On bijective functions of fixed variables in the Galois field of $p^k$ elements and on the ring of $p$-adic integers for an odd prime number $p$

In this paper there are given necessary and sufficient conditions under which a function of fixed variables $\psi{:} \mathbb{F}_{q}^{i+1}\to\mathbb{F}_{q}$ is bijective, where $ i\in\mathbb{N}\cup\{0\}$, $\mathbb{F}_{q}^{i+1} $ is the $(i+1)$-ary Cartesian power of the Galois field $\mathbb{F}_{q}$ of $ q=p^k $ elements, $ p $ is an odd prime number and $k\in\mathbb{N}$. In addition, such conditions of the bijective functions $\psi$ of fixed variables are used to write a criterion for the preserving Haar measure of functions from the important class of 1-Lipschitz functions in terms of its coordinate functions on the ring of $p$-adic integers $\mathbb{Z}_p, p\neq2$. In particular, the representation of 1-Lipschitz functions in terms of its coordinate functions on the ring of $2$-adic integers $ \mathbb{Z}_2$ turned out to be a general and useful tool for obtaining mathematical results applied in cryptography. In this work, the research of such representation of 1-Lipschitz functions on the ring of $p$-adic integers $ \mathbb{Z}_p,p\neq2$ is being continued, with special attention to the representation of bijective 1-Lipschitz functions in terms of its coordinate functions on $ \mathbb{Z}_p, p\neq2$.