Reliability and Latency Analysis for Wireless Communication Systems with a Secret-Key Budget

TL;DR

This work addresses secure wireless communication against a passive eavesdropper by modeling secret-key generation and consumption as a budgeted cash-flow. It adopts ruin-theoretic tools to analyze reliability (outage probability) and latency (time to rebuild the key budget) under two scheduling schemes: a deterministic, fixed-time scheme and a random-arrival scheme. Key contributions include: (i) proving almost-sure ruin for the deterministic scheme with a positive drift in net key usage, along with an integrodifference recursion and computable bounds; (ii) deriving conditions under which random transmissions can yield indefinite operation, via a negative drift threshold and a Lundberg-type bound for ultimate ruin, plus an integral equation for the exact ruin probability; and (iii) providing simple, interpretable latency expressions linking initial budget to expected inter-arrival times, validated with Rayleigh fading scenarios. The results offer practical guidance for designing secret-key budgets to meet specified reliability-latency requirements in next-generation wireless systems.

Abstract

We consider a wireless communication system with a passive eavesdropper, in which a transmitter and legitimate receiver generate and use key bits to secure the transmission of their data. These bits are added to and used from a pool of available key bits. In this work, we analyze the reliability of the system in terms of the probability that the budget of available key bits will be exhausted. In addition, we investigate the latency before a transmission can take place. Since security, reliability, and latency are three important metrics for modern communication systems, it is of great interest to jointly analyze them in relation to the system parameters. In particular, we show under what conditions the system may remain in an active state indefinitely, i.e., never run out of available secret-key bits. The results presented in this work will allow system designers to adjust the system parameters in such a way that the requirements of the application in terms of both reliability and latency are met.

Figures (6)

• Figure 1: Exemplary illustration of the temporal progress of the number of available bits $\bm{B}_t$. During the active state, both and transmission are performed. Once the budget is exhausted, the system switches to a recharge state, where only new key bits are generated until a certain threshold $b_0$ is reached. The latency $\bm{T}$ is defined as the number of time slots between two active states. In the shown example, we have $\bm{T}=10$.
• Figure 2: Illustration of the scheduling with a deterministic scheme during the active state. In each time slot, there is an block followed by a block. While the channels are assumed to remain constant for each individual phase, they change independently between the and blocks.
• Figure 3: Outage probability $\psi_{t}$ for different initial budgets $b_0$ over time for a system using the deterministic scheme. The channels to both Bob and Eve are Rayleigh fading with average $\mathbb{E}_{}\left[\bm{X}_i\right]=20\dB$ and $\mathbb{E}_{}\left[\bm{Y}_i\right]=10\dB$, respectively. The solid lines correspond to the numerically calculated outage probabilities according to \ref{['eq:prob-surv-recursive']} while the markers indicate results from simulations with $10^6$ samples (\ref{['ex:fixed-time-rayleigh']}).
• Figure 4: Required initial budget $b_0^{\tau}$ over the outage probability $\varepsilon$ for a system in the recharge state. The channels to both Bob and Eve are Rayleigh fading with average $\mathbb{E}_{}\left[\bm{X}_i\right]=20\dB$ and $\mathbb{E}_{}\left[\bm{Y}_i\right]=10\dB$, respectively. The solid lines correspond to the numerically calculated outage probabilities according to \ref{['eq:init-budget-inverse-ruin-prob']} while the markers indicate results from simulations with $10^6$ samples (\ref{['ex:fixed-time-init-budget']}).
• Figure 5: Illustration of the scheduling model with random blocks during the active system state. In each time slot $t$, there is a probability of $p_t$ that a message is transmitted. If no message is transmitted, is performed instead.

Theorems & Definitions (24)

• Lemma 1: Average Net Usage -- Deterministic Scheme
• proof
• Corollary 1: Probability of Ultimate Ruin -- Deterministic Scheme
• proof
• Remark
• Example
• Theorem 1: Worst-Case Bounds of the Ruin Probability -- Deterministic Scheme
• proof
• Remark 1
• Example 1: Rayleigh Fading