Cyber-Physical-Systems and Secrecy Outage Probability: Revisited

Abstract: This paper technically explores the secrecy rate $\Lambda$ and a maximisation problem over the concave version of the secrecy outage probability (SOP) as $\mathop{{\rm \mathbb{M}ax}}\limits_{\Delta } {\rm \; } \mathbb{P}\mathscr{r} \big( \Lambda \ge \lambda \big) $. We do this from a generic viewpoint even though we use a traditional Wyner's wiretap channel for our system model $-$ something that can be extended to every kind of secrecy modeling and analysis. We consider a Riemannian mani-fold for it and we mathematically define a volume for it as $\mathbb{V}\mathscr{ol}\big \lbrace \Lambda \big \rbrace$. Through achieving a new bound for the Riemannian mani-fold and its volume, we subsequently relate it to the number of eigen-values existing in the relative probabilistic closure. We prove in-between some novel lemmas with the aid of some useful inequalities such as the \textit{Finsler's} lemma, the generalised \textit{Young's} inequality, the generalised \textit{Brunn-Minkowski} inequality, the \textit{Talagrand's} concentration inequality. We additionally propose a novel Markov decision process based reinforcement learning algorithm in order to find the optimal policy in relation to the eigenvalue distributions $-$ something that is extended to a possibilisitically semi-Markov decision process for the case of periodic attacks.