Linear codes of Schubert cells and implementations of new quadratic public keys of Multivariate Cryptography

Abstract

Multivariate Cryptography and Code Base Cryptography together with three other directions form the list of core areas of Poat Quantum Cryptography. Secure quadratic multivariate cryptosystem from are able to establish the shortest digital signatures. An idea of multivariate cryptography algorithms with quadratic transformation induced by walks on Cellular Schubert Graphs was proposed recently. These graphs are defined via restriction of the incidence relations of finite projective geometry on two distinct Schubert cells. The method defines nonlinear transformation of the vector space of vertexes of one of this cells. In this paper the implementations of these multivariate cryptosystems are considered in the case of large finite fields of characteristic 2. Quadratic map of large order is combined with two affine transformations. The lower bound of polynomial degree of the inverse map and complexity estimates for private key decryption are introduced.