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} $R^{\mathrm{sc}}(D)$ within the stationary memoryless defense class $\Theta_{\mathrm{iid}}$ and provide its complete characterization. We prove that $R^{\mathrm{sc}}(D)$ 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 $\Theta_{\mathrm{iid}}$. For binary equal-prior tasks, $D_{\max} = \tfrac{1}{2}W_1(P_0,P_1)$ via Kantorovich--Rubinstein duality. On real-world website-fingerprinting defenses, the framework locates Front ($\Delta_{\mathrm{gap}}{=}0.028$\,bits), WTF-PAD ($0.034$\,bits), and TrafficSliver ($0.124$\,bits) above the theoretical curve, quantifying their suboptimality gaps.