Cyclic groups of quadratic transformations of affine spaces overgeneral commutative ring K will be constructed. These subgroupsof affine Cremona group have a property of quadratic stability, i. e.,all nonidentical elements of subgroups are quadratic transformations.In the case $K = F_q$ generators of the group have large order. Moreprecisely, for each vector space of kind $V_n = F_q^(n)$ we define their bijectivequadratic transformation, which generates a cyclic subgroup $G_n$ oforder at least $q^{-1} - 1$ with the property of quadratic stability. It means that multivariate Diffe - Hellman key exchange protocol anda new multivariate version of shifted El Gamal algorithm have goodcomplexity estimated in the case of the group $G_n$ and its conjugates.For the generation of stable maps $G_n$ the techniques of symboliccomputations in linguistic graphs were used. The algorithm of generation of nonlinear maps of bounded degree via mixing of families ofstable transformation of the large order is suggested.
@article{446,
title = {On the families of stable quadratic multivariate transformations of exponential order and their cryptographical applications},
journal = {Tatra Mountains Mathematical Publications},
volume = {70},
year = {2018},
language = {EN},
url = {http://dml.mathdoc.fr/item/446}
}
Ustimenko, Vasyl. On the families of stable quadratic multivariate transformations of exponential order and their cryptographical applications. Tatra Mountains Mathematical Publications, Tome 70 (2018) . http://gdmltest.u-ga.fr/item/446/