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Towards Secure Over-The-Air Computation

Matthias Frey, Igor Bjelaković, Sławomir Stańczak

TL;DR

This paper studies secure Over-The-Air computation of an arithmetic mean over a multi-user channel by introducing a friendly jammer whose signal is stronger at the legitimate receiver than at the eavesdropper. It builds a DFA-J framework linking secure computation to compound-channel coding and channel resolvability for continuous alphabets, and proves a Gaussian AWGN main result with explicit finite-$n$ bounds. The key contributions include a theorem that bounds the legitimate receiver's mean-squared error via $ \sigma_{\mathrm{eff},\mathfrak{B}}^2 \Psi(2/\sigma_{\mathrm{eff},\mathfrak{B}}) $ (up to exponentially small terms) and establishes a secrecy lower bound for the eavesdropper via $ \sigma_{\mathrm{eff},\mathfrak{E}}^2 \Psi(2/\sigma_{\mathrm{eff},\mathfrak{E}}) $, under a jamming condition $ h_{JB}/\sigma_B > h_{JE}/\sigma_E$. The framework extends to cost-constrained Gaussian fading compound channels and outlines steps toward semantic-security guarantees and practical code design for secure OTA computation. Overall, the work advances secure analog computation by connecting resolvability and compound-channel coding to function computation under eavesdropping and introduces techniques for reliable jam reconstruction at the legitimate receiver. The results have potential implications for scalable, privacy-preserving distributed computation in wireless networks.

Abstract

We propose a new method to protect Over-The-Air (OTA) computation schemes against passive eavesdropping. Our method uses a friendly jammer whose signal is -- contrary to common intuition -- stronger at the legitimate receiver than it is at the eavesdropper. We focus on the computation of arithmetic averages over an OTA channel. The derived secrecy guarantee translates to a lower bound on the eavesdropper's mean square error while the question of how to provide operationally more significant guarantees such as semantic security remains open for future work. The key ingredients in proving the security guarantees are a known result on channel resolvability and a generalization of existing achievability results on coding for compound channels.

Towards Secure Over-The-Air Computation

TL;DR

This paper studies secure Over-The-Air computation of an arithmetic mean over a multi-user channel by introducing a friendly jammer whose signal is stronger at the legitimate receiver than at the eavesdropper. It builds a DFA-J framework linking secure computation to compound-channel coding and channel resolvability for continuous alphabets, and proves a Gaussian AWGN main result with explicit finite- bounds. The key contributions include a theorem that bounds the legitimate receiver's mean-squared error via (up to exponentially small terms) and establishes a secrecy lower bound for the eavesdropper via , under a jamming condition . The framework extends to cost-constrained Gaussian fading compound channels and outlines steps toward semantic-security guarantees and practical code design for secure OTA computation. Overall, the work advances secure analog computation by connecting resolvability and compound-channel coding to function computation under eavesdropping and introduces techniques for reliable jam reconstruction at the legitimate receiver. The results have potential implications for scalable, privacy-preserving distributed computation in wireless networks.

Abstract

We propose a new method to protect Over-The-Air (OTA) computation schemes against passive eavesdropping. Our method uses a friendly jammer whose signal is -- contrary to common intuition -- stronger at the legitimate receiver than it is at the eavesdropper. We focus on the computation of arithmetic averages over an OTA channel. The derived secrecy guarantee translates to a lower bound on the eavesdropper's mean square error while the question of how to provide operationally more significant guarantees such as semantic security remains open for future work. The key ingredients in proving the security guarantees are a known result on channel resolvability and a generalization of existing achievability results on coding for compound channels.

Paper Structure

This paper contains 25 sections, 15 theorems, 73 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Prior Work
  3. OTA computation
  4. Coding for compound channels
  5. Channel Resolvability and Semantic Security
  6. Friendly Jamming
  7. Physical Layer Security
  8. Computational Wiretap Channels
  9. Summary of the Main Contributions and Outline
  10. System Model and Main Result
  11. Distributed Function Approximation with Jamming (DFA-J)
  12. AWGN Channel Model
  13. Main Result
  14. Special case ${K}=1$
  15. Coding for the Compound Channel
  16. ...and 10 more sections

Key Result

Lemma 1

Let $\mathcal{T} := [a, b]$, let $g: {\mathcal{S}}_1 \times \ldots \times {\mathcal{S}}_{K} \rightarrow \mathcal{T}$ be an eavesdropper's objective and $\delta \geq 0$ a real number. Assume that $g({s}_1, \dots, {s}_{K})$ is uniformly distributed on $\mathcal{T}$. Then, any scheme that is $\delta$-s

Figures (3)

  • Figure 1: Table of symbols.
  • Figure 2: System model for described in Section \ref{['sec:dfaj']}.
  • Figure 3: Illustration of the guarantees of Theorem \ref{['theorem:awgn-scheme-real']}. The dashed line is the which an eavesdropper would have without any received signal (i.e., guessing the middle of the interval).

Theorems & Definitions (35)

  • Definition 1
  • Definition 2
  • Definition 3
  • Lemma 1
  • proof
  • Theorem 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • proof : Proof of Lemma \ref{['lemma:awgn-scheme-ideal']}
  • ...and 25 more