Privacy Leakage in Discrete Time Updating Systems

TL;DR

This work studies the privacy-utility trade-off in discrete-time status-update systems under maximal leakage, with updates arriving as a rate-$λ$ Bernoulli process. It analyzes three server policies—MBT, DAD, and RAD—deriving both the maximal leakage and AoI for each, including closed-form expressions and asymptotic leakage rates, e.g., $\mathcal{L}(X^n\to Y^n)= n\log(1+\mu)$ for MBT/RAD and $\lim_{n\to\infty} \mathcal{L}(X^n\to Y^n)/n= \frac{\log 2}{\tau}$ for DAD. The results show that DAD yields the best AoI at a given leakage but operates at discrete age-leakage points, while MBT and RAD provide continuous trade-offs through $\mu$ and arrival thinning $\alpha$, offering flexible privacy-timeliness tuning. These insights guide privacy-preserving design of status-update networks and extend prior continuous-time analyses to a discrete-time setting with a random accumulate-and-dump variant. Overall, the paper clarifies how different service policies shape the interplay between timeliness and information leakage in practical update systems.

Abstract

A source generates time-stamped update packets that are sent to a server and then forwarded to a monitor. This occurs in the presence of an adversary that can infer information about the source by observing the output process of the server. The server wishes to release updates in a timely way to the monitor but also wishes to minimize the information leaked to the adversary. We analyze the trade-off between the age of information (AoI) and the maximal leakage for systems in which the source generates updates as a Bernoulli process. For a time slotted system in which sending an update requires one slot, we consider three server policies: (1) Memoryless with Bernoulli Thinning (MBT): arriving updates are queued with some probability and head-of-line update is released after a geometric holding time; (2) Deterministic Accumulate-and-Dump (DAD): the most recently generated update (if any) is released after a fixed time; (3) Random Accumulate-and-Dump (RAD): the most recently generated update (if any) is released after a geometric waiting time. We show that for the same maximal leakage rate, the DAD policy achieves lower age compared to the other two policies but is restricted to discrete age-leakage operating points.

Figures (5)

• Figure 1: A source sends status updates through a server to a monitor. An adversary (Adv) observes transmissions from the server to the monitor.
• Figure 2: The source sends fresh updates to the server in slots $N_k=n_k$, inducing the age process $A_i(n)$ at the input to the server. The server sends samples of the most recent update to the monitor in time slots $S_k=s_k$, inducing the age process $A_m(n)$ at the monitor.
• Figure 3: The age vs. maximal leakage rate for the MBT $(\alpha=1)$, DAD, and RAD servers with arrival rate $\lambda=0.5$. The service rate $\mu$ varies over $[0.524,1]$ for Geo/Geo/1 and over $[0.05,1]$ for RAD. For DAD, $\tau$ varies from 1 to 39.
• Figure 4: The age vs. maximal leakage rate for the RAD and DAD policies for $\lambda=0.1$, $\lambda=0.5$ and $\lambda=0.9$.
• Figure 5: The age vs. maximal leakage rate for the MBT system: for $\alpha=1$, the $\lambda=0.1$, $\lambda=0.5$ and $\lambda=0.9$ age-leakage trade-off curves are obtained by varying $\mu$ over the interval $[\lambda+\epsilon,1]$. With $\lambda$ fixed, for each $\mu$ (which specifies the leakage), the $\alpha\in(0,1]$ that minimizes $\mathop{\mathrm{E}}\nolimits[*]{A_{\text{MBT}}}$ in (\ref{['eqn:GeoAge']}) is also calculated. This yields the solid blue-orange-green trade-off curve. The blue segment shows $\alpha\lambda\in [0.054,0.1]$ and is achievable for $\lambda\ge 0.1$. The orange segment shows $\alpha\lambda\in [0.1,0.5]$ and the green segment shows $\alpha\lambda\in [0.5,0.9]$; these segments are achievable for $\lambda\ge 0.5$ and $\lambda\ge 0.9$ respectively.

Theorems & Definitions (10)

• Definition 1: Issa et al. 8943950
• Definition 2: Kaul et al. kaul2012real
• Lemma 1
• Theorem 1
• proof
• Lemma 2
• proof
• Theorem 2
• proof
• Theorem 3