Nonlinear permutations of a space over a finite field induced by linear transformations of a module over a Galois ring

Nonlinear permutations of $m$-dimensional vector space $P^{(m)}$ over a finite field $P=\mathrm{GF}(q)$ induced by linear transforms of a module $R^{(m)}$ over a Galois ring $R=\mathrm{GR}(q^2,p^2)$, $q=p^r$, are constructed. The transforms constructed by iteration of linear recurrent transforms are studied separately. Some applications in cryptography are discussed.