Dispensing with Noise Forward in the "Weak" Relay-Eavesdropper Channel

TL;DR

This paper introduces a new decoding scheme for the weak relay-eavesdropper channel that reduces delay and simplifies analysis by eliminating noise forwarding, while optimizing secrecy rates through tailored codebook and bin size choices.

Contribution

It proposes a novel sliding window decoding method with reduced delay that does not rely on noise forwarding, enhancing practical applicability and analytical tractability.

Findings

Reduced decoding delay with sliding window scheme

Maximized secrecy rate through optimized codebook and bin sizes

Multiblock equivocation calculations for security analysis

Abstract

The "weak" relay-eavesdropper channel was first studied by Lai and El Gamal, whose achievable scheme introduced noise forwarding (NF) and used backward decoding. We suggest a novel sliding window decoding scheme with a two block decoding delay, where the relay uses compress-forward with Wyner-Ziv (WZ) binning but does not use NF. Wireless engineers will welcome the reduced decoding delay. Sliding window decoding mandates multiblock equivocation calculations; dispensing with NF enables it. We identify nine regimes and develop a case-by-case choice of relay channel codebook and WZ bin sizes to maximize the secrecy rate. The multiblock equivocation calculations may be of independent interest.

Equations

$$\text{reliability: }{P}_e^{(n)}\le ϵ$$

$$\text{(weak) secrecy: }H(W^{[B-1]}|{\mathbf{Z}}^{[B]})\ge n(B-1)R_1-n(B-1)ϵ$$

$$W{Z}^{Bob}\stackrel{\text{def}}{=}I({\hat{Y}}_2;X_1,Y_3|X_2),W{Z}^{Eve}\stackrel{\text{def}}{=}I({\hat{Y}}_2;X_1,Z|X_2)$$

$$\hat{R}$$

$$R_1+{\tilde{R}}_1$$

$$+[I({\hat{Y}}_2;Y_3|X_2)+I(X_2;Y_3)-\hat{R}]$$

$$R_1+{\tilde{R}}_1$$

$$+[I({\hat{Y}}_2;Y_3|X_2)+R_2-\hat{R}]$$

$$H(W^{[B-1]}|{\mathbf{Z}}^{[B]})$$

$$\text{Term}\text{framebox}\text{eqref}eq:equivcal1:a(i):\text{ i.e. }H({\mathbf{X}}_1^{[B]}|{\mathbf{Z}}^{[B]})$$

$$\text{Term }\text{framebox}\text{eqref}eq:equivcal1:aa(i):\text{ i.e. }H(({\mathbf{X}}_1,{\mathbf{X}}_2,{\hat{\mathbf{Y}}}_2,\mathbf{Z}{)}^{[B]})$$

$$\text{In }\text{eqref}eq:equivcal1:aaa,T_j=\underset{(a)}{H({\mathbf{X}}_{2,j}|({\mathbf{X}}_1,{\mathbf{X}}_2,{\hat{\mathbf{Y}}}_2,\mathbf{Z}{)}^{[j-1]})}+\underset{(b)}{H({\mathbf{X}}_{1,j}|{\mathbf{X}}_{2,j},({\mathbf{X}}_1,{\mathbf{X}}_2,{\hat{\mathbf{Y}}}_2,\mathbf{Z}{)}^{[j-1]})}$$

$$+\underset{(c)}{H({\mathbf{Z}}_j|({\mathbf{X}}_1,{\mathbf{X}}_2{)}_j,({\mathbf{X}}_1,{\mathbf{X}}_2,{\hat{\mathbf{Y}}}_2,\mathbf{Z}{)}^{[j-1]})}+\underset{(d)}{H({\hat{\mathbf{Y}}}_{2,j}|({\mathbf{X}}_1,{\mathbf{X}}_2,\mathbf{Z}{)}_j,({\mathbf{X}}_1,{\mathbf{X}}_2,{\hat{\mathbf{Y}}}_2,\mathbf{Z}{)}^{[j-1]})}$$

$$\text{Terms }\text{framebox}\text{eqref}eq:equivcal1:aaaa(a)=0,\text{framebox}\text{eqref}eq:equivcal1:aaaa(b)=n[R_1+{\tilde{R}}_1],\text{framebox}\text{eqref}eq:equivcal1:aaaa(c)=H({\mathbf{Z}}_j|{\mathbf{X}}_{1,j},{\mathbf{X}}_{2,j}),\text{framebox}\text{eqref}eq:equivcal1:aaaa(d)=n[\hat{R}-I({\hat{Y}}_2;X_1,Z|X_2)]$$

$$\text{(Foregoing evaluated in main text). We can write: }{T}_j=n[R_1+{\tilde{R}}_1+H({\mathbf{Z}}_j|{\mathbf{X}}_{1,j},{\mathbf{X}}_{2,j})+\hat{R}-I({\hat{Y}}_2;X_1,Z|X_2)]$$

$$\text{Term }\text{framebox}\text{eqref}eq:equivcal1:aa(ii)\text{ i.e. }H(({\mathbf{X}}_2,{\hat{\mathbf{Y}}}_2{)}^{[B]}|({\mathbf{X}}_1,\mathbf{Z}{)}^{[B]})=\sum _{j=1}^B\underset{{\tilde{T}}_j}{H(({\mathbf{X}}_2,{\hat{\mathbf{Y}}}_2{)}_j|({\mathbf{X}}_2,{\hat{\mathbf{Y}}}_2{)}^{[j-1]},{\mathbf{X}}_1^{[B]},{\mathbf{Z}}^{[B]})}$$

$$\text{In }\text{eqref}eq:equivcal1:aaaaa,{\tilde{T}}_j=\underset{(=0,\text{ for the same reason as Term }\text{framebox}\text{eqref}eq:equivcal1:aaaa(a))}{H({\mathbf{X}}_{2,j}|({\mathbf{X}}_2,{\hat{\mathbf{Y}}}_2{)}^{[j-1]},({\mathbf{X}}_1,\mathbf{Z}{)}^{[B]})}+\underset{(\tilde{b})\text{ (evaluated in main text) }}{H({\hat{\mathbf{Y}}}_{2,j}|{\mathbf{X}}_{2,j},({\mathbf{X}}_2,{\hat{\mathbf{Y}}}_2{)}^{[j-1]},({\mathbf{X}}_1,\mathbf{Z}{)}^{[B]})}$$

$$=n[\hat{R}-\min \{R_2,I(X_2;Z|X_1)\}-I({\hat{Y}}_2;X_1,Z|X_2)]$$

$$\text{Term }\text{framebox}\text{eqref}eq:equivcal1:aa(iii)\text{ i.e. }H({\mathbf{Z}}^{[B]})=\sum _{j=1}^{B}H({\mathbf{Z}}_j|{\mathbf{Z}}^{[j-1]})\le \sum _{j=1}^{B}H({\mathbf{Z}}_j)$$

$$T_j-{\tilde{T}}_j-H({\mathbf{Z}}_j|{\mathbf{Z}}^{[j-1]})\ge T_j-{\tilde{T}}_j-H({\mathbf{Z}}_j)=n[R_1+{\tilde{R}}_1+\min \{R_2,I(X_2;Z|X_1)\}-I({\mathbf{X}}_{1,j},{\mathbf{X}}_{2,j};{\mathbf{Z}}_j)]$$

$$\text{From }\text{eqref}eq:equivcal1:aa,\text{ we have }H({\mathbf{X}}_1^{[B]}|{\mathbf{Z}}^{[B]})$$

$$\text{Above and }\text{eqref}eq:equivcal1:a\text{ give }H({\mathbf{W}}^{[B-1]}|{\mathbf{Z}}^{[B]})$$

$$I({\mathbf{X}}_{1,j},{\mathbf{X}}_{2,j};{\mathbf{Z}}_j)$$

$$\le H({\mathbf{X}}_{2,j})+nI(X_1;Z|X_2)+nϵ$$

$$\le n{R}_2+nI(X_1;Z|X_2)+nϵ\text{(u.b. II)}⟹$$

$$I({\mathbf{X}}_{1,j},{\mathbf{X}}_{2,j};{\mathbf{Z}}_j)$$

$$n{R}_1+n{\tilde{R}}_1+n\min \{R_2,I(X_2;Z|X_1)\}]$$

$$-n\min \{R_2,I(X_2;Z)\}-nI(X_1;Z|X_2)-nϵ$$

$$H(W^{[B-1]}|{\mathbf{Z}}^B)$$

$$\ge H(({\mathbf{X}}_1,{\mathbf{X}}_2,\mathbf{Z}{)}^{[B]})-H(({\mathbf{X}}_1,{\mathbf{X}}_2{)}^{[B]}|W^{[B-1]},{\mathbf{Z}}^{[B]})-H({\mathbf{Z}}^{[B]})$$

$$=\underset{(a)}{\underbrace{\sum _{j=1}^B\underset{T_j}{H(({\mathbf{X}}_1,{\mathbf{X}}_2,\mathbf{Z}{)}_j|({\mathbf{X}}_1,{\mathbf{X}}_2,\mathbf{Z}{)}^{[j-1]})}}}-\underset{(b)}{\underbrace{H(({\mathbf{X}}_1,{\mathbf{X}}_2{)}^{[B]}|W^{[B-1]},{\mathbf{Z}}^{[B]})}}-\underset{(c)}{\underbrace{\sum _{j=1}^{B}H({\mathbf{Z}}_j|{\mathbf{Z}}^{[j-1]})}}$$

$$T_j=\underset{(i)}{H({\mathbf{X}}_{2,j}|({\mathbf{X}}_1,{\mathbf{X}}_2,\mathbf{Z}{)}^{[j-1]})}+\underset{(ii)}{H({\mathbf{X}}_{1,j}|{\mathbf{X}}_{2,j},({\mathbf{X}}_1,{\mathbf{X}}_2,\mathbf{Z}{)}^{[j-1]})}+\underset{(iii)}{H({\mathbf{Z}}_j|{\mathbf{X}}_{1,j},{\mathbf{X}}_{2,j},({\mathbf{X}}_1,{\mathbf{X}}_2,\mathbf{Z}{)}^{[j-1]})}$$

$$\text{Using (see main text) }\text{Term }\text{framebox}\text{eqref}eq:equivcal2:aa(i)=n{R}_2,\text{ Term }\text{framebox}\text{eqref}eq:equivcal2:aa(ii)=n[R_1+{\tilde{R}}_1],\text{ Term }\text{framebox}\text{eqref}eq:equivcal2:aa(iii)=H({\mathbf{Z}}_j|{\mathbf{X}}_{1,j},{\mathbf{X}}_{2,j})$$

$$\text{Together with }\text{eqref}eq:equivcal2:aa,\text{the above gives }{T}_j=n{R}_2+n[R_1+{\tilde{R}}_1]+H({\mathbf{Z}}_j|{\mathbf{X}}_{1,j},{\mathbf{X}}_{2,j})$$

$$H(W^{[B-1]}|{\mathbf{Z}}^B)$$

$$\stackrel{\text{eqref}eq:equivcal1:mt}{\ge }\sum _{j=1}^{B}n[R_2+R_1+{\tilde{R}}_1-\min \{R_2,I(X_2;Z)\}-I(X_1;Z|X_2)-ϵ]-\underset{(b)}{H(({\mathbf{X}}_1,{\mathbf{X}}_2{)}^{[B]}|W^{[B-1]},{\mathbf{Z}}^{[B]})}$$

Full text

I Introduction

The “weak” relay-eavesdropper channel with a trusted relay and external eavesdropper (Eve) has been the focus of interest since [1, Theorems $3,4$]. By “weak” is meant a relay that does not decode-and-forward its received sequence, but one which compresses and sends the bin index of the compressed sequence [2]. [1] introduced the idea of NF and generalized NF (aka GNF), wherein the relay transmits some random codewords to confuse both the intended receiver (Bob) and Eve. Eve is affected more than Bob; leading to an increased secrecy rate. In [1], the relay did not perform WZ binning. A useful improvement to [1]’s scheme was developed by [3], who showed that dispensing with the requirement that given the message, Eve be able to decode the “relay’s” codeword leads to an increase in secrecy rate. Note that in pure NF [1, Theorem $3$] and in the helping interferer [3], the “relay” actually doesn’t relay; it acts as an oblivious i.e., deaf helper.

The relay channel has itself been the focus of continued research interest. Recently, [4] have developed a new decoding scheme for the “weak” relay channel. While no rate improvement was achieved for the canonical relay channel, they showed an improvement for multi-relay networks. In our scheme, we use the decoding technique from [4] together with ideas from [3] and [5].

Both [4], [5] indicate that too high a compression sequence rate, or too low a relay channel codeword rate can reduce the message rate. This brings us to the main idea of our paper i.e., to use compression sequences themselves to interfere with and confuse Eve by choosing appropriate compression rates and relay channel codeword, equivalently, WZ binning rates.

Sliding window decoding uses multiblock correlations to decode, entailing that the equivocation calculation must necessarily also be multiblock. A previous paper by the author [6] also uses multiblock equivocation (MBEq) in a (dedicated and trusted) four node relay broadcast channel with two receivers, each requiring an independent message to be kept secret from the other. The relay in [6] is a “strong” relay that decodes-and-forwards the messages to their destinations.

In related work, [7] have considered a “weak” relay-eavesdropper with long message noisy network coding (LM-NNC) scheme that also dispenses with NF as we have done by making use of the insight that judiciously choosing the number of compression sequences can impact the secrecy rate achievable. However, they do not employ Wyner-Ziv binning, and the decoding delay of LM-NNC is $B$ blocks.

II Model: “Weak” Relay-Eavesdropper Channel

The notation is standard, but we describe it here for completeness. The system model is depicted in Fig. 1. We assume a two receiver discrete memoryless relay-eavesdropper channel with a confidential message intended for one of the receivers, with the other acting as an eavesdropper. The finite sets $\mathcal{X}_{1},\mathcal{X}_{2}$, $\mathcal{Y}_{2},\mathcal{Y}_{3},\mathcal{Z}$ respectively represent the channel’s input at node $1$ (the transmitter), at node $2$ (the relay), the channel’s output at nodes $2$, $3$ (legitimate receiver aka Bob) and $4$ (eavesdropper aka Eve). Finally, $\mathcal{\hat{Y}}_{2}$ represents the alphabet of the compression random variable. The channel is described by the conditional probability distribution $P_{Y_{2},Y_{3},Z|X_{1},X_{2}}$, where RVs $X_{i}\in\mathcal{X}_{i},\,i=1,2$ and $Y_{i}\in\mathcal{Y}_{i},\,i=2,3$ and $Z\in\mathcal{Z}$. The transmitter intends to send an independent message $W\in\{1,2,\dots,2^{nR_{1}}\}\overset{\Delta}{=}\mathcal{W}$ to the receiver Rx $Y_{3}$ in $n$ channel uses while ensuring information theoretic secrecy, defined below. The channel is memoryless and without feedback i.e. $\forall(\mathbf{x}_{1},\mathbf{x}_{2})\in\prod_{t=1}^{2}\mathcal{X}_{t}^{n},\mathbf{y}_{t}\in\mathcal{Y}_{t}^{n},\,t=2,3,\mathbf{z}\in\mathcal{Z}^{n}$, $\begin{aligned} P(\mathbf{y}_{2},\mathbf{y}_{3},\mathbf{z}|\mathbf{x}_{1},\mathbf{x}_{2})=\prod_{i=1}^{n}P_{Y_{2},Y_{3},Z|X_{1},X_{2}}(y_{2i},y_{3i},z_{i}|x_{1i},x_{2i})\end{aligned}$

The decoding function at Bob $\equiv Y_{3}$ is a map $\phi_{3}:\mathcal{Y}_{3}^{n}\times\mathcal{Y}_{3}^{n}\rightarrow\mathcal{W}\times\widetilde{\mathcal{W}}$.

A $(2^{nR_{1}},n,P_{e}^{(n)})$ code for the relay-eavesdropper channel with compress-forward (CF) consists of a (stochastic) encoding function at the Tx and an encoding function at the relay, and a decoding function $\phi_{3}$ at the destination and error probability $\begin{aligned} P_{e}^{(n)}=\sum_{(w,\tilde{w})}\frac{\Pr[\phi_{3}(\mathbf{Y}_{3}^{n}\times\mathbf{Y}_{3}^{n})\neq(w,\tilde{w})|(w,\tilde{w})]}{2^{nR_{1}}\times 2^{n\tilde{R}_{1}}}\end{aligned}$

A secrecy rate $R_{1}$ is said to be achievable for the DM relay-eavesdropper channel if, for any $\epsilon>0$, $\exists(2^{n[R_{1}+\tilde{R}_{1}]},n,P_{e}^{(n)})$ code s.t. the following requirements are satisfied:

[TABLE]

We use the notation $[x]^{+}\overset{\text{def}}{=}\max\{x,0\}$.

III Inner Bound

The main result of our paper is presented below, namely, that judicious independent choice of the compression sequence rate $\hat{R}$ and WZ binning rate $\hat{R}-R_{2}$,111Equivalently, of $\hat{R}$ and $R_{2}$ enables us to maximize the secrecy rate.

Theorem 1**.**

We first define:

[TABLE]

*A (pure secrecy) rate $R_{1}$ is achievable if there exists distributions $P_{X_{1}}P_{X_{2}}P_{\hat{Y}_{2}|X_{2},Y_{2}}$ s.t. the following hold: *

Case $1$**: $I(X_{2};Z)+WZ^{Eve}<I(X_{2};Y_{3})+WZ^{Bob}$.

In all Case $1$ sub-cases, we choose $\hat{R}\in(I(X_{2};Z)+WZ^{Eve},I(X_{2};Y_{3})+WZ^{Bob})$ and $R_{2}>I(X_{2};Z_{3})$. So Eve cannot uniquely decode $\mathbf{x}_{2}$, and either decodes it nonuniquely or treats it as noise, depending on the sub-cases.

Case $1(a)$**: $I(X_{2};Z)+WZ^{Eve}<I(X_{2};Y_{3})+I(\hat{Y}_{2};Y_{3}|X_{2})$

Case $1(a)(i)$**: $I(X_{2};Z|X_{1})<I(X_{2};Y_{3})+I(\hat{Y}_{2};Y_{3}|X_{2})$. The choices $R_{2}=\max\{I(X_{2};Y_{3}),I(X_{2};Z|X_{1})\}+\epsilon$ and $\hat{R}=I(X_{2};Y_{3})+I(\hat{Y}_{2};Y_{3}|X_{2})-\epsilon>I(X_{2};Z)+WZ^{Eve}$, enable the secrecy rate $R_{1}=[I(X_{1};\hat{Y}_{2},Y_{3}|X_{2})-I(X_{1};Z)]^{+}$.

Case $1(a)(ii)$**: $I(X_{2};Y_{3})+I(\hat{Y}_{2};Y_{3}|X_{2})<I(X_{2};Z|X_{1})$. The choices $R_{2}=\max\{I(X_{2};Y_{3}),I(X_{2};Z)\}+\epsilon$, $\hat{R}$ as before, enable R_{1}=\Big{[}I(X_{1};\hat{Y}_{2},Y_{3}|X_{2})-[I(X_{1},X_{2};Z)-R_{2}]\Big{]}^{+}. The penalty term can be reduced by choosing $\epsilon>0$ s.t. $R_{2}\uparrow\hat{R}$.

Case $1(b)$**: $I(X_{2};Y_{3})+I(\hat{Y}_{2};Y_{3}|X_{2})<I(X_{2};Z)+WZ^{Eve}<I(X_{2};Y_{3})+WZ^{Bob}$

Case $1(b)(i)$**: $I(X_{2};Z|X_{1})<I(X_{2};Y_{3})+WZ^{Bob}$. The choices $R_{2}=\max\{I(X_{2};Z|X_{1}),I(X_{2};Y_{3})\}+\epsilon(>I(X_{2};Z))$ and $\hat{R}\in(I(X_{2};Z)+WZ^{Eve},I(X_{2};Y_{3})+WZ^{Bob})$ enable the secrecy rate R_{1}=\Big{[}I(X_{1};\hat{Y}_{2},Y_{3}|X_{2})+I(X_{2};Y_{3})+I(\hat{Y}_{2};Y_{3}|X_{2})-\hat{R}-I(X_{1};Z)\Big{]}^{+}. By choosing $\hat{R}\downarrow\max\{I(X_{2};Z|X_{1}),I(X_{2};Y_{3})\}$, we can obtain rate R_{1}=\Big{[}I(X_{1};\hat{Y}_{2},Y_{3}|X_{2})+I(X_{2};Y_{3})+I(\hat{Y}_{2};Y_{3}|X_{2})-\max\{I(X_{2};Z|X_{1}),I(X_{2};Y_{3})\}-I(X_{1};Z)\Big{]}^{+}.

Case $1(b)(ii)$**: $I(X_{2};Y_{3})+WZ^{Bob}<I(X_{2};Z|X_{1})$. The choices $R_{2}>\max\{I(X_{2};Y_{3}),I(X_{2};Z)\}+\epsilon$ and $\hat{R}\in(I(X_{2};Z)+WZ^{Eve},I(X_{2};Y_{3})+WZ^{Bob})$ enable the secrecy rate R_{1}=\Big{[}I(X_{1};\hat{Y}_{2},Y_{3}|X_{2})+I(X_{2};Y_{3})+I(\hat{Y}_{2};Y_{3}|X_{2})-\hat{R}-[I(X_{1},X_{2};Z)-R_{2}]\Big{]}^{+}. Choosing $\hat{R}\downarrow I(X_{2};Z)+WZ^{Eve},\,R_{2}\uparrow\hat{R}$ enables rate R_{1}=\Big{[}I(X_{1};\hat{Y}_{2},Y_{3}|X_{2})+I(X_{2};Y_{3})+I(\hat{Y}_{2};Y_{3}|X_{2})-I(X_{1},X_{2};Z)\Big{]}^{+}

Case $2$**: $I(X_{2}:Y_{3})+WZ^{Bob}<I(X_{2};Z)+WZ^{Eve}$

Case $2(a)$**: $WZ^{Eve}<WZ^{Bob}(\implies I(X_{2};Y_{3})<I(X_{2};Z))$

Case $2(a)(i)$**: $WZ^{Eve}<I(\hat{Y}_{2};Y_{3}|X_{2})$. The choices $R_{2}\in(I(X_{2};Y_{3}),I(X_{2};Z)),\,\hat{R}-R_{2}>WZ^{Eve},\hat{R}<I(X_{2};Y_{3})+I(\hat{Y}_{2};Y_{3}|X_{2})$ can be made consistent and enable secrecy rate $R_{1}=[I(X_{1};\hat{Y}_{2},Y_{3}|X_{2})-I(X_{1};Z|X_{2})]^{+}$.

Case $2(a)(ii)$**: $I(\hat{Y}_{2};Y_{3}|X_{2})<WZ^{Eve}<WZ^{Bob}$. Choosing $R_{2}>I(X_{2};Y_{3}),\hat{R}-R_{2}>WZ^{Eve},\hat{R}<I(X_{2};Y_{3})+WZ^{Bob}$ enables secrecy rate R_{1}=\Big{[}[I(X_{1};Y_{3}|X_{2})-I(X_{1};Z|X_{2})]+[WZ^{Bob}-WZ^{Eve}]\Big{]}^{+}.

Case $2(b)$**: $WZ^{Bob}<WZ^{Eve}$

Case $2(b)(i)$**: $I(X_{2};Y_{3})<I(X_{2};Z)$. Choosing $R_{2}<I(X_{2};Y_{3}),\,\hat{R}>I(X_{2};Z)+WZ^{Eve}\implies\hat{R}-R_{2}>WZ^{Eve}$ and hence also $\hat{R}-R_{2}>WZ^{Bob}$, which enables the secrecy rate $R_{1}=[I(X_{1};Y_{3}|X_{2})-I(X_{1};Z|X_{2})]^{+}$.

*Case $2(b)(ii):I(X_{2};Z)<I(X_{2};Y_{3})$**. *

$(A)$: $I(X_{2};Z)<I(X_{2};Y_{3})<I(X_{2};Z|X_{1})$. The choices $R_{2}\in(I(X_{2};Z),I(X_{2};Y_{3})),\,\hat{R}-R_{2}>WZ^{Eve}$ enables secrecy rate R_{1}=\Big{[}I(X_{1};Y_{3}|X_{2})-[I(X_{1},X_{2};Z)-R_{2}]\Big{]}^{+}. The penalty term is reduced by choosing $R_{2}\uparrow I(X_{2};Y_{3})$

$(B)$: $I(X_{2};Z)<I(X_{2};Z|X_{1})<I(X_{2};Y_{3})$. The choices $R_{2}\in(I(X_{2};Z|X_{1}),I(X_{2};Y_{3}))$ and $\hat{R}-R_{2}$ as before, enables secrecy rate $R_{1}=[I(X_{1};Y_{3}|X_{2})-I(X_{1};Z)]^{+}$.

IV Achievability Scheme

As is common in secrecy scenarios, the transmitter’s codebook is divided into bins (aka subcodebooks), with the bin size intended to confuse Eve. We use [4]’s decoding technique.

*Codebook Generation: * There are $B$ blocks of transmission. An independent set of codebooks is generated for each block, and these are known to all parties involved. Aside from an additional binning structure, the codebook construction at the transmitter is almost identical to that in the canonical relay channel [2, 8]. The transmitter codebook in block $b$ is given by $\begin{aligned} \mathcal{C}_{1}^{(b)}=\{\mathbf{x}_{1}^{(b)}(w,\tilde{w})|w\in[2^{nR_{1}}],\tilde{w}\in[2^{n\tilde{R}_{1}}])\}\end{aligned}$, with the codewords generated independently using distribution $P_{X_{1}}$ i.e. $\begin{aligned} \mathbf{x}_{1}^{(b)}(w,\tilde{w})\sim\prod_{i=1}^{n}P_{X_{1}}(x_{1i}^{(b)})\end{aligned}$. For a fixed $w=1$ (say), the set $\mathbf{x}_{1}(1,\tilde{w}),\tilde{w}\in[2^{n\tilde{R}_{1}}]$ can be thought of as a bin. The relay channel codebook $\mathcal{C}_{2}^{(b)}$ and compression codebook associated with each relay channel codeword is exactly identical to that in [2, 8]. $|\mathcal{C}_{2}^{(b)}|=2^{nR_{2}}$. The compression codebook consists of $2^{nR_{2}}$ – i.e., one per relay codeword – satellite codebooks; each is of size $2^{n\hat{R}}$, and is divided into $2^{nR_{2}}$ WZ bins of size $2^{n[\hat{R}-R_{2}]}$. The generation of the codewords is identical to that in [2, 8] and is omitted due to space. See [9] for details.

Remark 2**.**

There are two differences with the codebooks used for NF and GNF in [1]. In [1], the relay does not WZ bin the compression sequences $\mathbf{\hat{y}}_{2}$. Further, each $\mathbf{\hat{y}}_{2}$ is associated with a set of different possible relay codewords $\mathbf{x}_{2}$ – this is called NF. Once $\mathbf{\hat{y}}_{2}$ is known, the choice of $\mathbf{x}_{2}$ to transmit is determined by the relay’s private randomness. In our scheme, on the other hand, the relay does WZ bin the $\mathbf{\hat{y}}_{2}$s, and the $\mathbf{x}_{2}$ transmitted is wholly determined by the WZ bin of the previous block’s $\mathbf{\hat{y}}_{2}$ – private randomness not needed. This determinism is crucial for the success of the equivocation calculations. Since this gives a greater degree of control to the transmitter, we also have reason to believe that this could lead to an improvement in achievable secrecy rate.

Transmitter Encoding: To send $W_{j}\in[1:2^{nR_{1}}]$ in a block $b\in{1,2,\dots,B-1}$, the transmitter chooses uniformly at random from among $2^{n\tilde{R}_{1}}$ possibilities for $\tilde{w}$ inside the corresponding bin $W_{j}$. The corresponding codeword $\mathbf{x}_{1}$ is transmitted.

Relay Encoding: The relay encoding is standard [2, 8] i.e. it looks for a compression codeword that satisfies the joint typicality condition \big{(}\mathbf{x}_{2},\mathbf{\hat{y}}_{2},\mathbf{y}_{2}\big{)}\in\mathcal{T}^{(n)}_{\epsilon} and then determines $\mathbf{\hat{y}}_{2}$’s WZ bin index. The relay codeword corresponding to this WZ bin is transmitted in the next block.

*Decoding at Bob: * We use the decoding techniques developed by [4]. See [9]. These give rise to:

[TABLE]

Remark 3**.**

Consider the canonical “weak” relay with no secrecy requirement i.e., $\tilde{R}_{1}=0$, as in [4]. We observe that:

• •

If $\hat{R}\overset{\alpha>0}{=}I(X_{2};Y_{3})+I(\hat{Y}_{2};Y_{3}|X_{2})+\alpha$, then $R_{1}<I(X_{1};\hat{Y}_{2},Y_{3}|X_{2})-\alpha$ i.e. too many compression sequences reduce achievable rate $R_{1}$. (**[4]**, **[5]**).

• •

If $\hat{R}<I(X_{2};Y_{3})+I(\hat{Y}_{2};Y_{3}|X_{2})$, if in addition $R_{2}\overset{\beta>0}{=}\hat{R}-I(\hat{Y}_{2};Y_{3}|X_{2})-\beta$, then again $R_{1}<I(X_{1};\hat{Y}_{2},Y_{3}|X_{2})-\beta$ i.e. too few relay codewords also reduce achievable rate.

• •

If $\hat{R}-R_{2}>WZ^{Bob}$, then Bob cannot uniquely decode $\mathbf{\hat{y}}_{2}$.

Remark 4**.**

In the “weak” relay-eavesdropper channel, $\tilde{R}_{1}\neq 0$; $\hat{R}$ & $\hat{R}-R_{2}$ chosen to maximize secrecy rate: $R_{1}$

Remark 5**.**

If either $\eqref{eq:bob_dec_1a}$ (Bob’s proxy constraint) or $R_{2}<I(X_{2};Y_{3})$ is satisfied, then Bob can decode $\mathbf{x}_{2}$ uniquely. In Cases $1(a)(i),1(a)(ii),2(a)(i)$, $\because R_{2}>I(X_{2};Y_{3}),\hat{R}<I(X_{2};Y_{3})+I(\hat{Y}_{2};Y_{3}|X_{2})$, of the three constraints on $R_{1}+\tilde{R}_{1}$, the tightest constraint is $\eqref{eq:bob_dec_1b}$. In Cases $1(b)(i),1(b)(ii),2(a)(ii)$, $\because R_{2}>I(X_{2};Y_{3}),\hat{R}>I(X_{2};Y_{3})+I(\hat{Y}_{2};Y_{3}|X_{2})$, of the three constraints on $R_{1}+\tilde{R}_{1}$, the tightest constraint is $\eqref{eq:bob_dec_1c}$. In Cases $2(b)(i),2(b)(ii)$, constraint $\eqref{eq:bob_dec_1a}$ is violated, but $R_{2}<I(X_{2};Y_{3})$ still holds. So Bob uses $(\mathbf{x}_{2},\mathbf{y}_{3})\in\mathcal{T}^{(n)}_{\epsilon}$ i.e., single block information to decode $\mathbf{x}_{2}$, and then conditions on $\mathbf{x}_{2}$ to decode $\mathbf{x}_{1}$ using $(\mathbf{x}_{1},\mathbf{x}_{2},\mathbf{y}_{3})\in\mathcal{T}^{(n)}_{\epsilon}$ at rate $R_{1}+\tilde{R}_{1}\approx I(X_{1};Y_{3}|X_{2})$. It can be s.t. decoding $\mathbf{\hat{y}}_{2}$ nonuniquely would be suboptimal – details omitted.

Remark 6**.**

In Case $1$ i.e., $I(X_{2};Z)+WZ^{Eve}<I(X_{2};Y_{3})+WZ^{Bob}$, we will always choose $\hat{R}\in(I(X_{2};Z)+WZ^{Eve},I(X_{2};Y_{3})+WZ^{Bob})$, ensuring that Eve’s proxy constraint, to wit, $\hat{R}<I(X_{2};Z)+WZ^{Eve}$ is violated, and by additionally choosing $R_{2}>I(X_{2};Z)$, Eve cannot decode $\mathbf{x}_{2}$ uniquely and consequently cannot decode $\mathbf{\hat{y}}_{2}$ either.

Remark 7**.**

In Case $1$ scenarios, the $\mathbf{x}_{2}$ in the first block is known to all parties, including Eve. So, in the first block, Eve can decode $\mathbf{x}_{1}$ at a rate $\tilde{R}_{1}\approx I(X_{1};Z|X_{2})$. However, this holds only for the first block, creating a boundary effect that does not affect the overall secrecy rate achievable.

Remark 8**.**

In all Case $2$ scenarios i.e., $I(X_{2};Y_{3})+WZ^{Bob}<I(X_{2};Z)+WZ^{Eve}$, we will choose $\hat{R}-R_{2}>WZ^{Eve}$, thus ensuring that Eve cannot decode $\mathbf{\hat{y}}_{2}$ uniquely. But in Case $2a(i),(ii),2b(i),R_{2}<I(X_{2};Z)$, enabling Eve to decode $\mathbf{x}_{2}$ uniquely. Consequently, in these cases only, Eve can decode $\mathbf{x}_{1}$ at rate $\tilde{R}_{1}\approx I(X_{1};Z|X_{2})$.

Remark 9**.**

In all other cases, whether Eve decodes $\mathbf{x}_{2}$ nonuniquely or treats $\mathbf{x}_{2}$ as noise will be determined by the exact choices of $\hat{R}$ and $\hat{R}-R_{2}$; the choices in turn depend on the sub-case. Now there are the following possibilities:

$R_{2}\in(I(X_{2};Z),I(X_{2};Z|X_{1}))$. Given the message $W$, Eve decodes $\mathbf{x}_{2}$ nonuniquely and decodes $\mathbf{x}_{1}$ uniquely at rate $\tilde{R}_{1}=I(X_{1},X_{2};Z)-R_{2}$. 2. 2.

$R_{2}>I(X_{2};Z|X_{1})$. Given the message $W$, treat $\mathbf{x}_{2}$ as noise and decode $\mathbf{x}_{1}$ at rate $\tilde{R}_{1}=I(X_{1};Z)$

V Equivocation Calculations

We have separate equivocation calculations for Cases $1$, $2(a),2(b)(ii)$. Calculation for $2(b)(i)$ is identical to Case $2(a)$.

Case $1$: $I(X_{2};Z)+WZ^{Eve}<I(X_{2};Y_{3})+WZ^{Bob}$ In this regime, we will always choose $\hat{R}\in(I(X_{2};Z)+WZ^{Eve},I(X_{2};Y_{3})+WZ^{Bob})$, and $R_{2}>I(X_{2};Z)$.

We s.t. Term (2)(ii) “small”. The rate of a transmitter bin is $\tilde{R}_{1}\approx I(X_{1};Z)\mathbb{I}\{R_{2}>I(X_{2};Z|X_{2})\}+[I(X_{1},X_{2};Z)-R_{2}]\mathbb{I}\{R_{2}\in(I(X_{2};Z),I(X_{2};Z|X_{1}))\}$. Conditioning on transmitted $W_{j}$ is equivalent to knowing the bin and reduces the number of possibilities for $\mathbf{X}_{1,j}$ from $2^{n[R_{1}+\tilde{R}_{1}]}$ to $2^{n\tilde{R}_{1}}$, and knowing $\mathbf{Z}_{j}$, enables us to decode $\mathbf{X}_{1,j}$ inside the bin correctly whp, using, in every block $j=1,2,\dots,B-1$:

• •

either $(\mathbf{x}_{1,j},\mathbf{z}(j))\in\mathcal{T}^{(n)}_{\epsilon}$ i.e. treating $\mathbf{x}_{2,j}$ as noise if $R_{2}>I(X_{2};Z|X_{1})$.

• •

or $(\mathbf{x}_{1,j},\mathbf{x}_{2,j},\mathbf{z}(j))\in\mathcal{T}^{(n)}_{\epsilon}$ for a unique $\mathbf{x}_{1}$ and some $\mathbf{x}_{2}$ at a rate $I(X_{1},X_{2};Z)-R_{2}$ if $R_{2}\in(I(X_{2};Z),I(X_{2};Z|X_{1}))$.

In the last block no new message is transmitted and thus $\mathbf{X}_{1,B}$ is known. A standard application of Fano’s inequality gives: $\framebox{\text{Term }\eqref{eq:equivcal1:a}(ii)}\leq(B-1)\times n\epsilon$.

Term $\eqref{eq:equivcal1:aaaa}(a)$ = 0. No noise-forwarding i.e. $\mathbf{X}_{2j}$ is a function of $(\mathbf{X}_{2,j-1},\mathbf{\hat{Y}}_{2,j-1})$ present in the conditioning.

Term $\eqref{eq:equivcal1:aaaa}(b)$= $n[R_{1}+\tilde{R}_{1}]$. $\mathbf{X}_{1,j}$ is uniformly chosen from $2^{n[R_{1}+\tilde{R}_{1}]}$ codewords; choice unaffected by the conditioning.

Term $\eqref{eq:equivcal1:aaaa}(c)$ $=H(\mathbf{Z}_{j}|\mathbf{X}_{1,j},\mathbf{X}_{2,j})$. $\because$ Channel is DMC,

$\mathbf{Z}_{j}\leftrightarrow(\mathbf{X}_{1,j},\mathbf{X}_{2,j})\leftrightarrow(\mathbf{X}_{1},\mathbf{X}_{2},\mathbf{\hat{Y}}_{2},\mathbf{Z})^{[j-1]}$ are Markov RVs.

$\begin{aligned} \text{Term }\framebox{\eqref{eq:equivcal1:aaaa}(d)}=n[\hat{R}-I(\hat{Y}_{2};X_{1},Z|X_{2})]\end{aligned}$. The first three quantities in the conditioning, namely, $(\mathbf{X}_{1},\mathbf{X}_{2},\mathbf{Z})_{j}$ can be used to reduce the number of possibilities for $\mathbf{\hat{y}}_{2,j}$ as follows. Knowledge of $\mathbf{X}_{2,j}$ reduces the number of possibilities for the compression sequence down to $2^{n\hat{R}}$. Now by the condition \begin{aligned} \big{(}\mathbf{x}_{1,j},\mathbf{x}_{2,j},\mathbf{\hat{y}}_{2}(|\mathbf{x}_{2,j}),\mathbf{z}_{j}\big{)}\in\mathcal{T}^{(n)}_{\epsilon}\end{aligned}, we can create an equiprobable list of size $2^{n[\hat{R}-I(\hat{Y}_{2};X_{1},Z|X_{2})]}$.

Term $\eqref{eq:equivcal1:aaaaaa}(\tilde{b})$. Knowledge of $\mathbf{X}_{2,j}$ reduces the possibilities for $\mathbf{\hat{Y}}_{2,j}$ down to $2^{n\hat{R}}$. Given $(\mathbf{X}_{1,j+1},\mathbf{Z}_{j+1})$, we can further reduce the number of possible codewords $\mathbf{X}_{2,j+1}$ in block $j+1$ down to $2^{n[R_{2}-I(X_{2};Z|X_{1})]}$. Each such codeword corresponds to a separate WZ bin, each of size $2^{n[\hat{R}-R_{2}]}$. Now the size of the list of possible $\mathbf{\hat{y}}_{2}$ is $\begin{aligned} 2^{n[R_{2}-I(X_{2};Z|X_{1})]}\times 2^{n[\hat{R}-R_{2}]}\end{aligned}$ where the first term will be $1$ if $R_{2}<I(X_{2};Z|X_{1})$. In terms of rate, we have $\begin{aligned} n[\cancel{R_{2}}-\min\{R_{2},I(X_{2};Z|X_{1})\}]+n[\hat{R}-\cancel{R_{2}}]\end{aligned}$. Conditioning on $(\mathbf{X}_{2,j},\mathbf{X}_{1,j},\mathbf{Z}_{j})$ gives us a further reduction by: $2^{nI(\hat{Y}_{2};X_{1},Z|X_{2})}$ so that, finally, the Term

$\eqref{eq:equivcal1:aaaaaa}(\tilde{b})$ $\begin{aligned} =n[\hat{R}\!-\!\min\{R_{2},\!I(X_{2};Z|X_{1})\}\!-\!I(\hat{Y}_{2};X_{1},Z|X_{2})]\end{aligned}$.

We now lower bound the RHS of $\eqref{eq:equivcal1:aaaaaaaa}$ by first computing two upper bounds for $I(\mathbf{X}_{1,j},\mathbf{X}_{2,j};\mathbf{Z}_{j})$. We have (see [3] and [10])

[TABLE]

We use the above to obtain a lower bound for the RHS of $\eqref{eq:equivcal1:aaaaaaaa}$ to obtain, for a single block:

[TABLE]

• •

If $R_{2}>I(X_{2};Z|X_{1})>I(X_{2};Z)$, then $\tilde{R}_{1}\approx I(X_{1};Z)$, and the above becomes $nR_{1}-n\epsilon$.

• •

If $R_{2}\in(I(X_{2};Z),I(X_{2};Z|X_{1}))$, then $\tilde{R}_{1}=I(X_{1},X_{2};Z)-R_{2}$, and we again have $nR_{1}-n\epsilon$.

Using this lower bound for RHS of $\eqref{eq:equivcal1:aaaaaaaa}$ in $\eqref{eq:equivcal1:aaaaaaaaa}$, summing over all $B$ blocks and noting that the $B$th block does not contribute, and using the lower bound on $\eqref{eq:equivcal1:a}(ii)$, we get $H(W^{[B-1]}|\mathbf{Z}^{[B]})\geq(B-1)\times n[R_{1}-2\epsilon]$.

Case $2$: $I(X_{2};Y_{3})+WZ^{Bob}<I(X_{2};Z)+WZ^{Eve}$ Case $2(a)$: $WZ^{Eve}\!<\!WZ^{Bob}$ ($\implies\!I(X_{2};Y_{3})\!<\!I(X_{2};Z)$)

Term \eqref{eq:equivcal2:aa}(i)$$=nR_{2}. Note that $\mathbf{X}_{2,j}$ is a function of $(\mathbf{X}_{2,j-1},\mathbf{\hat{Y}}_{2,j-1})$. $\mathbf{\hat{Y}}_{2,j-1}$ is determined by $(\mathbf{X}_{2,j-1},\mathbf{Y}_{2,j-1})$, and $\mathbf{Y}_{2,j-1}\leftrightarrow(\mathbf{X}_{1,j-1},\mathbf{X}_{2,j-1})\leftrightarrow(\mathbf{X}_{1},\mathbf{X}_{2},\mathbf{Z},\mathbf{Y}_{2},\mathbf{\hat{Y}}_{2})^{[j-2]}$ form a Markov chain, implying that $\begin{aligned} \mathbf{X}_{2,j}\leftrightarrow(\mathbf{X}_{1,j-1},\mathbf{X}_{2,j-1},\mathbf{Z}_{j-1})\leftrightarrow(\mathbf{X}_{1},\mathbf{X}_{2},\mathbf{Z})^{[j-2]}\end{aligned}$ form a Markov chain. Conditioning on $(\mathbf{X}_{1},\mathbf{X}_{2},\mathbf{Z})^{[j-1]}$ reduces the possibilities for $\mathbf{\hat{y}}_{2}$ per WZ bin from $2^{n[\hat{R}-R_{2}]}$ down to $2^{n[\hat{R}-R_{2}-WZ^{Eve}]}$ which – by our choices – is exponentially large in all Case $2$ regimes. Hence every WZ bin is possible in block $j-1$, and so is every $\mathbf{X}_{2,j}$.

As in Case $1$, Term \eqref{eq:equivcal2:aa}(ii)$$=n(R_{1}+\tilde{R}_{1}). Since $R_{2}<I(X_{2};Z)$, and $\tilde{R}_{1}\approx I(X_{1};Z|X_{2})$, and since block $B$ does not contribute, the RHS of inequality $\eqref{eq:equivcal2:aaaaa}$ becomes $n(B-1)[R_{1}\!-\!\epsilon]\!-\!H((\mathbf{X}_{1},\!\mathbf{X}_{2})^{[B]}|W^{[B-1]},\!\mathbf{Z}^{[B]})$. Now we need only show that: $\begin{aligned} &H((\mathbf{X}_{1},\mathbf{X}_{2})^{[B]}|W^{[B-1]},\mathbf{Z}^{[B]})\end{aligned}$ is “small”. $\because R_{2}<I(X_{2};Z)$, conditioning on $\mathbf{Z}_{j}$ enables Eve to decode $\mathbf{X}_{2,j}$ correctly whp using $(\mathbf{x}_{2,j},\mathbf{z}_{j})\in\mathcal{T}^{(n)}_{\epsilon}$. Given $W_{j}$, the number of possibilities for $\mathbf{X}_{1,j}$ reduces from $2^{n[R_{1}+\tilde{R}_{1}]}\rightarrow 2^{n[\tilde{R}_{1}]}$. Knowing both $\mathbf{X}_{2,j}$ and $\mathbf{Z}_{j}$ enables Eve to decode $\mathbf{X}_{1,j}$ using $(\mathbf{x}_{1,j},\mathbf{x}_{2,j},\mathbf{z}_{j})\in\mathcal{T}^{(n)}_{\epsilon}$ at a rate $I(X_{1};Z|X_{2})$, which suffices, since $\tilde{R}_{1}=I(X_{1};Z|X_{2})-\epsilon$. By Fano: $H((\mathbf{X}_{1},\mathbf{X}_{2})^{[B]}|W^{[B-1]},\mathbf{Z}^{[B]})\approx(B-1)\times n\epsilon$. So we finally have: $H(W^{[B-1]}|\mathbf{Z}^{[B]})\geq(B-1)nR_{1}-(B-1)n\epsilon$.

Case $2(b)$: $WZ^{Bob}<WZ^{Eve}$ Case $2(b)(i)$: $I(X_{2};Y_{3})<I(X_{2};Z)$ The calculation is virtually identical to Case $2(a)$ and is omitted.

Case $2(b)(ii):I(X_{2};Z)<I(X_{2};Y_{3})$

We choose $R_{2}\in(I(X_{2};Z),I(X_{2};Y_{3}))$, $\hat{R}>I(X_{2};Z)+WZ^{Eve}$, and $\hat{R}-R_{2}>WZ^{Eve}>WZ^{Bob}$. As in Case $2(b)(i)$, neither Bob nor Eve can decode $\mathbf{\hat{y}}_{2}$. Our choice of $R_{2}<I(X_{2};Y_{3})$ ensures that Bob can decode $\mathbf{x}_{2}$. Since both $R_{2}>I(X_{2};Z)\text{and }\hat{R}>I(X_{2};Z)+WZ^{Eve}$, Eve cannot decode $\mathbf{x}_{2}$ uniquely. Whether Eve decodes $\mathbf{x}_{2}$ nonuniquely or treats it as noise depends on two further sub-cases, depending on whether $I(X_{2};Y_{3})<\text{ or }>I(X_{2};Z|X_{1})$. Expanding by the chain rule, term $\eqref{eq:equivcal2:aaaaa}(b)$ $\begin{aligned} &=\sum_{j=1}^{B}H((\mathbf{X}_{1},\mathbf{X}_{2})_{j}|(\mathbf{X}_{1},\mathbf{X}_{2})^{[j-1]},W^{[B-1]},\mathbf{Z}^{[B]})\\ &\leq\sum_{j=1}^{B}H(\mathbf{X}_{1,j},\mathbf{X}_{2,j}|W_{j},\mathbf{Z}_{j})\end{aligned}$.

$W_{j}$ in the conditioning reduces the possibilities for $\mathbf{X}_{1,j}$ from $2^{n[R_{1}+\tilde{R}_{1}]}\rightarrow 2^{n\tilde{R}_{1}}$. There are $2^{nR_{2}}$ possibilities for $\mathbf{X}_{2,j}$. Conditioning on $\mathbf{Z}_{j}$ reduces the possibilities for the pair $(\mathbf{X}_{1,j},\mathbf{X}_{2,j})$ from $2^{n[\tilde{R}_{1}+R_{2}]}\rightarrow 2^{n[\tilde{R}_{1}+R_{2}-I(X_{1},X_{2};Z)]}$. So we have: $H(\mathbf{X}_{1,j},\mathbf{X}_{2,j}|W_{j},\mathbf{Z}_{j})\approx n[\tilde{R}_{1}+R_{2}-I(X_{1},X_{2};Z)]$. Using this in $\eqref{eq:equivcal2:aaaaa}$, and since $R_{2}>I(X_{2};Z)$ and block $B$ does not contribute, we obtain: $\begin{aligned} H(W^{[B-1]}|\mathbf{Z}^{[B]})\geq(B-1)nR_{1}-(B-1)n\epsilon\end{aligned}$

VI Conclusion and Future Work

It remains to be proven that our scheme achieves a higher secrecy rate than [1]. NF achieves secrecy improvement by attacking both Bob and Eve. Since Eve is affected more, Bob’s secrecy rate can improve in certain regimes. In our scheme, Alice has greater control over the relay’s channel codeword in the next block. Our scheme’s decoding delay of two blocks is to be preferred to [1]’s backward decoding delay of $B$ blocks. Lastly, our MBEq calculation is, besides [6] by the author, one of the first of its kind – all other MBEq calculations, such as [7], to the best of our knowledge, were made in the context of LM-NNC without WZ binning. Thus our equivocation calculations may be of independent interest. The next problem to be tackled is the four node dedicated relay broadcast channel with mutual secrecy requirement where the relay is “strong” with respect to one receiver and decode-forwards its intended message, and “weak” (or possibly untrusted) wrt the other receiver, and applies a version of compress-forward to its intended message. We foresee substantial technical challenges, but believe the effort will be worth it.

VII Acknowledgements

The author is grateful for the support of the Bharti Centre for Communication, IIT Bombay.