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The families of bijective transformations $G_n$ of affine space $K^n$ overgeneral commutative ring $K$ of increasing order with the property of stabilitywill be constructed. Stability means that maximal degree of elements of cyclicsubgroup generated by the transformation of degree $d$ is bounded by $d$. Inthe case $K=F_q$ these transformations of $K^n$ can be of an exponential order.We introduce large groups formed by quadratic transformations and numericalencryption algorithm protected by secure protocol of NoncommutativeCryptography. The construction of transformations is presented in terms ofwalks on Double Schubert Graphs.

On computations with Double Schubert Automaton and stable maps of Multivariate Cryptography