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

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.