On the flag geometry of simple group of Lie type and multivariate cryptography
We propose some multivariate cryptosystems based on finite \(BN\)-pair \(G\) defined over the fields \(F_q\). We convert the adjacency graph for maximal flags of the geometry of group \(G\) into a finite Tits automaton by special colouring of arrows and treat the largest Schubert cell \({\rm Sch}\)...
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| Дата: | 2018 |
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| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Lugansk National Taras Shevchenko University
2018
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| Онлайн доступ: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1181 |
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| Назва журналу: | Algebra and Discrete Mathematics |
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oai:ojs.admjournal.luguniv.edu.ua:article-11812018-05-17T07:50:53Z On the flag geometry of simple group of Lie type and multivariate cryptography Ustimenko, Vasyl Multivariate Cryptography, flag variety, Geometry of Simple Group of Lie type, Schubert cell, symbolic walks We propose some multivariate cryptosystems based on finite \(BN\)-pair \(G\) defined over the fields \(F_q\). We convert the adjacency graph for maximal flags of the geometry of group \(G\) into a finite Tits automaton by special colouring of arrows and treat the largest Schubert cell \({\rm Sch}\) isomorphic to vector space over \(F_q\) on this variety as a totality of possible initial states and a totality of accepting states at a time. The computation (encryption map) corresponds to some walk in the graph with the starting and ending points in \({\rm Sch}\). To make algorithms fast we will use the embedding of geometry for \(G\) into Borel subalgebra of corresponding Lie algebra.We also consider the notion of symbolic Tits automata. The symbolic initial state is a string of variables \(t_{\alpha}\in F_q\), where roots \(\alpha\) are listed according Bruhat's order, choice of label will be governed by special multivariate expressions in variables \(t_{\alpha}\), where \(\alpha\) is a simple root.Deformations of such nonlinear map by two special elements of affine group acting on the plainspace can produce a computable in polynomial time nonlinear transformation. The information on adjacency graph, list of multivariate governing functions will define invertible decomposition of encryption multivariate function. It forms a private key which allows the owner of a public key to decrypt a ciphertext formed by a public user. We also estimate a polynomial time needed for the generation of a public rule. Lugansk National Taras Shevchenko University 2018-05-17 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1181 Algebra and Discrete Mathematics; Vol 19, No 1 (2015) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1181/670 Copyright (c) 2018 Algebra and Discrete Mathematics |
| institution |
Algebra and Discrete Mathematics |
| collection |
OJS |
| language |
English |
| topic |
Multivariate Cryptography flag variety Geometry of Simple Group of Lie type Schubert cell symbolic walks |
| spellingShingle |
Multivariate Cryptography flag variety Geometry of Simple Group of Lie type Schubert cell symbolic walks Ustimenko, Vasyl On the flag geometry of simple group of Lie type and multivariate cryptography |
| topic_facet |
Multivariate Cryptography flag variety Geometry of Simple Group of Lie type Schubert cell symbolic walks |
| format |
Article |
| author |
Ustimenko, Vasyl |
| author_facet |
Ustimenko, Vasyl |
| author_sort |
Ustimenko, Vasyl |
| title |
On the flag geometry of simple group of Lie type and multivariate cryptography |
| title_short |
On the flag geometry of simple group of Lie type and multivariate cryptography |
| title_full |
On the flag geometry of simple group of Lie type and multivariate cryptography |
| title_fullStr |
On the flag geometry of simple group of Lie type and multivariate cryptography |
| title_full_unstemmed |
On the flag geometry of simple group of Lie type and multivariate cryptography |
| title_sort |
on the flag geometry of simple group of lie type and multivariate cryptography |
| description |
We propose some multivariate cryptosystems based on finite \(BN\)-pair \(G\) defined over the fields \(F_q\). We convert the adjacency graph for maximal flags of the geometry of group \(G\) into a finite Tits automaton by special colouring of arrows and treat the largest Schubert cell \({\rm Sch}\) isomorphic to vector space over \(F_q\) on this variety as a totality of possible initial states and a totality of accepting states at a time. The computation (encryption map) corresponds to some walk in the graph with the starting and ending points in \({\rm Sch}\). To make algorithms fast we will use the embedding of geometry for \(G\) into Borel subalgebra of corresponding Lie algebra.We also consider the notion of symbolic Tits automata. The symbolic initial state is a string of variables \(t_{\alpha}\in F_q\), where roots \(\alpha\) are listed according Bruhat's order, choice of label will be governed by special multivariate expressions in variables \(t_{\alpha}\), where \(\alpha\) is a simple root.Deformations of such nonlinear map by two special elements of affine group acting on the plainspace can produce a computable in polynomial time nonlinear transformation. The information on adjacency graph, list of multivariate governing functions will define invertible decomposition of encryption multivariate function. It forms a private key which allows the owner of a public key to decrypt a ciphertext formed by a public user. We also estimate a polynomial time needed for the generation of a public rule. |
| publisher |
Lugansk National Taras Shevchenko University |
| publishDate |
2018 |
| url |
https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1181 |
| work_keys_str_mv |
AT ustimenkovasyl ontheflaggeometryofsimplegroupoflietypeandmultivariatecryptography |
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2024-04-12T06:25:09Z |
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2024-04-12T06:25:09Z |
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