Construction of a class of functions on finite fields using linear recurrences over Galois rings

The paper deals with a class of functions over a finite field $\mathrm{GF}(q)$ constructed on the basis of linear recurring sequences (LRS) over a ring $\mathrm{GR}(q^n,p^n)$ with a distinguished characteristic polynomial. The order of the arguments of the functions in this class is obtained from the set of LRS over the finite field, and the values of the functions are obtained from the complicated LRS over the ring. When some conditions are met, for the proximity $C(f)$ of the studied functions $f$ in $m$ variables to the class of affine functions, the estimate $C(f)\le q^{(m+n-1)/2}(p^{n-1}-1)(q-1)^{1/2}$ is proved. The power of a class of functions and its automaton implementation are also studied.