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

2026-04-21

AI Summary

This work addresses the trade-off between semantic security and reliability for finite-length secrecy coding over degraded eavesdropped binary symmetric channels. It proposes an empirical comparison framework based on the uniform distinguishing advantage metric and, for the first time, introduces PAC codes as coset codes in the wiretap channel setting, proving their equivalence to polar codes over the eavesdropper’s synthesized bit channels. The study further translates strong secrecy bounds into semantic security guarantees for symmetric wiretap channels. Leveraging the Tal–Vardy construction, coset encoding, and a randomness extractor framework—combined with closed-form semantic security bounds—experiments demonstrate that PAC codes achieve significantly lower frame error rates at the legitimate receiver while maintaining security levels comparable to polar codes. Moreover, both polar and PAC codes yield tighter finite-length security guarantees than those provided by invertible extractors.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Game Theory and Economic Paradigms: Imperfect Information
• Reasoning under Uncertainty: Relational Probabilistic Models

Application Category

• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems

Abstract

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.

Problem

• wiretap channel
• finite-blocklength
• semantic secrecy
• polar codes
• PAC codes

Innovation

• Polar codes
• PAC codes
• Invertible extractor
• Semantic secrecy
• Finite-blocklength analysis