On the families of stable quadratic multivariate transformations of exponential order and their cryptographical applications
Abstract
Cyclic groups of quadratic transformations of affine spaces over
general commutative ring K will be constructed. These subgroups
of 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. More
precisely, for each vector space of kind
$V_n = F_q^(n)$ we define their bijective
quadratic transformation, which generates a cyclic subgroup $G_n$ of
order at least $q^{-1} - 1$ with the property of quadratic stability.
It means that multivariate Diffe - Hellman key exchange protocol and
a new multivariate version of shifted El Gamal algorithm have good
complexity estimated in the case of the group $G_n$ and its conjugates.
For the generation of stable maps $G_n$ the techniques of symbolic
computations in linguistic graphs were used. The algorithm of generation of nonlinear maps of bounded degree via mixing of families of
stable transformation of the large order is suggested.
general commutative ring K will be constructed. These subgroups
of 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. More
precisely, for each vector space of kind
$V_n = F_q^(n)$ we define their bijective
quadratic transformation, which generates a cyclic subgroup $G_n$ of
order at least $q^{-1} - 1$ with the property of quadratic stability.
It means that multivariate Diffe - Hellman key exchange protocol and
a new multivariate version of shifted El Gamal algorithm have good
complexity estimated in the case of the group $G_n$ and its conjugates.
For the generation of stable maps $G_n$ the techniques of symbolic
computations in linguistic graphs were used. The algorithm of generation of nonlinear maps of bounded degree via mixing of families of
stable transformation of the large order is suggested.