On new multivariate cryptosystems with nonlinearity gap

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.