Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Detecting Symmetrizability in Physical Systems

Florian Seitz, Janis Nötzel

TL;DR

This work investigates symmetrizability for arbitrarily varying channels (AVCs) and the impossibility of a universal Turing decision for exact symmetrizability. By introducing approximate symmetrizability (ε-SYM) and formulating it as a linear feasibility problem, the authors provide a polynomial-time procedure to certify non-symmetrizability under finite jammer states and energy constraints. They connect these results to physical channels, notably a lossy bosonic channel with M-PSK encoding, showing that exact symmetrizability is exceptional and that energy limitations restore tractable analysis. The study moreover demonstrates that finite discretization of continuous jammer inputs preserves approximate symmetrizability, enabling reliable capacity assessments for realistic, energy-bounded DOS scenarios.

Abstract

We study the problem of data transmission under the influence of a jammer, which is typical for wireless systems and commonly modeled as an arbitrarily varying channel (AVC) in information theory. AVC fulfilling a certain set of linear equations are called symmetrizable and are known to be prone to denial of service attacks. Recent work has shown that deciding if a given AVC is symmetrizable or not is a non-Turing computable problem. By relaxing the formulation of symmetrizability, we show the existence of a polynomial-time algorithm that determines whether a given AVC is non-symmetrizable, but displays a critical dependence on the number of jammer input states. We then show how imposing an energy constraint on the jammer allows the same algorithm to efficiently identify large classes of AVCs which are non-symmetrizable.

Detecting Symmetrizability in Physical Systems

TL;DR

This work investigates symmetrizability for arbitrarily varying channels (AVCs) and the impossibility of a universal Turing decision for exact symmetrizability. By introducing approximate symmetrizability (ε-SYM) and formulating it as a linear feasibility problem, the authors provide a polynomial-time procedure to certify non-symmetrizability under finite jammer states and energy constraints. They connect these results to physical channels, notably a lossy bosonic channel with M-PSK encoding, showing that exact symmetrizability is exceptional and that energy limitations restore tractable analysis. The study moreover demonstrates that finite discretization of continuous jammer inputs preserves approximate symmetrizability, enabling reliable capacity assessments for realistic, energy-bounded DOS scenarios.

Abstract

We study the problem of data transmission under the influence of a jammer, which is typical for wireless systems and commonly modeled as an arbitrarily varying channel (AVC) in information theory. AVC fulfilling a certain set of linear equations are called symmetrizable and are known to be prone to denial of service attacks. Recent work has shown that deciding if a given AVC is symmetrizable or not is a non-Turing computable problem. By relaxing the formulation of symmetrizability, we show the existence of a polynomial-time algorithm that determines whether a given AVC is non-symmetrizable, but displays a critical dependence on the number of jammer input states. We then show how imposing an energy constraint on the jammer allows the same algorithm to efficiently identify large classes of AVCs which are non-symmetrizable.

Paper Structure

This paper contains 8 sections, 38 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. Approximate Symmetrizability
  4. Symmetrizability of Random Channels
  5. A Physical Channel Model
  6. Runtime Estimates under Energy Constraints
  7. Conclusion
  8. Appendix

Figures (3)

  • Figure 1: A surface plot showing the results of a numerical experiment to determine values of $p_\mathrm{sym}(X,Y,S,\varepsilon)$ for $X=Y=4$, $S = 2,...,14$, and $\varepsilon \in [2^{-15},2^{-3}]$. For each set of values, ten thousand samples were drawn.
  • Figure 2: Visualization of the complex Gaussian outcome distributions for the different messages in case there is no jamming. The parameters in this case are $M=6$, $E=16$ and $N_A = N_S = 1$.
  • Figure 3: Minimal symmetrization error $F(\mathfrak W)$ according to \ref{['def:F(W)']} of a lossy bosonic channel with thermal noise, for different transmittivities $\eta$. For all curves the number of messages is $M=6$.