Discrete Memoryless Interference and Broadcast Channels with Confidential Messages: Secrecy Rate Regions

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.Source: §I · p. 1, B. The Interference Channel with… · p. 2, C. The Broadcast Channel · p. 3

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.Source: §I · p. 1, A. Interference Channel with… · p. 3

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.Source: §I · pp. 1–2

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.Source: §I · p. 2, §III · p. 3

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.Source: §III · p. 3

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.Source: A. Interference Channel with… · p. 3

• The outer bound and inner bound for the broadcast channel are unions of rate pairs over distributions in πBC.Source: B. Broadcast Channel with Confidential… · p. 4
• The broadcast-channel outer bound has the same mutual-information expression as the interference-channel outer bound, but uses a different input-distribution class.Source: B. Broadcast Channel with Confidential… · p. 4
• Double binning combines Gel’fand-Pinsker binning with random binning to support joint encoding while preserving confidentiality.Source: B. Broadcast Channel with Confidential… · p. 4

Less noisy broadcast channel

• For less noisy broadcast channels, only the better user can obtain a non-zero secrecy rate under the stated specialization.Source: B. Broadcast Channel with Confidential… · p. 4

• If both stated mutual-information inequalities hold for some πBC distribution, both receivers can achieve strictly positive secrecy rates.Source: B. Broadcast Channel with Confidential… · p. 4

• In the switch channel, each receiver independently chooses between its own and the other transmitter’s signal at each symbol time.Source: C. Switch Channel · p. 4
• The switch state is i.i.d. and available at each receiver, so it can be treated as part of the channel output.Source: C. Switch Channel · p. 5
• Theorem 5 states the switch-channel secrecy capacity region as the union of rate pairs over distributions in πIC−I.Source: C. Switch Channel · p. 5
• When τ1 = τ2 = 1, the switch channel reduces to two independent parallel channels without secrecy constraints.Source: C. Switch Channel · p. 5

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.Source: C. Switch Channel · p. 5

• Transmitter 2 splits its power into message power P2,M and artificial-noise power P2,A.Source: 3) Artificial Noise · p. 6
• Artificial noise can spoil receiver 2’s signal and thereby protect transmitter 1’s confidential message without exchanging confidential messages.Source: 3) Artificial Noise · p. 6
• The artificial noise cannot be predicted or subtracted by either receiver.Source: 3) Artificial Noise · p. 6
• The artificial-noise achievable region is optimized over power-control parameters β1 and β2 and the power-splitting parameter λ.Source: 3) Artificial Noise · p. 6
• Reversing the transmitter roles can further increase the achievable region.Source: 3) Artificial Noise · p. 6

• The artificial-noise strategy supports communication over larger rates than time-sharing and multiplexed transmission in both numerical results.Source: GIC, R[M] · p. 6
• The outer-bound proof begins from reliable transmission and the information-theoretic secrecy constraint.Source: §IV · p. 7
• The derivation bounds equivocation in two ways before introducing a time-sharing random variable Q to obtain a single-letter rate bound.Source: §IV · p. 7, A. First Bound · pp. 7–8
• The first equivocation bound uses Fano’s inequality.Source: A. First Bound · pp. 7–8

• The second bound on R1 uses successful decoding of both confidential messages and therefore applies Fano’s inequalities at both receivers.Source: B. Second Bound · p. 8
• The outer-bound analysis produces individual constraints for R1 and R2 and two forms of sum-rate constraint.Source: C. Outer Bound and Discussion · p. 8
• For the interference channel, the joint distribution factors according to independent channel inputs from the two encoders.Source: C. Outer Bound and Discussion · p. 8
• R1 + R2 ≤ min[∆1 + Θ2, ∆2 + Θ1] is tighter than the separate sum-rate bounds R1 + R2 ≤ Θ1 + Θ2.Source: C. Outer Bound and Discussion · p. 8
• The difference between the two bounds is generally non-zero and equals I(V1; V2|Y2, U) − I(V1; V2|Y1, U).Source: C. Outer Bound and Discussion · p. 8
• The achievable IC-CM construction uses an auxiliary U and one stochastic equivocation codebook for each message.Source: A. Interference Channel with… · pp. 8–9

Error probability

• For any ǫ0 > 0, the total error probability is at most ǫ0 for sufficiently large n when (R1, R2) ∈ RIC(πIC−I).Source: 1) Error Probability Analysis · p. 9

Equivocation

• The secrecy condition for W1 is satisfied by bounding conditional entropy terms through a Markov chain, decoding analysis, and Fano’s inequality.Source: 2) Equivocation Calculation · pp. 9–10
• The analysis concludes that the security conditions for both confidential messages hold as blocklength grows.Source: 2) Equivocation Calculation · p. 10

Code construction

• The BC-CM construction combines Gel’fand-Pinsker binning with random binning in a double-binning scheme.Source: B. Broadcast Channel with Confidential… · p. 10
• A joint encoder generates one equivocation codeword for each message and maps both codewords into the channel input.Source: B. Broadcast Channel with Confidential… · p. 10

Double binning

• Each transmitter codebook is partitioned into message bins, and each message bin is further divided into sub-bins.Source: B. Broadcast Channel with Confidential… · p. 11

Encoding

• To encode, the scheme randomly chooses a sub-bin and selects a jointly typical pair of codewords before generating the channel input.Source: B. Broadcast Channel with Confidential… · p. 11

Decoding

• Decoders identify the transmitted message using unique joint typicality between the corresponding codeword and received sequence.Source: B. Broadcast Channel with Confidential… · p. 11

Encoding

• The double-binning construction requires the randomization rate R† to exceed I(V1; V2|U) for successful jointly typical codeword selection.Source: 1) Error Probability Analysis · p. 11

Error probability

• The receiver-1 decoding condition is 1 + R† < I(V1; Y1|U).Source: 1) Error Probability Analysis · p. 11
• For rate pairs in RBC(πBC), the total error probability is at most ǫ0 for sufficiently large n.Source: 1) Error Probability Analysis · p. 11
• The receiver-2 decoding condition is 2 + R† < I(V2; Y2|U).Source: 1) Error Probability Analysis · p. 11

Secrecy

• The proof establishes the secrecy condition for W1 and, by the same approach, for W2.Source: 2) Equivocation Calculation · pp. 11–12

Conclusion

• The paper concludes that artificial-noise transmission outperforms time-sharing and simultaneous transmission, while practical wiretap-code construction remains challenging.Source: §VI · p. 12

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.Source: C. Outer Bound and Discussion · p. 8, APPENDIX · pp. 12–13

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.Source: APPENDIX · pp. 12–13

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.Source: APPENDIX · p. 13
• 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.Source: APPENDIX · p. 13
• 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.Source: APPENDIX · p. 13