On Schubert cells in Grassmanians and new algorithms of multivariate cryptography

V. A. Ustimenko

Trudy Instituta matematiki, Tome 23 (2015) no. 2, pp. 137-148

Résumé

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 $K[x_1,x_2,\dots,x_k]^k$ (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.