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Discrete Memoryless Interference and Broadcast Channels with Confidential Messages: Secrecy Rate Regions
Ruoheng Liu, Ivana Maric, Predrag Spasojevic, Roy D. Yates
TL;DR
The paper addresses secure transmission of independent confidential messages over discrete memoryless interference and broadcast channels. It derives outer and inner secrecy-rate bounds using mutual-information analysis and random binning, including joint encoding for the broadcast channel. The bounds meet for the switch channel, while Gaussian schemes using artificial noise outperform time-sharing and simple multiplexing.
Problem
The paper studies how to transmit independent messages to two receivers while keeping each receiver ignorant of the other receiver’s message.
Method
The paper derives mutual-information outer bounds and random-binning inner bounds, using joint encoding for broadcast channels and stochastic encoders for interference channels.
Results
The bounds meet for the switch channel, and an artificial-noise Gaussian scheme outperforms time-sharing and simple multiplexing.
Takeaways & Limitations
The results provide secrecy-rate regions for both channel models and identify a switch-channel case with a characterized capacity region.
Abstract
from arXiv · showhide
We study information-theoretic security for discrete memoryless interference and broadcast channels with independent confidential messages sent to two receivers. Confidential messages are transmitted to their respective receivers with information-theoretic secrecy. That is, each receiver is kept in total ignorance with respect to the message intended for the other receiver. The secrecy level is measured by the equivocation rate at the eavesdropping receiver. In this paper, we present inner and outer bounds on secrecy capacity regions for these two communication systems. The derived outer bounds have an identical mutual information expression that applies to both channel models. The difference is in the input distributions over which the expression is optimized. The inner bound rate regions are achieved by random binning techniques. For the broadcast channel, a double-binning coding scheme allows for both joint encoding and preserving of confidentiality. Furthermore, we show that, for a special case of the interference channel, referred to as the switch channel, the two bound bounds meet. Finally, we describe several transmission schemes for Gaussian interference channels and derive their achievable rate regions while ensuring mutual information-theoretic secrecy. An encoding scheme in which transmitters dedicate some of their power to create artificial noise is proposed and shown to outperform both time-sharing and simple multiplexed transmission of the confidential messages.
I. INTRODUCTION
The paper studies information-theoretically secure interference and broadcast channels with independent confidential messages, deriving bounds and achievable schemes for both models. It also identifies a capacity-achieving switch-channel case and Gaussian schemes using artificial noise.
- Problem setting: The study considers discrete memoryless interference and broadcast channels where each receiver decodes its own message while remaining ignorant of the other receiver’s message.Secrecy is measured through equivocation at the eavesdropping receiver.
- Bounds: The paper derives outer bounds with a common mutual information expression, optimized over independent inputs for interference channels and joint encoding distributions for broadcast channels.The distributional distinction reflects independent transmitters versus a single jointly encoding transmitter.
- Achievable schemes: For the broadcast channel, double binning supports joint precoding while preserving confidentiality, whereas secrecy rules out partial decoding and classical rate splitting for the interference channel.The encoders therefore use stochastic encoding rather than the rate-splitting approach used for classical interference channels.
- Gaussian schemes: For Gaussian interference channels, artificial-noise transmission schemes achieve secrecy rate regions and outperform time-sharing and simple multiplexing.Transmitters dedicate part of their power to artificial noise that neither receiver can predict and subtract.
- Special case: The outer and inner bounds meet for the switch channel, yielding its capacity region.This special case is identified as an interference channel with confidential messages.
- Achievable schemes: Achievable inner bounds use random binning; the interference-channel construction employs an auxiliary U and independent stochastic encoders with secrecy penalty terms.The rate for one confidential message includes I(V1; Y2|V2, U) as an eavesdropper-channel penalty.
B. Broadcast Channel with Confidential Messages
The broadcast-channel results provide outer and inner secrecy-rate bounds optimized over a specified class of input distributions. Joint encoding uses double binning, and under stated conditions both receivers can achieve positive confidential rates.
- The outer bound and inner bound for the broadcast channel are unions of rate pairs over distributions in πBC.
- The broadcast-channel outer bound has the same mutual-information expression as the interference-channel outer bound, but uses a different input-distribution class.
- Double binning combines Gel’fand-Pinsker binning with random binning to support joint encoding while preserving confidentiality.
- Less noisy broadcast channel: For less noisy broadcast channels, only the better user can obtain a non-zero secrecy rate under the stated specialization.
- If both stated mutual-information inequalities hold for some πBC distribution, both receivers can achieve strictly positive secrecy rates.
C. Switch Channel
The switch channel limits each receiver to one of two transmissions at each symbol time, with switching probabilities governing desired reception and interception. Its secrecy capacity region is characterized by a theorem, and the bounds meet for this special interference-channel case.
- In the switch channel, each receiver independently chooses between its own and the other transmitter’s signal at each symbol time.
- The switch state is i.i.d. and available at each receiver, so it can be treated as part of the channel output.
- Theorem 5 states the switch-channel secrecy capacity region as the union of rate pairs over distributions in πIC−I.
- When τ1 = τ2 = 1, the switch channel reduces to two independent parallel channels without secrecy constraints.
- Noiseless memoryless switch channel: For the noiseless memoryless switch channel, the capacity region applies under τ1 + τ2 ≥1, with τ1 + τ2 −1 measuring transmission unseen by the other receiver.
3) Artificial Noise:
The artificial-noise scheme splits one transmitter’s power between its confidential message and artificial noise. The resulting achievable region can be enlarged by reversing transmitter roles.
- Transmitter 2 splits its power into message power P2,M and artificial-noise power P2,A.
- Artificial noise can spoil receiver 2’s signal and thereby protect transmitter 1’s confidential message without exchanging confidential messages.
- The artificial noise cannot be predicted or subtracted by either receiver.
- The artificial-noise achievable region is optimized over power-control parameters β1 and β2 and the power-splitting parameter λ.
- Reversing the transmitter roles can further increase the achievable region.
GIC, R[M]
The Gaussian interference-channel analysis compares achievable secrecy regions through numerical examples and develops outer bounds from reliability and secrecy constraints. The supplied passages emphasize the proof framework rather than the full bound expressions.
- The artificial-noise strategy supports communication over larger rates than time-sharing and multiplexed transmission in both numerical results.
- The outer-bound proof begins from reliable transmission and the information-theoretic secrecy constraint.
- The derivation bounds equivocation in two ways before introducing a time-sharing random variable Q to obtain a single-letter rate bound.
- The first equivocation bound uses Fano’s inequality.
B. Second Bound
The second outer-bound approach combines equivocation bounds derived using Fano’s inequalities and yields rate and sum-rate constraints for the interference channel.
- The second bound on R1 uses successful decoding of both confidential messages and therefore applies Fano’s inequalities at both receivers.
- The outer-bound analysis produces individual constraints for R1 and R2 and two forms of sum-rate constraint.
- For the interference channel, the joint distribution factors according to independent channel inputs from the two encoders.
- R1 + R2 ≤ min[∆1 + Θ2, ∆2 + Θ1] is tighter than the separate sum-rate bounds R1 + R2 ≤ Θ1 + Θ2.
- The difference between the two bounds is generally non-zero and equals I(V1; V2|Y2, U) − I(V1; V2|Y1, U).
- The achievable IC-CM construction uses an auxiliary U and one stochastic equivocation codebook for each message.
1) Error Probability Analysis:
The interference-channel construction uses stochastic encoders and typicality decoding, then establishes vanishing error probability and the secrecy conditions for achievable rate pairs.
- Error probability: For any ǫ0 > 0, the total error probability is at most ǫ0 for sufficiently large n when (R1, R2) ∈ RIC(πIC−I).
- Equivocation: The secrecy condition for W1 is satisfied by bounding conditional entropy terms through a Markov chain, decoding analysis, and Fano’s inequality.
- Equivocation: The analysis concludes that the security conditions for both confidential messages hold as blocklength grows.
B. Broadcast Channel with Confidential Messages
The broadcast-channel scheme uses a joint encoder with two equivocation codewords and double binning to support confidential-message transmission.
- Code construction: The BC-CM construction combines Gel’fand-Pinsker binning with random binning in a double-binning scheme.
- Code construction: A joint encoder generates one equivocation codeword for each message and maps both codewords into the channel input.
- Double binning: Each transmitter codebook is partitioned into message bins, and each message bin is further divided into sub-bins.
- Encoding: To encode, the scheme randomly chooses a sub-bin and selects a jointly typical pair of codewords before generating the channel input.
- Decoding: Decoders identify the transmitted message using unique joint typicality between the corresponding codeword and received sequence.
1) Error Probability Analysis:
The broadcast-channel analysis shows that the double-binning encoder succeeds with high probability, decoding errors vanish for the achievable region, and both secrecy requirements hold.
- Encoding: The double-binning construction requires the randomization rate R† to exceed I(V1; V2|U) for successful jointly typical codeword selection.
- Error probability: The receiver-1 decoding condition is 1 + R† < I(V1; Y1|U).
- Error probability: For rate pairs in RBC(πBC), the total error probability is at most ǫ0 for sufficiently large n.
- Error probability: The receiver-2 decoding condition is 2 + R† < I(V2; Y2|U).
- Secrecy: The proof establishes the secrecy condition for W1 and, by the same approach, for W2.
- Conclusion: The paper concludes that artificial-noise transmission outperforms time-sharing and simultaneous transmission, while practical wiretap-code construction remains challenging.
APPENDIX
The appendix establishes an error-bound result through typicality arguments and proves that the switch-channel outer and inner bounds coincide under specified conditions.
- Switch-channel result: The proof combines intermediate inequalities involving ∆1, ∆2, Θ1, and Θ2, including min[∆1 + Θ2, ∆2 + Θ1] ≤ ∆1 + ∆2 = Θ1 + Θ2.These inequalities support the equality needed to complete the switch-channel argument.
- Error analysis: Typical-sequence and joint-typicality arguments bound decoding-error probabilities, with sufficiently small auxiliary error terms for large blocklength.The proof introduces typical sets, indicator functions, and conditional mutual-information expansions before combining intermediate bounds.
- Switch-channel result: The switch channel is treated as a special interference-channel case, reducing the comparison to the corresponding outer and inner bounds.The proof focuses on bounds (9) and (10) for the SC-CM case.
- Switch-channel result: For the switch channel, the outer bound meets the inner bound when the required conditions hold.The argument first addresses auxiliary-variable independence and then verifies the relevant equalities and conditions in the outer bound.
- Switch-channel result: Functional dependence in the switch model yields conditional independence relations used to establish the required equalities.Because each switch output depends only on the selected channel input, the proof derives I(W1; W2|Ui) = 0 and obtains the desired result.