolina Montoya Juan Andres |
| author_facet | Mejia Carolina Montoya Juan Andres |
| author_sort | Mejia Carolina |
| collection | DOAJ |
| description | Dougherty et al. introduced the common information (CI) method as a method to produce non-Shannon inequalities satisfied by linear random variables, which are called linear rank inequalities. This method is based on the fact that linear random variables have CI. Dougerthy et al. asked whether this method is complete, in the sense that it can be used to produce all linear rank inequalities. We study this question, and we attack it using the theory of secret sharing schemes. To this end, we introduce the notions of Abelian secret sharing scheme and Abelian capacity. We prove that: If there exists an access structure whose Abelian capacity is smaller than its linear capacity, then the CI method is not complete. We investigate the existence of such an access structure. |
| first_indexed | 2024-04-13T01:22:21Z |
| format | Article |
| id | doaj.art-83d4e33becc44c3bab0741e58aaaa907 |
| institution | Directory Open Access Journal |
| issn | 1862-2984 |
| language | English |
| last_indexed | 2024-04-13T01:22:21Z |
| publishDate | 2022-08-01 |
| publisher | De Gruyter |
| record_format | Article |
| series | Journal of Mathematical Cryptology |
| spelling | doaj.art-83d4e33becc44c3bab0741e58aaaa9072022-12-22T03:08:44ZengDe GruyterJournal of Mathematical Cryptology1862-29842022-08-0116123325010.1515/jmc-2022-0020Abelian sharing, common informations, and linear rank inequalitiesMejia Carolina0Montoya Juan Andres1Mathematics Department, Universidad Nacional de Colombia, Bogotá, ColombiaMathematics Department, Universidad Nacional de Colombia, Bogotá, ColombiaDougherty et al. introduced the common information (CI) method as a method to produce non-Shannon inequalities satisfied by linear random variables, which are called linear rank inequalities. This method is based on the fact that linear random variables have CI. Dougerthy et al. asked whether this method is complete, in the sense that it can be used to produce all linear rank inequalities. We study this question, and we attack it using the theory of secret sharing schemes. To this end, we introduce the notions of Abelian secret sharing scheme and Abelian capacity. We prove that: If there exists an access structure whose Abelian capacity is smaller than its linear capacity, then the CI method is not complete. We investigate the existence of such an access structure.https://doi.org/10.1515/jmc-2022-0020secret sharingpolymatroidslinear rank inequalities94d99 |
| spellingShingle | Mejia Carolina Montoya Juan Andres Abelian sharing, common informations, and linear rank inequalities Journal of Mathematical Cryptology secret sharing polymatroids linear rank inequalities 94d99 |
| title | Abelian sharing, common informations, and linear rank inequalities |
| title_full | Abelian sharing, common informations, and linear rank inequalities |
| title_fullStr | Abelian sharing, common informations, and linear rank inequalities |
| title_full_unstemmed | Abelian sharing, common informations, and linear rank inequalities |
| title_short | Abelian sharing, common informations, and linear rank inequalities |
| title_sort | abelian sharing common informations and linear rank inequalities |
| topic | secret sharing polymatroids linear rank inequalities 94d99 |
| url | https://doi.org/10.1515/jmc-2022-0020 |
| work_keys_str_mv | AT mejiacarolina abeliansharingcommoninformationsandlinearrankinequalities AT montoyajuanandres abeliansharingcommoninformationsandlinearrankinequalities |