PaperlayerView exact revision on arXiv

Source-linked AI summary

The General Gaussian Multiple Access and Two-Way Wire-Tap Channels: Achievable Rates and Cooperative Jamming

Ender Tekin, Aylin Yener

arXiv:cs/0702112v2cs.ITcs.CR

TL;DR

The paper studies secrecy in Gaussian multiple-access and two-way wire-tap channels, where several users communicate despite an external eavesdropper. It defines a collective secrecy constraint, derives achievable secrecy regions and optimal power allocations, and introduces cooperative jamming. The results show that users can improve one another’s secrecy, with especially strong benefits in the two-way setting and when jamming is needed to overcome a stronger eavesdropper.

  • Problem

    The paper addresses how multiple users can communicate confidentially over Gaussian multiple-access and two-way channels when an external eavesdropper observes their transmissions.

  • Method

    The authors define a collective secrecy constraint, derive achievable secrecy rate regions, optimize secrecy sum-rate power allocations, and introduce cooperative jamming by otherwise non-transmitting users.

  • Results

    The achievable regions and power policies show that users can help one another achieve secrecy, while cooperative jamming increases secrecy rates and can enable communication when ordinary secret communication is not possible.

  • Takeaways & Limitations

    Multiple-access structure can improve system secrecy because users’ signals can hide one another’s messages, with the benefit especially clear in two-way channels.

Abstract

from arXiv · show

The General Gaussian Multiple Access Wire-Tap Channel (GGMAC-WT) and the Gaussian Two-Way Wire-Tap Channel (GTW-WT) are considered. In the GGMAC-WT, multiple users communicate with an intended receiver in the presence of an eavesdropper who receives their signals through another GMAC. In the GTW-WT, two users communicate with each other over a common Gaussian channel, with an eavesdropper listening through a GMAC. A secrecy measure that is suitable for this multi-terminal environment is defined, and achievable secrecy rate regions are found for both channels. For both cases, the power allocations maximizing the achievable secrecy sum-rate are determined. It is seen that the optimum policy may prevent some terminals from transmission in order to preserve the secrecy of the system. Inspired by this construct, a new scheme, \ital{cooperative jamming}, is proposed, where users who are prevented from transmitting according to the secrecy sum-rate maximizing power allocation policy "jam" the eavesdropper, thereby helping the remaining users. This scheme is shown to increase the achievable secrecy sum-rate. Overall, our results show that in multiple-access scenarios, users can help each other to collectively achieve positive secrecy rates. In other words, cooperation among users can be invaluable for achieving secrecy for the system.

I. INTRODUCTION

The paper situates secrecy in Gaussian multiple-access and two-way channels, then formulates GGMAC-WT and GTW-WT models with collective secrecy constraints. It develops achievable secrecy regions and power-allocation strategies, including cooperative jamming to improve secrecy.

  • Research setting: The study considers multiple transmitters using a shared physical channel while an external eavesdropper receives their signals through a Gaussian multiple-access channel.The models represent wireless settings such as ad-hoc networks.
  • System assumptions: The transmitters send separate secret and open messages, with no requirement that open messages be decodable by the eavesdropper.The channel parameters, codebooks, and coding scheme are assumed known to the eavesdropper.
  • Secrecy formulation: The paper uses a collective secrecy constraint based on normalized conditional entropy of the secret messages given the eavesdropper’s signal.The authors state that satisfying this constraint implies secrecy for all users.
  • Main analysis: Achievable secrecy rate regions are derived for both the GGMAC-WT and GTW-WT, extending information-theoretic secrecy analysis to these multi-terminal Gaussian settings.The work builds on prior wire-tap-channel secrecy measures and results for Gaussian and multi-user channels.
  • Power allocation and cooperation: The paper determines sum-secrecy-rate-maximizing power allocations and proposes cooperative jamming by users excluded from transmission under those allocations.The proposed jamming lets disadvantaged users interfere with the eavesdropper and can increase the secrecy rate of transmitting users.

B. The Gaussian Two-Way Wire-Tap Channel

The Gaussian two-way wire-tap channel has two transmitter-receiver pairs communicating over a common channel while an eavesdropper observes their transmissions. Each receiver decodes the other user’s messages, and both users seek reliable communication with secret and open messages.

  • Channel structure: Two transmitter-receiver pairs communicate with each other over a common channel, while the eavesdropper receives their signals through a multiple-access channel.Receiver k decodes the messages transmitted by the other user.
  • Communication objective: Each user aims to communicate open and secret messages with arbitrarily low error probability while maintaining secrecy of the secret messages.The model uses the same power constraints as the multiple-access setting with K = 2.
  • Standardization: The GTW-WT is represented in a standardized form by scaling codewords, maximum powers, transmitter gains, and eavesdropper gains.The listed transformations define the normalized parameters used in the model.

C. Preliminary Definitions

The paper defines collective secrecy for multi-access wire-tap systems and develops achievable GGMAC-WT rate regions using superposition coding, TDMA, and their convex closure. It also characterizes secrecy limitations and shows how stronger or non-transmitting users can assist weaker users by confusing the eavesdropper.

  • Secrecy constraint: Collective secrecy uses the normalized joint conditional entropy of all transmitted messages given the eavesdropper’s received signal.Perfect secrecy of the system implies perfect secrecy for every user group and individual user.
  • User classifications: A user is called strong when h_k ≤ 1 and weak when h_k > 1, indicating whether its single-user secrecy capacity is positive or zero.The paper also defines single-user decodability at the eavesdropper and notes that secrecy cannot be guaranteed for such user groups.
  • Achievable regions: The achievable GGMAC-WT region combines superposition and TDMA regions through convex closure.Superposition uses Gaussian codebooks carrying secret, open, and randomization components, while TDMA schedules users who can achieve single-user secrecy.
  • Achievability: Theorem 1 states that the constructed rate region is achievable for the GGMAC-WT, beginning with superposition encoding for a fixed power allocation.The proof then establishes achievability of the superposition region before incorporating the broader construction.
  • Limitations: If the relevant secrecy expression is zero for a user group, the scheme cannot achieve secrecy for that group; if the main-channel sum-capacity is below the eavesdropper’s, system secrecy is impossible.The paper assumes the relevant quantity is positive before proceeding with its construction.
  • Cooperation and rate trade-offs: Stronger users can help weaker users by contributing extra randomness, provided the weak user is not single-user decodable at the eavesdropper.The stronger users may sacrifice some of their own rate, and codewords that confuse the eavesdropper can still carry meaningful information to the intended receiver.

B. The Gaussian Two-Way Wire-Tap Channel

The GTW-WT admits an achievable superposition region whose secrecy structure resembles the two-user GGMAC-WT, while the two-way main channel supports larger secrecy regions through self-interference cancellation and mutual codeword assistance.

  • The GTW-WT achievable rate region is constructed using a superposition-coding approach analogous to that used for the GGMAC-WT.
  • Figure 6 shows an achievable secrecy region parameterized by transmit powers, with higher powers producing a larger region.
  • Each transmitter subtracts its own codeword, decomposing the Gaussian two-way channel into two parallel channels.
  • As an eavesdropper-facing channel, the GTW-WT remains a two-user GMAC requiring extra shared codewords when users are not single-user decodable.
  • Mutual codeword knowledge acts as a secret key between legitimate users, reducing the extraneous codewords needed to confuse the eavesdropper.
  • Two-way transmission provides an advantage over the corresponding single-user channels, while TDMA does not enlarge the secrecy region.

IV. MAXIMIZATION OF SUM RATE

The sum-rate maximization problem selects transmit powers based on standardized eavesdropper gains, ordered from weakest to strongest.

  • The achievable regions depend on transmit powers, motivating optimization of the power allocation that maximizes total secrecy sum-rate.
  • Users are ordered by increasing standardized eavesdropper channel gain, with higher h_k indicating a better eavesdropper channel.

A. GGMAC-WT

For the GGMAC-WT, secrecy sum-rate maximization yields a threshold structure in which only an initial subset of users transmits, with the remaining users assigned zero power.

  • The GGMAC-WT secrecy sum-rate maximizing allocation has a limiting user T, with P*_k = 0 for users k > T.
  • Only a subset of the ordered users transmits, determined by threshold conditions involving h_T and the preceding users’ powers.
  • The optimal transmitting set consists of an initial ordered segment because if a user transmits, users with smaller eavesdropper gains must also transmit.
  • For two users, the allocation is either full power for both, full power for user 1 only, or zero power for both, depending on the channel gains.
  • The achievable secrecy sum-rate is the maximum of the superposition and TDMA-region solutions, and users with h_k ≥ 1 should not transmit under TDMA.

B. GTW-WT

For the GTW-WT, the secrecy sum-rate maximizing allocation has a two-user threshold form and permits secrecy over a broader range of eavesdropper channel gains than the GGMAC-WT.

  • The allocation is derived by optimizing the GTW-WT secrecy sum-rate using a Lagrangian and derivative conditions.
  • The GTW-WT optimum allocates full power to both users, full power to user 1 only, or zero power to both according to channel-gain conditions.
  • If a user is single-user decodable by the eavesdropper, it cannot transmit with non-zero secrecy and only reduces the secrecy sum-rate.
  • Compared with the GGMAC-WT allocation, the GTW-WT permits transmission over a larger range of channel gains, even when the eavesdropper channel is not very weak.

V. SECRECY THROUGH COOPERATIVE JAMMING

The cooperative-jamming scheme reallocates users excluded by secrecy-optimal transmission policies into jammers that impair the eavesdropper. It applies to the superposition region, where weaker users can improve secrecy by jamming.

  • V. SECRECY THROUGH COOPERATIVE JAMMING: Users whose secrecy-optimal transmit powers are zero can instead jam the eavesdropper to help transmitting users.The scheme partitions users into transmitting and jamming sets; jamming users transmit Gaussian noise rather than codewords.
  • V. SECRECY THROUGH COOPERATIVE JAMMING: The analysis restricts cooperative jamming to the superposition region because TDMA gives each user a dedicated time slot.
  • V. SECRECY THROUGH COOPERATIVE JAMMING: For the GTW-WT, receivers know the jamming sequence, so jamming harms the eavesdropper without harming intended receivers.This yields a higher secrecy sum-rate advantage than in the GGMAC-WT.
  • V. SECRECY THROUGH COOPERATIVE JAMMING: Each user is assumed either to transmit information or to jam, because splitting its power between both actions is suboptimal for secrecy sum-rate maximization.

A. GGMAC-WT

For the GGMAC-WT, secrecy-sum-rate maximization selects transmitting and jamming users according to standardized eavesdropper gains. Weaker users can jam when doing so improves the effective channel for transmitting users.

  • A. GGMAC-WT: Power splitting between transmission and jamming is suboptimal: a user should allocate its power entirely to the action favored by its channel gain.
  • A. GGMAC-WT: A nonzero secrecy sum-rate requires φK(P) ≤ φTc(P), while cooperative jamming improves over no jamming only when φTc(P) > 1.
  • A. GGMAC-WT: Transmitting users with positive power use maximum power when hj < φK(P*), while jamming users satisfy hj ≥ φK(P*).
  • A. GGMAC-WT: The optimal allocation separates users into full-power transmitters, silent users, and jammers, with the jamming boundary determined by hJ = ΦT(P*).The candidate transmitting sets can be reduced to K(K − 1) possibilities for exhaustive comparison.
  • A. GGMAC-WT: For two users, the weaker user should jam when it is not single-user decodable and has enough power to make the other user strong in the effective channel.

B. GTW-WT

In the GTW-WT, cooperative jamming is optimized by assigning users either to transmit or to jam at maximum power. Because the communicating receiver knows the jamming sequence, jamming does not reduce the intended transmission rate.

  • B. GTW-WT: A user is assigned exclusively to transmission or jamming without capacity loss because the jamming user is also the receiver and knows the transmitted signal.
  • B. GTW-WT: A GTW-WT jamming user transmits at maximum power, and the possible jamming sets for two users are ∅, {1}, or {2}.
  • B. GTW-WT: Jamming is advantageous for a user j only when hj > 1, provided the jamming enables the other user to transmit securely.
  • B. GTW-WT: The weaker user is sufficient as jammer when h2 P̄2 > h1 P̄1, a condition corresponding to higher eavesdropper SNR for that user.
  • B. GTW-WT: The two-user allocation has both users transmit when h1 < h2 ≤ 1, while user 1 transmits and user 2 jams in the stated higher-gain regimes.

VI. NUMERICAL RESULTS

Numerical results illustrate secrecy-rate gains from cooperative jamming and show a larger secrecy region and broader protection for the GTW-WT than for the GGMAC-WT. The results also identify channel and power regimes governing these gains.

  • VI. NUMERICAL RESULTS: The GTW-WT achieves a larger secrecy-rate region than the GGMAC-WT and offers more protection to weak users.
  • VI. NUMERICAL RESULTS: Figures 7 and 8 show GGMAC-WT secrecy-rate improvement when user 2 jams with varying power and channel parameters.The plots correspond to user 1’s single-user secrecy capacity because only user 1 transmits.
  • VI. NUMERICAL RESULTS: For the GGMAC-WT, secrecy is zero when h1 ≥ 1 unless user 2 has enough power to reduce user 1’s re-standardized gain below 1.
  • VI. NUMERICAL RESULTS: In the GTW-WT, jamming is always optimal for user 2 when it enables user 1 to transmit, because the receiver knows the jamming sequence.
  • VI. NUMERICAL RESULTS: GTW-WT jamming supports secrecy over a much larger eavesdropper-location area because the jamming signal does not hurt the intended receiver.

VII. CONCLUSIONS AND FUTURE WORK

The paper shows that multiple-access structure can improve secrecy through achievable rates, cooperative jamming, and users’ shared randomness. It also identifies important limits: secrecy-capacity regions remain open, practical codes are unavailable, and channel-parameter estimates are needed for design.

  • Achievable secrecy rates were established for both the GGMAC-WT and GTW-WT, showing that multiple-access structure can improve system secrecy.
  • Users with zero single-user wire-tap capacity may still communicate securely when the eavesdropper is disadvantaged overall by the users’ combined randomness.
  • Perfect secrecy in the two-way setting remains possible even when the eavesdropper’s channel gain is stronger, provided single-user decoding is unavailable.
  • Cooperative jamming lets a disadvantaged user jam the eavesdropper, producing potentially significant gains when the eavesdropper has much higher SNR than the receivers.
  • The secrecy-capacity regions remain open, and achievable secrecy sum-rates coincide with the upper bound only in the degraded GGMAC-WT case.
  • The results are mainly theoretical because practical multi-access wire-tap codes are not known, while code design also requires accurate eavesdropper-channel estimates.
Loading cs/0702112v2…