Shifted products of independent random variables with values in finite groups

We consider sequences of random variables
$$ \varkappa^{(N)}=\zeta_1\zeta_2\ldots\zeta_N, \quad \omega^{(N)}=\xi_1\zeta_1\xi_2\zeta_2\ldots\xi_N\zeta_N, \quad N\ge 1, $$
where $(\xi_N,\zeta_N)$, $N\ge 1$, is a sequence of independent identically distributed random variables with values in the Cartesian product $G\times G$ of a finite group $(G;\cdot)$. We investigate the degree of dependence of the random variables $\varkappa^{(N)}$ and $\omega^{(N)}$. Such problems arise in the study of a class of information security algorithms. In connection to this problem, we study the random variable $\omega_a^{(N)}$ with values in $G$ whose distribution coincides with the conditional distribution of the random variable $\omega^{(N)}$ under condition that $\varkappa^{(N)}=a$, where $a\in G$ is such that $\mathbf P\{\varkappa^{(N)}=a\}>0$. We give conditions of convergence and limit distributions of $\omega_a^{(s_N)}$ as $N\to\infty$, where $s_N$ is a sequence of integers tending to infinity in such a way that $\mathbf P\{\varkappa^{(s_N)}=a\}>0$.