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On Schubert cells in Grassmanians and new algorithms of multivariate cryptography
V. A. Ustimenko^{} ^{} Maria CurieSkłodowska University, Lublin
Abstract:
The partition of projective geometry over the field $F_q$ into Schubert sets allows to convert an incidence
graph to symbolic Grassman automaton. Special symbolic computations of these automata produce bijective
transformation of the largest Schubert cell. Some of them are chosen as maps which are used in new
cryptosystems.
The natural analogue of projective geometry over $F_q$ and related Grassman automaton can be defined
over a general commutative ring $K$ and used for applications. In case of finite ring these automata allows to
define a symmetric encryption algorithm, which uses plainspace $(K)^{k(k+1)}$ and key space formed by special
tuple of elements from $K[x_1,x_2,…,x_k]^k$ (governing functions). The length of the password tuple can be
chosen by users. Every encryption map corresponding to chosen tuple is a multivariate map, its degree is
defined by degrees of multivariate governing functions. These degrees can be chosen in a way that the value
of corresponding multivariate map given in standard form can be computed in a polynomial time.
It will be shown that bijectivity of the last governing function guaranties bijectivity of the transformation
of space $(K)^{k(k+1)}$. So this symmetric algorithm can be used for the extention of the bijective polynomial
map $h: K^k\to K^k$ to the bijective nonlinear map $E(h): (K)^{k(k+1)}\to (K)^{k(k+1)}$. Transformations of kind $G=T_1E(h)T_2$, where $T_1$ and $T_2$ are affine bijections can be used in cryptography. In the case when all governing functions are linear the transformation $G$ will be quadratic.
We consider examples of quadratic cryptosystems $E(h)$ over special fields, where h is an encryption
function of Imai Matsumoto algorithm. Finally we suggest multivariate algorithms of Postquantum Cryptography
which use hidden discrete logarithm problem and hidden factorisation problems for integers. In case
of factorization the last governing function ia chosen as a nonbijective map.
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UDC:
519.1 Received: 02.11.2015
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Citation:
V. A. Ustimenko, “On Schubert cells in Grassmanians and new algorithms of multivariate cryptography”, Tr. Inst. Mat., 23:2 (2015), 137–148
Citation in format AMSBIB
\Bibitem{Ust15}
\by V.~A.~Ustimenko
\paper On Schubert cells in Grassmanians and new algorithms of multivariate cryptography
\jour Tr. Inst. Mat.
\yr 2015
\vol 23
\issue 2
\pages 137148
\mathnet{http://mi.mathnet.ru/timb251}
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http://mi.mathnet.ru/eng/timb251 http://mi.mathnet.ru/eng/timb/v23/i2/p137
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This publication is cited in the following articles:

Vasyl Ustimenko, “On new multivariate cryptosystems with nonlinearity gap”, Algebra Discrete Math., 23:2 (2017), 331–348

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