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Algebra and Discrete Mathematics, 2015, Volume 19, Issue 1, Pages 130–144 (Mi adm512)  

This article is cited in 5 scientific papers (total in 5 papers)

RESEARCH ARTICLE

On the flag geometry of simple group of Lie type and multivariate cryptography

Vasyl Ustimenko

Maria Curie-Sklodowska University, Lublin
Full-text PDF (366 kB) Citations (5)
References:
Abstract: 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.
Keywords: multivariate cryptography, flag variety, geometry of simple group of Lie type, Schubert cell, symbolic walks.
Received: 23.01.2015
Revised: 21.02.2015
Bibliographic databases:
Document Type: Article
Language: English
Citation: Vasyl Ustimenko, “On the flag geometry of simple group of Lie type and multivariate cryptography”, Algebra Discrete Math., 19:1 (2015), 130–144
Citation in format AMSBIB
\Bibitem{Ust15}
\by Vasyl~Ustimenko
\paper On the flag geometry of simple group of Lie type and multivariate cryptography
\jour Algebra Discrete Math.
\yr 2015
\vol 19
\issue 1
\pages 130--144
\mathnet{http://mi.mathnet.ru/adm512}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=3376345}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=000209846200012}
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  • This publication is cited in the following 5 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Algebra and Discrete Mathematics
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    Abstract page:192
    Full-text PDF :91
    References:45
     
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