Rate-Distortion Function for Encrypted Traffic Side-Channel Defense

Parameter selection for encrypted traffic defense has long relied on empirical tuning, yet the fundamental question – \emph{given a QoS cost budget $D$, how low can the leakage rate go under sustained observation?} – lacks a provable, computable baseline. Taking the semantic label sequence $X^n$ as the source, the defended feature sequence $Y^n$ as the observation, and Wasserstein-1 distance as the defense cost, we define the \emph{side-channel rate-distortion function} $R^{\mathrm{s}\mathrm{c}}(D)$ within the stationary memoryless defense class ${\mathrm{\Theta }}_{\mathrm{i}\mathrm{i}\mathrm{d}}$ and provide its complete characterization. We prove that $R^{\mathrm{s}\mathrm{c}}(D)$ is monotone decreasing, convex, and continuous, with exact endpoints; the optimal defense has an exponential-tilting (Boltzmann) structure governed by KKT conditions; and the curve constitutes the exact Pareto frontier within ${\mathrm{\Theta }}_{\mathrm{i}\mathrm{i}\mathrm{d}}$. For binary equal-prior tasks, $D_{\max }=\frac{1}{2}{W}_1(P_0,P_1)$ via Kantorovich–Rubinstein duality. On real-world website-fingerprinting defenses, the framework locates Front (${\mathrm{\Delta }}_{\mathrm{g}\mathrm{a}\mathrm{p}}=0.028$,bits), WTF-PAD ($0.034$,bits), and TrafficSliver ($0.124$,bits) above the theoretical curve, quantifying their suboptimality gaps.