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Capacity Region for Covert Secret Key Generation over Multiple Access Channels

Yingxin Zhang, Lin Zhou, Qiaosheng Zhang

TL;DR

This work introduces covert secret key generation over a binary-input two-user MAC with one-way public discussion, aiming to generate two secret keys with Charlie while remaining undetectable by Willie. The authors derive inner and outer bounds on the CSK capacity region, showing that with covertness the key rates scale as $O(\, rac{1}{\, ^{1/2}})$ and providing explicit region expressions parameterized by $oldsymbol{\rho}$ through $R_{in}$ and $R_{out}$. They connect CSK to the wiretap secret key capacity when the covertness constraint is removed and discuss a duality with covert communication over the same MAC, supported by numerical examples. The achievability combines a covert process with a likelihood-encoder-based auxiliary scheme to balance reliability, secrecy, and covertness, while the converse adapts multiterminal Csiszár–Narayan arguments under covertness. The results illuminate the fundamental trade-offs imposed by covertness in multiterminal key generation and pave the way for extensions to continuous alphabets and other multiuser channels.

Abstract

We study covert secret key generation over a binary-input two-user multiple access channel with one-way public discussion and derive bounds on the capacity region. Specifically, in this problem, there are three legitimate parties: Alice, Bob and Charlie. The goal is to allow Charlie to generate a secret key with Alice and another secret key with Bob, reliably, secretly and covertly. Reliability ensures that the key generated by Alice and Charlie is the same and the key generated by Bob and Charlie is the same. Secrecy ensures that the secret keys generated are only known to specific legitimate parties. Covertness ensures that the key generation process is undetectable by a warden Willie. As a corollary of our result, we establish bounds on the capacity region of wiretap secret key generation without the covertness constraint and discuss the impact of covertness. Our results generalize the point-to-point result of Tahmasbi and Bloch (TIFS 2020) to the setting of multiterminal communication.

Capacity Region for Covert Secret Key Generation over Multiple Access Channels

TL;DR

This work introduces covert secret key generation over a binary-input two-user MAC with one-way public discussion, aiming to generate two secret keys with Charlie while remaining undetectable by Willie. The authors derive inner and outer bounds on the CSK capacity region, showing that with covertness the key rates scale as and providing explicit region expressions parameterized by through and . They connect CSK to the wiretap secret key capacity when the covertness constraint is removed and discuss a duality with covert communication over the same MAC, supported by numerical examples. The achievability combines a covert process with a likelihood-encoder-based auxiliary scheme to balance reliability, secrecy, and covertness, while the converse adapts multiterminal Csiszár–Narayan arguments under covertness. The results illuminate the fundamental trade-offs imposed by covertness in multiterminal key generation and pave the way for extensions to continuous alphabets and other multiuser channels.

Abstract

We study covert secret key generation over a binary-input two-user multiple access channel with one-way public discussion and derive bounds on the capacity region. Specifically, in this problem, there are three legitimate parties: Alice, Bob and Charlie. The goal is to allow Charlie to generate a secret key with Alice and another secret key with Bob, reliably, secretly and covertly. Reliability ensures that the key generated by Alice and Charlie is the same and the key generated by Bob and Charlie is the same. Secrecy ensures that the secret keys generated are only known to specific legitimate parties. Covertness ensures that the key generation process is undetectable by a warden Willie. As a corollary of our result, we establish bounds on the capacity region of wiretap secret key generation without the covertness constraint and discuss the impact of covertness. Our results generalize the point-to-point result of Tahmasbi and Bloch (TIFS 2020) to the setting of multiterminal communication.

Paper Structure

This paper contains 21 sections, 9 theorems, 58 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Main Contributions
  3. Organization of the Paper
  4. Problem Formulation
  5. Covert Secret Key Generation
  6. Performance Metric
  7. Result and Discussions
  8. Covert Secret Key Generation
  9. Wiretap Secret Key Generation
  10. Achievability Proof of Theorem \ref{['theorem:main']}
  11. Covert Process
  12. Auxiliary Coding Scheme
  13. Key Generation Scheme
  14. Duality with Covert Communication
  15. Converse Proof of Theorem \ref{['theorem:main']}
  16. ...and 6 more sections

Key Result

Theorem 1

The capacity region $C_{\mathrm{csk}}$ for CSK generation satisfies

Figures (4)

  • Figure 1: System model for covert secret key generation over a two-user multiple access channel.
  • Figure 2: Plot of inner and outer bounds for CSK capacity region $C_{\mathrm{csk}}$ over MACs with channel matrices $W^{(1)}_{Y|X_1X_2}$ and $W^{(1)}_{Z|X_1X_2}$ specified in Table \ref{['tab:parameters']}.
  • Figure 3: System model for wiretap secret key generation over multiple access channels with non-interactive public discussion.
  • Figure 4: System model for the auxiliary coding problem.

Theorems & Definitions (17)

  • Definition 1
  • Remark 1
  • Remark 2
  • Definition 2
  • Definition 3
  • Theorem 1
  • Example 1
  • Example 2
  • Corollary 2
  • Definition 4: Covert Process
  • ...and 7 more