Optimizing Secrecy Codes Using Gradient Descent

TL;DR

This work tackles finite-blocklength secrecy coding for the binary erasure wiretap channel by casting coset-code design as a gradient-descent optimization over a continuous code-definition vector $q$. It introduces continuously valued performance functions that map $q$ to equivocation and $\chi^2$ divergence, enabling constraint-aware gradient updates via a movement-cost framework and a bounded update mechanism. Across a wide range of code dimensions and blocklengths, codes produced by gradient descent outperform published capacity-based and finite-blocklength constructions, with $\chi^2$-based optimization correlating strongly with low equivocation loss. The findings underscore the practical viability of subspace-decomposition approaches for secrecy coding at moderate-to-large blocklengths and suggest broader applicability of the continuous, geometry-informed optimization framework to other coding-theoretic problems.

Abstract

Recent theoretical developments in coset coding theory have provided continuous-valued functions which give the equivocation and maximum likelihood (ML) decoding probability of coset secrecy codes. In this work, we develop a method for incorporating these functions, along with a complex set of constraints, into a gradient descent optimization algorithm. This algorithm employs a movement cost function and trigonometric update step to ensure that the continuous-valued code definition vector ultimately reaches a value which yields a realizable coset code. This algorithm is used to produce coset codes with blocklength up to a few thousand. These codes were compared against published codes, including both short-blocklength and capacity-achieving constructions. For most code sizes, codes generated using gradient descent outperformed all others, especially capacity-achieving constructions, which performed significantly worse than randomly-generated codes at short blocklength.

Figures (23)

• Figure 1: Binary erasure wiretap channel.
• Figure 2: Geometric interpretation of the boundary-compliant update procedure.
• Figure 3: Geometric representation of the boundary-compliant mean adjustment procedure.
• Figure 4: Gradient descent path minimizing equivocation loss for $n=64$ and $\kappa=8$.
• Figure 5: Gradient descent path minimizing $\chi^2$ divergence for $n=64$ and $\kappa=8$.