A New Achievable Region of the $K$-User MAC Wiretap Channel with Confidential and Open Messages Under Strong Secrecy

TL;DR

The paper studies a $K$-user DM MAC-WT channel where users transmit both secret and open messages to Bob while Eve targets the secrets. It introduces a new achievable region by allowing auxiliary messages and by partitioning users into a secrecy-enabled subset ${\\cal K}'$ and its complement, then eliminating auxiliary rates via a Fourier–Motzkin-based induction that scales with $K$. The main contributions are a resolvability-based strong secrecy achievability proof, the demonstration that zero-secret users can play multiple nonredundant roles, and a convex-hull construction over all ${\\cal K}'$ that strictly improves prior results (e.g., xu2022achievable). The two-user binary-input real adder channel serves as a concrete example where the new region strictly enlarges the previous region, underscoring the practical impact on spectral efficiency and security. The work advances the understanding of open-message roles in MAC-WT settings and lays groundwork for future outer-bounding and capacity-characterization efforts under strong secrecy.

Abstract

This paper investigates the achievable region of a $K$-user discrete memoryless (DM) multiple access wiretap (MAC-WT) channel, where each user transmits both secret and open messages. All these messages are intended for Bob, while Eve is only interested in the secret messages. In the achievable coding strategy, the confidential information is protected by open messages and also by the introduction of auxiliary messages. When introducing an auxiliary message, one has to ensure that, on one hand, its rate is large enough for protecting the secret message from Eve and, on the other hand, the resulting sum rate (together with the secret and open message rate) does not exceed Bob's decoding capability. This yields an inequality structure involving the rates of all users' secret, open, and auxiliary messages. To obtain the rate region, the auxiliary message rates must be eliminated from the system of inequalities. A direct application of the Fourier-Motzkin elimination procedure is elusive since a) it requires that the number of users $K$ is explicitly given, and b) even for small $K = 3, 4, \ldots$, the number of inequalities becomes extremely large. We prove the result for general $K$ through the combined use of Fourier-Motzkin elimination procedure and mathematical induction. This paper adopts the strong secrecy metric, characterized by information leakage. To prove the achievability under this criterion, we analyze the resolvability region of a $K$-user DM-MAC channel. In addition, we show that users with zero secrecy rate can play different roles and use different strategies in encoding their messages. These strategies yield non-redundant rate inequalities. By considering all possible coding strategies, we provide a new achievable region for the considered channel, and show that it strictly improves those already known in the existing literature by considering a specific example.

Figures (15)

• Figure 1: Block diagram of a $K$-user DM MAC-WT channel with secret (i.e., confidential) and open (i.e., non-confidential) messages. Both secret and open messages are intended for Bob, while Eve is interested only in the secret messages.
• Figure 2: A division of subcodebook ${\mathpzc c}_k (m_k^{\text{s}})$ of user $k \in {\cal K}'$.
• Figure 3: Codebook ${\mathpzc c}_k$ of user $k \in {\cal K}'$.
• Figure 4: Illustration of a two-user binary-input real adder channel.
• Figure 5: Achievable region ${\hat{\mathscr R}} (X_{\cal K}, \{1, 2\})$ in different cases.

Theorems & Definitions (11)

• Definition 1
• Theorem 1
• Lemma 1
• Remark 1
• Theorem 2
• Theorem 3
• Remark 2
• Lemma 2
• Remark 3
• Theorem 4