RankGuardPolar Private Public Finite Length Polar Codes with Rank-Certified Leakage

We introduce \textbf{RankGuard-Polar}, a framework for safely publishing a subset of polar codeword coordinates over shared public resources. We assume a strong eavesdropper who has access to the channel input, i.e., the transmitted codeword coordinates published on a public resource access model. Working over $\mathbb F_2$ and focusing on time-shared public/private BEC uses, we show that leakage from a published index set $\mathbf{P}$ admits an exact algebraic characterization comes from an information-theoretic viewpoint, and we construct an explicit linear extractor ($R$) that identifies the leaked linear combinations. Building on this identity, we (i) give efficient procedures to compute and certify leakage for any $\mathbf{P}$, (ii) propose a practical fast algorithm with provable efficiency.