Secrecy in Cooperative Relay Broadcast Channels

E. Ekrem; S. Ulukus

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Secrecy in Cooperative Relay Broadcast Channels

E. Ekrem, S. Ulukus

arXiv:0811.1317v1Nov 9, 2008cs.IT

TL;DR

The paper asks how user cooperation affects simultaneous secrecy in broadcast channels. It combines Marton coding with compress-and-forward relaying and derives single-letter outer bounds. It shows that cooperation can enlarge the secrecy region, including positive secrecy rates for both users in a Gaussian CRBC, while the achievable secrecy of the helped user is zero under a specified degradedness condition.

Problem
The paper studies whether cooperation can improve simultaneous individual confidentiality in broadcast channels, where each receiver must keep its message secret from the other.
Method
The paper combines Marton’s coding scheme for broadcast channels with Cover and El Gamal’s compress-and-forward scheme for relay channels, and uses auxiliary random variables to derive single-letter outer bounds.
Results
User cooperation can enlarge the secrecy region, and both users can have positive secrecy rates in a Gaussian CRBC; compress-and-forward helps without requiring the cooperating user to decode the message.
Takeaways & Limitations
Cooperation can improve secrecy even when the cooperating party is untrusted, provided the cooperation method does not require that party to decode the confidential message.
Takeaways & Limitations
For the achievable scheme considered, if user 2’s channel is degraded with respect to user 1’s, user 2’s equivocation rate is zero; the paper also lacks an input-output-only outer bound for user 1.

Abstract

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We investigate the effects of user cooperation on the secrecy of broadcast channels by considering a cooperative relay broadcast channel. We show that user cooperation can increase the achievable secrecy region. We propose an achievable scheme that combines Marton's coding scheme for broadcast channels and Cover and El Gamal's compress-and-forward scheme for relay channels. We derive outer bounds for the rate-equivocation region using auxiliary random variables for single-letterization. Finally, we consider a Gaussian channel and show that both users can have positive secrecy rates, which is not possible for scalar Gaussian broadcast channels without cooperation.

1 Introduction

The paper studies how cooperation changes secrecy in broadcast channels using the cooperative relay broadcast channel, with single- and two-sided cooperative links. It develops achievable schemes and outer bounds, and shows Gaussian cooperation can produce positive secrecy rates for both users.

Problem: The paper investigates simultaneous individual confidentiality in a two-user cooperative relay broadcast channel with single-sided or two-sided cooperation.Removing cooperation or setting one user’s rate to zero recovers established confidential broadcast, relay, and eavesdropper channel models.
Method: The proposed achievable scheme combines Marton’s broadcast-channel coding with Cover and El Gamal’s compress-and-forward relaying.User 1 sends a quantized version of its observation to help user 2 decode its message.
Bounds: The paper derives single-letter outer bounds for the rate-equivocation region using suitable auxiliary random variables.For the helped user, it also develops an outer bound depending only on channel inputs and outputs.
Gaussian result: In the Gaussian CRBC, both users can achieve positive secrecy rates through cooperation, unlike the corresponding scalar Gaussian broadcast channel without cooperation.The paper uses auxiliary-variable assignments with dirty-paper-coding interpretations and combines jamming with relaying when the relaying user is weak.

4 An Outer Bound

The paper develops auxiliary-variable and input-output outer bounds for the CRBC rate-equivocation region, with degraded channels forcing user 2’s equivocation rate to zero.

Auxiliary-variable outer bound: Theorem 2 gives an outer bound on the CRBC rate-equivocation region using auxiliary random variables and a specified Markov-chain constraint.Its equivocation bounds use mutual-information differences involving V1, V2, and U.
Auxiliary-variable outer bound: Theorem 2’s equivocation expressions match those for broadcast channels with per-user secrecy constraints, but its broader Markov-chain class yields a larger region.The larger region is consistent with the CRBC achievable region containing the broadcast-channel achievable region.
Input-output outer bound: A second outer bound for user 2 uses only channel inputs and outputs, without auxiliary random variables.The paper presents this as a simpler bound alongside the auxiliary-variable outer bound.
Degraded channels: When the channel is degraded according to X →(X1, Y1) →Y2, user 2’s equivocation rate is zero.The conclusion follows from the corresponding Markov chain.
Input-output outer bound: The input-output outer bound is generally loose because it assumes user 2 completely accesses user 1’s observation, but it can become close when the cooperative link is strong.A strong link may let user 1 convey its observation precisely, making the bound close to the CAF-achievable rate.
Input-output outer bound: A simple input-output outer bound for user 1 is difficult because user 1 can encode X1 from Y1, creating across-time input-output correlations.The paper states that accounting for this correlation requires auxiliary variables.

5 An Example: Gaussian CRBC

The Gaussian CRBC example evaluates independent-input and DPC-based achievable schemes under a stronger user-1 channel. Cooperation enables positive secrecy rates for both users, with a trade-off between them.

Independent inputs: Proposition 1 achieves an equivocation region using independent Gaussian auxiliary inputs V1 and V2.The construction also compresses user 1’s observation for relay transmission.
Correlated inputs and DPC: Proposition 2 enlarges the achievable region by correlating V1 and V2, with a dirty-paper-coding interpretation.The transmitter treats user 2’s signal as non-causally known interference when encoding for user 1.
Cooperation effects: As the cooperative link improves, user 2’s achievable secrecy increases because equivocation is monotonically decreasing in compression noise Nc.The evaluated regions use P = 8, N1 = 1, and N2 = 2.
Cooperation effects: Both users achieve positive secrecy rates through cooperation, although increasing user 2’s secrecy reduces user 1’s secrecy.Without cooperation, user 2 has no positive secrecy rate for these channel parameters.
Asymptotic behavior: In the large-a limit, the maximum achievable equivocation rate for user 2 approaches the outer-bound limit.The paper attributes the limiting behavior to the transmitter-to-user-1 link.

6 Joint Jamming and Relaying

When the cooperating user is weaker, combining jamming with relaying can provide positive secrecy rates for both users. Gaussian constructions use independent or DPC-related auxiliary-variable assignments and show how jamming changes the achievable equivocation region.

Joint Jamming and Relaying: User 1 can combine jamming and relaying to provide both users with positive secrecy rates when it is the weaker user.Without cooperation, user 1 cannot have a positive secrecy rate in this setting.
Joint Jamming and Relaying: The achievable rate region is characterized by Theorem 4 under a distribution involving V1, V2, U, X, X1, and a compressed observation ˆY1.The theorem imposes a sum-rate bound involving I(V1; Y1|X1), I(V2; Y2, ˆY1|U), and I(V1; V2).
Joint Jamming and Relaying: User 1 sends a help signal U while X1 may include additional jamming that user 2 cannot fully decode and remove.This residual jamming can make decoding V1 at user 2 impossible, supporting user 1’s secrecy.
Gaussian Example: In the Gaussian construction, user 1 splits X1 into a jamming-noise component and a compressed-observation bin index, then sends a compressed version of Y1 to user 2.With sufficiently large power, the jamming makes user 2’s channel noisier than user 1’s, enabling positive secrecy rates for both users.
Gaussian Example: For P = 8, N1 = 2, and N2 = 1, Propositions 3 and 4 demonstrate achievable equivocation regions when user 1 jams and relays.The independent and correlated auxiliary-variable cases correspond to Figures 5 and 6; the latter admits a DPC interpretation.
Gaussian Example: For N1 = 1 and N2 = 2, the same propositions also apply when user 1 is stronger, and jamming substantially improves its maximum secrecy rate while preserving user 2’s maximum.At one reported operating point, user 1 reaches about 1.58 bits/channel use by using all its power to jam user 2.

8 Two-sided Cooperation

The paper develops an achievable region for a CRBC with two-sided cooperation, where each receiver can relay information to the other while maintaining individual secrecy.

Two-sided cooperation lets each user act as a relay for the other in a memoryless CRBC with two message sets and three channel inputs.
The achievable region is characterized by rate quadruples satisfying the conditions of Theorem 5.
The sum-rate constraint combines both users’ decoded outputs and reconstructed observations, conditioned on their relay variables, while subtracting the auxiliary-message dependence.
The construction uses auxiliaries, compressed observations, and a joint distribution coupling messages, channel inputs, relay inputs, and reconstructed outputs.
In the two-sided scheme, users cannot remove their own codewords before compression because sliding-window decoding delays recovery until the next block.

9 Gaussian Example for Two-sided Cooperation

The Gaussian example instantiates the two-sided achievable scheme under power and noise constraints and evaluates its secrecy region using jointly optimized jamming, relaying, and compression parameters.

Proposition 5 gives achievable equivocation rates for all (α, β1, β2) ∈ [0, 1]^3 under the Gaussian two-sided CRBC model.
The compression-noise variances Nc,1 and Nc,2 are constrained alongside complementary parameters ¯α, ¯β1, and ¯β2.
The Gaussian construction uses independent transmitter components V1 and V2, relay inputs Ui plus noise, and compressed observations with independent Gaussian compression noise.
P = 8, N1 = 1, and N2 = 2 define the numerical example used for the achievable equivocation region in Figure 9.
Two-sided cooperation significantly improves user 2’s secrecy through jamming and improves user 1’s secrecy through relaying, although user 1’s additional gain is modest when it uses all power for jamming.

10 Conclusions

The conclusions show that cooperation can increase secrecy, but the effect depends on how cooperation is implemented.

User cooperation can increase secrecy even when the cooperating party is untrusted.
Decode-and-forward can increase communication rate without improving secrecy because the cooperating eavesdropper must decode the forwarded message.
Compress-and-forward avoids requiring the cooperating party to decode the message, allowing it to assist rates beyond what it can itself decode.

A Proof of Theorem 2

The converse derives outer bounds on the CRBC rate-equivocation region and converts the resulting blocklength-n inequalities into single-letter expressions using auxiliary variables and time sharing.

The proof establishes outer bounds on the CRBC capacity-equivocation region through auxiliary random variables and Markov-chain relationships.
Fano’s lemma supplies the reliability terms in the bounds for both users’ achievable rates and equivocation rates.
Conditioning cannot increase entropy, and definitions of Ui, V1,i, and V2,i convert intermediate information expressions into bounds involving these auxiliaries.
The converse treats user 1 and user 2 symmetrically when deriving their rate and equivocation bounds, while accounting for the other user’s observations and channel inputs.
A uniformly distributed time-sharing variable J selects one channel use, yielding U, V1, V2, X, X1, Y1, and Y2 and completing single-letterization.

B Proof of Theorem 3

The proof derives a single-letter outer-bound expression by applying entropy and mutual-information inequalities, then introduces a uniformly distributed time index to complete single-letterization.

Proof of Theorem 3: The proof bounds multi-letter information quantities using Fano’s lemma, conditioning inequalities, independence, and Markov-chain properties.These steps successively produce the displayed inequalities leading to the outer bound.
Proof of Theorem 3: A uniformly distributed random variable J selects one channel use, yielding X = X_J and corresponding single-letter variables.The construction converts the n-letter expression into the single-letter expression in Theorem 3.

C Proof of Corollary 1

The proof establishes achievability of the Gaussian secrecy rate and derives an outer bound using entropy inequalities, Gaussian maximum-entropy properties, and the input power constraint.

Proof of Corollary 1: Letting a tend to infinity makes the secrecy rate in (49) achievable under Propositions 1 and 2.The section also notes that H(·) denotes differential entropy.
Proof of Corollary 1: The outer-bound derivation uses independent Gaussian noises, conditioning inequalities, Gaussian entropy maximization under a power constraint, and the power constraint on X.The resulting bound is valid for every α before selecting α to recover (49).

D Proof of Theorem 4

The proof constructs and analyzes a Marton-plus-compress-and-forward scheme for Theorem 4, including typicality decoding, list decoding, and equivocation bounds for both users.

Code construction: The transmitter combines Marton joint encoding with a compress-and-forward relay scheme at user 1, while user 2 uses list decoding to recover the relay description.The construction uses auxiliary codewords, compressed observations, and randomized relay codewords.
Decoding: User 1 decodes a typical codeword pair and compressed observation, while user 2 first decodes its auxiliary sequence and then uses list decoding with a cell intersection.The intersection is required to be unique and correct, producing the compression constraint in (58).
Equivocation computation: For user 2, the achievable equivocation lower bound is nI(V2; Y2, ˆY1|U) − nI(V2; Y1, V1|X1) − nǫ_n.The final term involving conditional entropy is driven to zero using side information and Fano’s lemma.
Other cases: Other rate cases are handled by reducing the total number of codewords while repeating the same equivocation argument.For example, user 1’s codeword rate is selected as R(V1) = R1 + I(V1; Y2, ˆY1|V2,U) + I(V1; V2).

E Proof of Theorem 5

The proof of Theorem 5 extends the cooperative construction symmetrically to both users, using separate relay auxiliaries, compressions, randomized help signals, list decoding, and equivocation bounds.

Code construction: Each user generates auxiliary sequences, compressed observations, and randomized relay inputs conditioned on its own cooperation variable.The product distribution separates the two users’ cooperation components while retaining the joint Marton variables.
Encoding and decoding: Each relay transmits a help codeword when its previous compressed observation maps into the corresponding cell, with randomization intended to confuse the other user.The decoding analysis is presented for user 1 and completed for user 2 by symmetry.
Decoding: User 1 decodes the other user’s cooperation variable, forms its own compressed-observation estimate, and list-decodes the other relay’s compression before decoding its auxiliary message.The list is intersected with a cell to obtain a unique compression index.
Equivocation computation: When R1 satisfies the stated case condition, the equivocation analysis yields nI(V1; Y1, ˆY2|X1, U2) − nI(V1; Y2, ˆY1, V2|X2, U1) − nǫ_n.Independence then rewrites the leakage term as a conditional mutual-information term plus I(V1; V2).
Other cases: The complementary rate case changes the total codeword count to R(V1) = R1 + I(V1; Y2, ˆY1|X2, V2, U1) + I(V1; V2).The subsequent equivocation calculation follows the same steps as the first case.