New investigations of the generalized $p$-radical functional equations with its application to the secret key in the client-server environment

Abstract

Let $p$ be a natural number, $k$ be a natural number such that $k \geq 2$, $a_i, b_i\in \mathbb{R}\setminus\{0\}$ for all $i=1, \ldots, k$  and $Y$ is a quasi-Banach space.
In this paper, we investigate the general solution of the functional equation of the form
\begin{equation}\label{eq 00}
f\left(\sqrt[p]{\sum_{i=1}^ka_ix_i^p}\right)=\sum_{i=1}^kb_if(x_i), \ \ \ x_1,\dots,x_k \in \mathbb{R}  \tag{A}
\end{equation}
where $f: \mathbb{R} \to Y$ is an unknown, which is called the generalized $p$-radical functional equation. The hyperstability results for the function equation (\ref{eq 00}) are proved, and theses results was applied in the hyperstability results for the inhomogeneous generalized $p$-radical functional equations. Moreover, we provide an application for generating secret keys in the client-server environment.