Finite-Length Empirical Comparison of Polar, PAC, and Invertible-Extractor Secrecy Codes over the Wiretap BSC

We compare three secrecy-coding schemes for the degraded wiretap binary symmetric channel (BSC) in the finite-blocklength regime: (i) polar wiretap coset codes, (ii) PAC codes used as wiretap coset codes, and (iii) the invertible-extractor (IE) framework of Bellare-Tessaro. Our comparison is empirical and uses a common semantic-secrecy metric (distinguishing advantage). For polar coset codes, we compute Eve's polarized bit-channel capacities (via the Tal-Vardy construction) to obtain explicit finite-length upper bounds on mutual-information leakage, yielding strong secrecy bounds. For PAC coset codes, we prove that Eve's synthesized bit-channels are equivalent to those of polar codes (up to a permutation), so the same leakage bounds apply; we then convert these strong-secrecy bounds into semantic-secrecy guarantees for symmetric wiretap channels. For the IE scheme, we use the closed-form semantic-secrecy bounds given in the reference work. Finally, we report finite-length results that jointly characterize (a) semantic-secrecy guarantees against Eve and (b) frame-error-rate performance at Bob, illustrating that PAC codes can significantly improve reliability without changing the secrecy bounds inherited from polar coding. Moreover, under the finite-length bounds considered in this work, polar/PAC secrecy codes provide tighter security guarantees than the invertible-extractor framework.