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Characterization of the Arithmetic Complexity of the Secrecy Capacity of Fast-Fading Gaussian Channels

Holger Boche, Andrea Grigorescu, Rafael F. Schaefer, H. Vincent Poor

TL;DR

This work investigates the algorithmic computability of the secrecy capacity for fast-fading Gaussian wiretap channels. By embedding the problem in computability theory and the arithmetical hierarchy, it shows that for some computable fading distributions the secrecy capacity is a non-computable real, implying no universal algorithm can compute the capacity to arbitrary precision. The authors further show that the capacity lies in $\underline{\Delta}_2$ (as a difference of two $\Sigma_1$ reals) and, in some cases, does not belong to $\Sigma_1$ or $\Pi_1$, meaning neither computable achievability nor computable converse bounds exist. These results underscore fundamental limits on algorithmic code design for secure communication and contrast with known computability properties of other channel models.

Abstract

This paper studies the computability of the secrecy capacity of fast-fading wiretap channels from an algorithmic perspective, examining whether it can be computed algorithmically or not. To address this question, the concept of Turing machines is used, which establishes fundamental performance limits of digital computers. It is shown that certain computable continuous fading probability distribution functions yield secrecy capacities that are non-computable numbers. Additionally, we assess the secrecy capacity's classification within the arithmetical hierarchy, revealing the absence of computable achievability and converse bounds.

Characterization of the Arithmetic Complexity of the Secrecy Capacity of Fast-Fading Gaussian Channels

TL;DR

This work investigates the algorithmic computability of the secrecy capacity for fast-fading Gaussian wiretap channels. By embedding the problem in computability theory and the arithmetical hierarchy, it shows that for some computable fading distributions the secrecy capacity is a non-computable real, implying no universal algorithm can compute the capacity to arbitrary precision. The authors further show that the capacity lies in (as a difference of two reals) and, in some cases, does not belong to or , meaning neither computable achievability nor computable converse bounds exist. These results underscore fundamental limits on algorithmic code design for secure communication and contrast with known computability properties of other channel models.

Abstract

This paper studies the computability of the secrecy capacity of fast-fading wiretap channels from an algorithmic perspective, examining whether it can be computed algorithmically or not. To address this question, the concept of Turing machines is used, which establishes fundamental performance limits of digital computers. It is shown that certain computable continuous fading probability distribution functions yield secrecy capacities that are non-computable numbers. Additionally, we assess the secrecy capacity's classification within the arithmetical hierarchy, revealing the absence of computable achievability and converse bounds.
Paper Structure (9 sections, 5 theorems, 61 equations, 1 figure, 1 table)

This paper contains 9 sections, 5 theorems, 61 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Channel Model
  3. Computability Framework
  4. Computable Numbers
  5. Computable Sequences
  6. Computable Functions
  7. Arithmetical Hierarchy
  8. Computability of Wiretap Fading Capacity
  9. Conclusions

Key Result

Theorem 1

The secrecy capacity of a fast-fading channel $(f_{\alpha_1},f_{\alpha_1},\sigma_1^2,\sigma_2^2)$ subject to an average power constraint $P$ is where the optimal input distribution is $\mathcal{N}(0,P)$.

Figures (1)

  • Figure 1: TM $\mathfrak{T}_{C_{P,f_\alpha,\sigma^2,\epsilon}}$ for the computation of wiretap fast-fading codes operating at $R\geq C(P,f_\alpha,\sigma^2)-\frac{1}{2^M}$.

Theorems & Definitions (33)

  • Definition 1
  • Definition 2
  • Definition 3
  • Theorem 1: lin2016fast
  • Remark 1
  • Definition 4
  • Definition 5
  • Definition 6
  • Remark 2
  • Definition 7
  • ...and 23 more