### Abstract:

The pair of families of bijective multivariate maps
of kind Fn and Fn
−1
on affine space Kn over finite commutative ring
K given in their standard forms has a nonlinearity gap if the degree
of Fn is bounded from above by independent constant d and degree
of F
−1
is bounded from below by c
n, c > 1. We introduce examples
of such pairs with invertible decomposition Fn = G1
nG2
n . . . Gk
n,
i.e. the decomposition which allows to compute the value of F
n−1
in given point p = (p1, p2, . . . , pn) in a polynomial time O(n
2
).
The pair of families Fn, F
′
n of nonbijective polynomial maps of
affine space Kn such that composition FnF
′
n leaves each element
of K∗n
unchanged such that deg(Fn) is bounded by independent
constant but deg(F
′
n
) is of an exponential size and there is a decom-
position G1
nG2
n . . . Gk
n of Fn which allows to compute the reimage
of vector from F(K∗n
) in time 0(n
2
). We introduce examples of
such families in cases of rings K = Fq and K = Zm.