# Non-Adaptive Coding for Two-Way Wiretap Channel with or without Cost
Constraints

Masahito Hayashi, , and Yanling Chen  Masahito Hayashi is with School of Data
Science, The Chinese University of Hong Kong, Shenzhen, Longgang District,
Shenzhen, 518172, China, International Quantum Academy (SIQA), Futian
District, Shenzhen 518048, China, and Graduate School of Mathematics, Nagoya
University, Chikusa-ku, Nagoya 464-8602, Japan. (e-mail:
<EMAIL_ADDRESS>hayashi@iqasz.cn).Yanling Chen is with Volkswagen
Infotainment GmbH, Germany. (e-mail: yanling.chen@volkswagen-
infotainment.com).MH is supported in part by the National Natural Science
Foundation of China (Grant No. 62171212).

###### Abstract

This paper studies the secrecy results for the two-way wiretap channel (TW-WC)
with an external eavesdropper under a strong secrecy metric. Employing non-
adaptive coding, we analyze the information leakage and the decoding error
probability, and derive inner bounds on the secrecy capacity regions for the
TW-WC under strong joint and individual secrecy constraints. For the TW-WC
without cost constraint, both the secrecy and error exponents could be
characterized by the conditional Rényi mutual information in a concise and
compact form. And, some special cases secrecy capacity region and sum-rate
capacity results are established, demonstrating that adaption is useless in
some cases or the maximum sum-rate that could be achieved by non-adaptive
coding. For the TW-WC with cost constraint, we consider the peak cost
constraint and extend our secrecy results by using the constant composition
codes. Accordingly, we characterize both the secrecy and error exponents by a
modification of Rényi mutual information, which yields inner bounds on the
secrecy capacity regions for the general discrete memoryless TW-WC with cost
constraint. Our method works even when a pre-noisy processing is employed
based on a conditional distribution in the encoder and can be easily extended
to other multi-user communication scenarios.

###### Index Terms:

Rényi mutual information, error exponent, constant composition codes,
information leakage, two-way channel

##  I Introduction

###  I-A Two-way channel

The two-way communications channel (TWC) was first introduced by Shannon in a
pioneering paper [1], where he established an inner bound and an outer bound
for the capacity region of a discrete memoryless TWC. 18 year later, Dueck [2]
provided an example showing that the capacity region of the TWC can exceed
Shannon’s inner bound. Thereafter, Schalkwijk [3] used a constructive coding
approach for the binary multiplier channel (BMC), attaining an achievable rate
pair that is beyond Shannon’s inner bound; whilst a more general coding scheme
for TWC was proposed by Han in [4]. Til now the BMC still serves as an example
of deterministic, binary, common output TWC where capacity in term of single-
letter characterization is not yet known so far.

Why does Shannon’s inner bound not coincide with Shannon’s outer bound in
general? The reason is that the inner bound is achieved using non-adaptive
encoding, where the channel input at each user is constructed from the message
to be transmitted only (and is independent of the past received signals).
Clearly, this non-adaptive encoding makes the channel inputs of the two users
independent. However, the outer bound allows the channel inputs of the two
users dependent, where the dependence could be introduced by the adaption of
channel input to the past received signals at each user.

Interestingly, there are certain special cases where adaptation is useless
from a capacity perspective, making Shannon’s inner bound coincide with
Shannon’s outer bound. These special cases include but not limited to:

  * •

the class of symmetric discrete memoryless TWCs [1];

  * •

the additive white Gaussian noise (AWGN) TWC [4];

  * •

the additive exponential noise TWC [5];

  * •

the finite-field additive noise TWC [6];

  * •

the class of injective semideterministic TWCs [7].

For these channels, the signals received in the past are not needed for the
encoding to achieve any rate pair within the capacity region, rendering the
TWC equivalent to two independent one-way channels. This feature is desirable
in practice for a simpler engineering system design.

###  I-B Two-way wiretap channel (preceding studies)

Information theoretic secrecy was first developed by Shannon in [8], where the
perfect secrecy is achieved if the eavesdropper’s observation $\mathbf{Z}$
provides no information on the transmitted message $M$ (i.e., the information
leakage to the eavesdropper $I(M;\mathbf{Z})=0$). Wyner in [9] applied this
concept to the one-way discrete memoryless channel, which is known as “wiretap
channel”; and generalized the “perfect secrecy” to an “asymptotic perfect
secrecy” by measuring the equivocation at the eavesdropper by the normalized
conditional entropy of the message $M$ given the eavesdropper’s observation
$\mathbf{Z}$. Wyner defined a secrecy capacity, up to which it is possible to
achieve the asymptotic perfect secrecy in the manner that the information
leakage rate to the eavesdropper is negligible as compared to the number of
channel uses $I(M;\mathbf{Z})/H(M)\to 0.$ Such a secrecy measured by
information leakage rate is often referred as “weak secrecy”, as it tolerates
the fact that the eavesdropper might obtain a substantial amount of
information in an absolute sense [10, 11], i.e., $I(M;\mathbf{Z})\nrightarrow
0.$

The two-way wiretap channel (TW-WC) was first considered in [12] for both the
Gaussian TW-WC (with details in [13]) and the binary additive TW-WC, where two
users are communicating with each other in the presence of an external
eavesdropper. Based on the observation that the eavesdropper receives signals
through a multi-access channel (MAC), the references [12, 13] showed that the
controlled interference between transmitted signals (i.e., cooperative
jamming) from the legitimate users could provide secrecy gains. Inner bounds
on the secrecy capacity regions (subject to weak secrecy constraint:
$I(M_{1},M_{2};\mathbf{Z})/H(M_{1},M_{2})\to 0$) for both channels were
derived. Note that these achievable secret-rate regions were established by
using non-adaptive coding, since the cooperative jamming strategy used therein
did not exploit the advantage of the previous received signals for the channel
input.

Can the inner bounds of the capacity regions be improved by utilizing these
previous received signals (i.e., adaptive coding)? The impact of these signals
was studied in [14]. It was demonstrated that adaptive coding could be highly
beneficial for certain Gaussian TW-WCs. Moreover, [15] considered the general
discrete memoryless TW-WC channel, and derived an inner bound of the capacity
region under weak secrecy by combining cooperative jamming and a secret-key
exchange mechanism (i.e., one user sacrifices part of its secret rate to
transmit a key to the other user). Remarkably, for the Gaussian TW-WC, the
region obtained in [15] was shown to be strictly larger than the regions given
in [13], demonstrating the effectiveness of adaptive coding (each user
utilizing a key received previously from the other user to encrypt part of its
message as described in [15]) in terms of enlarging the existing secret-rate
region achieved by non-adaptive coding in [13].

The above mentioned studies on TW-WCs focused on the weak secrecy constraint
that is defined by $\allowbreak
I(M_{1},M_{2};\mathbf{Z})\allowbreak/H(M_{1},M_{2})\to 0$ (which considers the
leakage rate of $M_{1},M_{2}$ jointly). Notably, there are also other secrecy
measures for multi-user communications [16]. Especially, for the TW-WCs, there
are one-sided secrecy [18] and individual secrecy [17]:

(WO)

one-sided secrecy considering the leakage of $M_{1}$ or $M_{2}$ that is
defined by $I(M_{1};\mathbf{Z})/H(M_{1})\to 0$ or
$I(M_{2};\mathbf{Z})/H(M_{2})\to 0$;

(WI)

individual secrecy considering the leakage of $M_{1}$ and $M_{2}$ that is
defined by $I(M_{1};\mathbf{Z})/H(M_{1})\to 0$ and
$I(M_{2};\mathbf{Z})/H(M_{2})\to 0$.

Inner bounds on the secrecy capacity regions for the TW-WCs under such one-
sided/individual secrecy constraints are established in [18] and [17],
respectively.

Besides these studies on TW-WCs under weak secrecy, the reference [19]
considered the general discrete memoryless TW-WC channel under a strong joint
secrecy constraint, where the information leakage to the eavesdropper (rather
than the leakage rate) is required to negligible (i.e.,
$I(M_{1},M_{2};\mathbf{Z})\to 0$). The reference [19] not only provided the
inner bounds on the secrecy capacity regions by using cooperation jamming only
(i.e., non-adaptive coding), but also recovered the achievable secret-rate
region obtained in [15] by using cooperative jamming and key exchange (i.e.,
adaptive coding as in [15]). A noteworthy observation is that the region by
adaptive coding therein did not improve the bound on the secrecy sum-rate that
could be achieved by simply using the non-adaptive coding.

###  I-C Two-way wiretap channel without cost constraint (our contributions)

In this paper, we study the TW-WC (see Fig. 1 subject to a strong secrecy
criterion that can be

(SJ)

joint secrecy defined by $I(M_{1},M_{2};Z^{n})\to 0$; Or

(SI)

individual secrecy defined by $I(M_{1};Z^{n})\to 0$ and $I(M_{2};Z^{n})\to 0$.

By definition, the joint secrecy implies the individual secrecy, resulting in
the first relationship as given in (19). The one-sided secrecy case is not
included here since the analysis is straightforward by following the approach
to the case of individual secrecy. Focusing on non-adaptive coding, we analyze
the information leakage and the decoding error probability and derive inner
bounds on the secrecy capacity regions for the TW-WC under strong joint and
individual secrecy constraints. Our inner bound under the joint secrecy
coincides with the inner bound established in [19, Corollary 1]; while our
inner bound under the individual secrecy is similar to what the reference [18]
obtained for weak secrecy. To this end, different approaches from [19] are
taken to analyze the information leakage and the decoding error probability.
Interestingly, both the secrecy and error exponents could be characterized by
the conditional Rényi mutual information in a concise and compact form. Thus,
the region is guaranteed with both information leakage and decoding error
probability decreasing exponentially in the code length. More specifically,
our technique for strong secrecy analysis reflects the channel resolvability
for the $k$-transmitter MAC, which was obtained in the reference [20] by
extending the method by the references [11, 21]; while our error exponent
could be regarded as a generalization of Gallager’s error exponent [22].

Taking a close look into the additive TW-WC over a finite field, we establish
the following special case results:

  * •

under certain condition (that the eavesdropper observes a degraded version of
the outputs at both legitimate users), the obtained inner bounds under the
strong joint secrecy actually equals to the joint secrecy capacity region
(also for weak secrecy) if only non-adaptive coding is permitted.

  * •

under certain condition (that given inputs, more noise is added to the
eavesdropper’s channel), the obtained inner bounds under the individual strong
secrecy coincides with the capacity region of the channel, thus serving as
also the capacity region under the individual weak secrecy. Since the inner
bounds be achieved by non-adaptive codes, so adaption is useless in this case.

Our work extends the existing results on additive TW-WC [12, 15] in several
directions, such as: considering a more general model (not limiting to the
binary case), considering other secrecy criteria (not necessarily joint
secrecy), and developing special case converse results and so on.

###  I-D Wiretap channel with cost constraint (preceding studies)

When we focus on Gaussian channel, it is natural to impose a cost constraint
on the code. In the case of wiretap channel, the sender uses a stochastic
encoder. That is, to send a specific message, the sender chooses the
transmitted input randomly among several possible inputs. We consider two
kinds of cost constraints. In the first case, the cost constraint is imposed
to all possible inputs, which is called the peak constraint. In the second
case, the cost constraint is imposed to the averaged inputs, which is called
the average constraint.

Clearly, the capacity under the peak constraint is upper bounded by the
capacity under the average constraint. In the case of Gaussian wiretap
channel, the reference [23] derived the capacity under the average constraint
with the weak secrecy. That is, the reference [23] showed the direct and
converse parts under this setting. The essential tool of the converse part
with the weak secrecy is the entropy power inequality [24].

Later, the reference [25] derived an achievable rate under the strong secrecy
with the peak constraint for a general discrete memoryless wiretap channel and
a Poisson wiretap channel. To make the desired code, they employed random
coding method by modifying the distribution to generate codes by taking the
cost constraint into account. Also, the references [21, Appendix D], [26]
derived similar results, and the reference [27] studied this case with the
second order asymptotics. Adding the time sharing to the achievable rate
obtained by [25], the reference [28] derived the capacity under the strong
secrecy with the peak constraint for a general discrete memoryless wiretap
channel. It showed the existence of a channel, in which, the achievable rate
by [25] does not match the capacity under the cost constraint while the
achievable rate by [25] matches the capacity under the cost constraint for
less noisy case [28, Corollary 1]. For the Gaussian wiretap channel, the
references [25] and [21, Appendix D-C] showed the achievability of the rate
obtained by the reference [23] under the strong secrecy with the average
constraint, and the reference [29] showed the same achievability under the
strong secrecy with the peak constraint when the constraint satisfies a
certain condition.

However, it is not so easy to extend their method to our problem setting.
Instead of their method, we focus on constant composition codes because the
cost of each codeword in a constant composition code always equals the average
cost with respect to the distribution to generate the code. Csisár and Körner
[30] studied the channel coding with constant composition codes. Also, the
reference [31] studied the channel coding with the random coding on constant
composition codes. The reference [21, Section XII] extended the coding with
constant composition codes by [30] to the wiretap channel. Recently, the
reference [32] studied quantum soft covering for constant composition codes,
which is equivalent to channel resolvability for constant composition codes.
Therefore, we can expect that the combination of the above two methods yields
another proof of the direct part of wiretap channel with cost constraint.
However, this simple combination does not work even for a single-sender
discrete memoryless wiretap channel due to the following reason. An encoder
for wiretap channel often requires the use of pre-noisy processing based on a
conditional distribution in the encoder. A constant composition code with the
pre-noisy processing does not satisfy the peak constraint, in general.
Therefore, when the wiretap channel is not degraded, we need a more advanced
method for the peak constraint.

###  I-E Two-way wiretap channel with cost constraint (our contributions)

Although several papers studied wiretap channel with peak and average cost
constraints for various channel models, preceding papers [12, 13, 14, 15]
studied two-way wiretap channel only under the average cost constraint and
with the weak secrecy for the Gaussian TW-WC. In this paper, we introduce the
peak cost constraint to the codes for a two-way wiretap channel, and extend
our results on the capacity regions for the TW-WC under strong joint and
individual secrecy constraints (the conditions (SJ) and (SI)) by using the
constant composition codes. Accordingly, based on the random coding on
constant composition codes, we characterize both the secrecy and error
exponents by a modification of Rényi mutual information, which yields inner
bounds of the above capacity regions. It is worth mentioning that the obtained
regions are valid for general discrete memoryless TW-WC which includes not
only the channels with finite-alphabet inputs and outputs, e.g. the additive
TW-WC over a finite field. Our method works even when a pre-noisy processing
is employed based on a conditional distribution in the encoder. Also, our
method with constant composition codes directly works only with finite-
alphabet inputs. However, since the continuous-alphabet inputs can be
approximated by finite-alphabet inputs, a modification of our method also
works with non-finite-alphabet inputs and outputs, e.g. Gaussion TW-WC.

In addition, similar to the reference [23], we also obtained an outer bound
for the capacity region for Gaussion TW-WC with a cost constraint by using the
entropy power inequality [24]. Combining the above results, we identify the
special case where our codes actually achieves the secrecy capacity of the
sum-rates under the joint secrecy criteria (also for weak secrecy) if only
non-adaptive coding is permitted.

###  I-F Outline

The rest of the paper is organized as follows. In Section II, we describe the
system model of a two-way wiretap channel, give definitions of a non-adaptive
or adaptive code (according to different setups at the encoder and decoder),
different secrecy constraints (weak/strong, individual/joint) and present our
main results on inner bounds for the respective secrecy capacity regions. In
Section III-A, we provide the definitions of Rényi mutual information and
conditional Rényi mutual information that are used to characterize the secrecy
and error exponents. In Section III, we derive our inner bounds on the
capacity regions for the TW-WC under strong joint/individual secrecy
constraint using non-adaptive coding when no cost constraint is imposed. More
specifically, we characterize both the secrecy and error exponents, which shed
some lights on the exponential decreasing rates in the leaked information and
the probability of decoding errors with respect to the number of channel uses
while using the non-adaptive coding. In Section IV, we extend our results in
the previous section by introducing cost constraints to the non-adaptive
codes. Using the constant composition codes, we show our inner bounds for the
respective secrecy capacity regions of a TW-WC under a cost constraint, and
provide the secrecy and error exponents accordingly. In Sections V and VI, we
take a close look into the additive TW-WC over a finite field and the Gaussian
TW-WC, respectively. Secrecy capacity results are established for certain
special cases. Also, these two sections derive outer bounds for the capacity
regions for these two special cases. Section VII concludes the paper by
discussing possible extensions of this work.

##  II Channel model and strong secrecy result

###  II-A Formulation

We consider that two legitimate users, User 1 and User 2 intend to exchange
messages with each other in the presence of an external eavesdropper. The two
legitimate users and the external eavesdropper are connected via a discrete
memoryless two-way wiretap channel, which is characterized by
$P_{Y_{1}Y_{2}Z|X_{1}X_{2}}.$ In this model, User $i$ has the input symbol set
${\cal X}_{i}$ and the output symbol set ${\cal Y}_{i}$ for $i=1,2$. The
external eavesdropper has the output symbol set ${\cal Z}$. The channel model
is shown in Fig. 1. When ${\cal X}_{2}$ and ${\cal Y}_{1}$ are singleton, the
above model is a one-to-one wiretap channel. Hence, the two-way wiretap
channel model includes the one-to-one wiretap channel model.

Figure 1: Two-Way wiretap channel with an external eavesdropper.

In this paper, we use the following notations $\times$ and $\cdot$ for
conditional distributions and distributions in the following way:

$\displaystyle P_{Z|X_{1}X_{2}}\times P_{X_{1}|V_{1}}(z,x_{1}|v_{1},x_{2})$
$\displaystyle:=P_{Z|X_{1}X_{2}}(z|x_{1},x_{2})P_{X_{1}|V_{1}}(x_{1}|v_{1}),$
(1) $\displaystyle P_{Z|X_{1}X_{2}}\cdot P_{X_{1}|V_{1}}(z|v_{1},x_{2})$
$\displaystyle:=\sum_{x_{1}\in{\cal
X}_{1}}P_{Z|X_{1}X_{2}}(z|x_{1},x_{2})P_{X_{1}|V_{1}}(x_{1}|v_{1}),$ (2)
$\displaystyle P_{Z|X_{1}X_{2}}\times P_{X_{1}|V_{1}}\times
P_{X_{2}}(z,x_{1},x_{2}|v_{1})$
$\displaystyle:=P_{Z|X_{1}X_{2}}(z|x_{1},x_{2})P_{X_{1}|V_{1}}(x_{1}|v_{1})P_{X_{2}}(x_{2}),$
(3) $\displaystyle P_{Z|X_{1}X_{2}}\cdot P_{X_{1}|V_{1}}\times
P_{X_{2}}(z,x_{2}|v_{1})$ $\displaystyle:=\sum_{x_{1}\in{\cal
X}_{1}}P_{Z|X_{1}X_{2}}(z|x_{1},x_{2})P_{X_{1}|V_{1}}(x_{1}|v_{1})P_{X_{2}}(x_{2}).$
(4)

Their messages $M_{i}$ are assumed to be uniformly distributed over the
message sets $\mathcal{M}_{i}=[1:e^{nR_{i}}]$ for $i=1,2$. For their
communication, they use the above channel $n$ times, and we denote User $i$’s
channel input and output by $X_{i}^{n}=(X_{i,1},\ldots,X_{i,n})\in{\cal
X}_{i}^{n}$ and $Y_{i}^{n}=(Y_{i,1},\ldots,Y_{i,n})\in{\cal Y}_{i}^{n}$,
respectively. Also, we denote the channel output at the eavesdropper by
$Z^{n}=(Z_{1},\ldots,Z_{n})\in{\cal Z}^{n}$.

Consider the encoder of User $i$ for $i=1,2$. We consider two types of
encoders for two users. The first type of encoder is a non-adaptive encoder
$\phi_{i}$, which stochastically assigns the whole input $X_{i}^{n}$ based on
the message $M_{i}$. The second type of encoder is an adaptive encoder
$\phi_{i}$, which stochastically assigns the $t$-th input $X_{i,t}$ based on
the message $M_{i}$ and the previous outputs $Y_{i,1},\ldots,Y_{i,t-1}$ for
$t=1,\ldots,n$.

Consider the decoder of User $i$ for $i=1,2$. We consider two types of
decoders for two users. In the 1st type decoder, the decoder $\psi_{i}$ of
User $i$ is given as a map from (${\cal M}_{i}$, ${\cal X}_{i}^{n}$, ${\cal
Y}_{i}^{n}$) to ${\cal M}_{i\oplus 1}.$ In the 2nd type decoder, the decoder
$\psi_{i}$ of User $i$ is given as a map from (${\cal X}_{i}^{n}$, ${\cal
Y}_{i}^{n}$) to ${\cal M}_{i\oplus 1}.$ Note that for two-way channels without
any secrecy constraint, deterministic encoders are often used where user $i$
can recover information $X_{i}^{n}$ from $M_{i}$ and $Y_{i}^{n}.$ Hence, the
1st type decoder $\psi_{i}$ of User $i$ reduces to a map from (${\cal M}_{i}$,
${\cal Y}_{i}^{n}$) to ${\cal M}_{i\oplus 1}$ as can be seen in previous works
which include but not limited to [1, 5, 33, 6, 7, 34]. However, for the TW-WC
with secrecy constraint, stochastic encoders are often used where User $i$
cannot recover information $X_{i}^{n}$ from $M_{i}$ and $Y_{i}^{n}$ only. This
is the reason why we consider the above two types of decoders.

When both encoders are non-adaptive encoders, we have the joint distribution
as

$\displaystyle
P_{M_{1}M_{2}X_{1}^{n}X_{2}^{n}Y_{1}^{n}Y_{2}^{n}Z^{n}}(m_{1},m_{2},x_{1}^{n},x_{2}^{n},y_{1}^{n},y_{2}^{n},z^{n})$
$\displaystyle=$ $\displaystyle
P_{M_{1}}(m_{1})P_{M_{2}}(m_{2})P_{X_{1}^{n}|M_{1}}(x_{1}^{n}|m_{1})P_{X_{2}^{n}|M_{2}}(x_{2}^{n}|m_{2})P_{Y_{1}^{n}Y_{2}^{n}Z^{n}|X_{1}^{n}X_{2}^{n}}(y_{1}^{n},y_{2}^{n},z^{n}|x_{1}^{n},x_{2}^{n}),$
(5)

which implies the Markovian chains

$\displaystyle M_{i}-X_{i}^{n}-(M_{i\oplus 1},Y_{i}^{n})$ (6)

for $i=1,2$. Hence, knowing $M_{i}$ does not help the decoder $\psi_{i}$ of
User $i$ (who holds (${\cal X}_{i}^{n}$, ${\cal Y}_{i}^{n}$)) gain any
information on $M_{i\oplus 1}$ (here $\oplus$ denotes the modulo-2 addition).
That is, it is sufficient to use the 2nd type decoder in this case.

Therefore, a $(e^{nR_{1}},e^{nR_{2}},n)$ adaptive (non-adaptive) secrecy code
$C_{n}$ with the 1st (2nd) type decoder for the two-way wiretap channel
consists of $2$ message sets ($\mathcal{M}_{1},\mathcal{M}_{2}$), $2$ adaptive
(non-adaptive) encoders ($\phi_{1}$, $\phi_{2}$), and $2$ decoders
($\psi_{1}$, $\psi_{2}$) of the 1st (2nd) type. When the encoder $\phi_{i}$
takes values only in a subset ${\cal S}_{i}\subset{\cal X}_{i}^{n}$ for
$i=1,2$, we say that a code $C_{n}$ is in ${\cal S}_{1}\times{\cal S}_{2}$. We
denote $(R_{1},R_{2})$ by $\kappa(C_{n})$.

To evaluate the reliability of the transmission to the legitimate receiver, we
consider the _average probability of decoding error_ at the legitimate
receiver that is defined by

$P_{e}^{n}(C_{n})=\frac{1}{2^{n[R_{1}+R_{2}]}}\sum_{m_{1},m_{2}}\Pr\left\\{\bigcup\limits_{i\in\\{1,2\\}}\\{m_{i}\neq\hat{m}_{i}\\}\Bigg{|}C_{n}\right\\}.$
(7)

To evaluate the secrecy of the transmission, we consider the strong secrecy
that measures the amount of information (of $(M_{1},M_{2})$ jointly or of
$M_{1},M_{2}$ individually) leaked to the eavesdropper (through his/her
observation $Z^{n}$). We say that the rate pair $(R_{1},R_{2})$ is _achievable
under the strong joint secrecy constraint by adaptive (non-adaptive) codes
with the 1st (2nd) type decoder_ , if there exists a sequence of
$(e^{nR_{1,n}},e^{nR_{2,n}},n)$ adaptive (non-adaptive) codes $\\{C_{n}\\}$
with the 1st (2nd) type decoder such that $R_{i,n}\to R_{i}$ for $i=1,2$ and

$\displaystyle P_{e}^{n}(C_{n})$ $\displaystyle\leq\epsilon_{n},$ (8)
$\displaystyle I(M_{1},M_{2};Z^{n}|C_{n})$ $\displaystyle\leq\tau_{n},$ (9)

with $\lim\limits_{n\to\infty}\epsilon_{n}=0$ and
$\lim\limits_{n\to\infty}\tau_{n}=0.$ We say that the rate pair
$(R_{1},R_{2})$ is _achievable under the strong individual secrecy constraint
by adaptive (non-adaptive) codes with the 1st (2nd) type decoder_ , if there
exists a sequence of $(e^{nR_{1,n}},e^{nR_{2,n}},n)$ adaptive (non-adaptive)
codes $\\{C_{n}\\}$ with the 1st (2nd) type decoder such that $R_{i,n}\to
R_{i}$ for $i=1,2$ and

$\displaystyle P_{e}^{n}(C_{n})$ $\displaystyle\leq\epsilon_{n},$ (10)
$\displaystyle I(M_{1};Z^{n}|C_{n})$ $\displaystyle\leq\tau_{n},$ (11)
$\displaystyle I(M_{2};Z^{n}|C_{n})$ $\displaystyle\leq\tau_{n}.$ (12)

We define the achievability under the _weak_ joint secrecy constraint by
adaptive (non-adaptive) codes by replacing $\tau_{n}$ by $n\tau_{n}$ in (9).
In the same way, we define the achievability under the _weak_ individual
secrecy constraint by adaptive (non-adaptive) codes by replacing $\tau_{n}$ by
$n\tau_{n}$ in (11) and (12).

###  II-B Capacity region (without cost constraint)

We have the following types of capacity regions.

$\displaystyle{\cal C}_{J,S(W),A(N),1(2)}$
$\displaystyle:=\left\\{(R_{1},R_{2})\left|\begin{array}[]{l}(R_{1},R_{2})\hbox{
is achievable under the strong (weak) joint secrecy}\\\ \hbox{constraint by
adaptive (non-adaptive) codes with the 1st (2nd) type
decoder}\end{array}\right.\right\\},$ (15) $\displaystyle{\cal
C}_{I,S(W),A(N),1(2)}$
$\displaystyle:=\left\\{(R_{1},R_{2})\left|\begin{array}[]{l}(R_{1},R_{2})\hbox{
is achievable under the strong (weak) individual secrecy}\\\ \hbox{constraint
by adaptive (non-adaptive) codes with the 1st (2nd) type
decoder}\end{array}\right.\right\\}.$ (18)

We have the following relations

$\displaystyle{\cal C}_{J,b,c,d}\subset{\cal C}_{I,b,c,d},~{}{\cal
C}_{a,S,c,d}\subset{\cal C}_{a,W,c,d},~{}{\cal C}_{a,b,N,d}\subset{\cal
C}_{a,b,A,d},~{}{\cal C}_{a,b,c,2}\subset{\cal C}_{a,b,c,1}$ (19)

for $a\in\\{I,J\\}$, $b\in\\{S,W\\}$, $c\in\\{N,A\\}$ and $d\in\\{1,2\\}.$ In
addition, the Markovian chain (6) implies the relation

$\displaystyle{\cal C}_{a,b,N,2}={\cal C}_{a,b,N,1}.$ (20)

In wiretap channel coding, we often employ such a pre-noisy processing given
by $P_{X_{i}|V_{i}}$ for $i=1,2$. That is, the transmission rate of wiretap
channel can be improved when the encoder applies the pre-noisy processing
given by a certain conditional distribution. Since we cannot deny this
possibility in a two-way wiretap channel, we employ such pre-noisy processings
to derive inner bounds of the above capacity regions.

Considering the strong joint secrecy criteria (9), we define the region

$\displaystyle{\cal
R}_{J,N}(P_{V_{1}V_{2}X_{1}X_{2}}):=\left\\{(R_{1},R_{2})\in\mathbb{R}_{+}^{2}\left|\begin{array}[]{l}R_{1}\leq
I(Y_{2};V_{1}|X_{2})-I(Z;V_{1}),\\\ R_{2}\leq
I(Y_{1};V_{2}|X_{1})-I(Z;V_{2}),\\\ R_{1}+R_{2}\leq
I(Y_{2};V_{1}|X_{2})+I(Y_{1};V_{2}|X_{1})-I(V_{1},V_{2};Z)\end{array}\right.\right\\}$
(24)

for a joint distribution $P_{V_{1}V_{2}X_{1}X_{2}}$, where $\mathbb{R}_{+}$ is
the set of positive real numbers.

We also define ${\cal
Q}:=\\{\prod\limits_{i\in\\{1,2\\}}P_{V_{i}}(v_{i})P_{X_{i}|V_{i}}(x_{i}|v_{i})\\}$.
Then, we define the region

$\displaystyle{\cal R}_{J,N}:=$ $\displaystyle
cl.\bigcup_{P_{V_{1}V_{2}X_{1}X_{2}}\in{\cal Q}}{\cal
R}_{J,N}(P_{V_{1}V_{2}X_{1}X_{2}}),$ (25)

where $cl.$ expresses the closure of the convex hull.

Considering the strong individual secrecy criteria (11) and (12), we define
the region

$\displaystyle{\cal
R}_{I,N}(P_{V_{1}V_{2}X_{1}X_{2}}):=\left\\{(R_{1},R_{2})\in\mathbb{R}_{+}^{2}\left|\begin{array}[]{l}R_{1}\leq
I(Y_{2};V_{1}|X_{2})-I(Z;V_{1}),\\\ R_{2}\leq
I(Y_{1};V_{2}|X_{1})-I(Z;V_{2}),\\\ \max\\{R_{1},R_{2}\\}\leq
I(Y_{1};V_{2}|X_{1})+I(Y_{2};V_{1}|X_{2})-I(V_{1},V_{2};Z)\end{array}\right.\right\\}$
(29)

for a joint distribution $P_{V_{1}V_{2}X_{1}X_{2}}.$ Then, we define the
region

$\displaystyle{\cal R}_{I,N}:=$ $\displaystyle
cl.\bigcup_{P_{V_{1}V_{2}X_{1}X_{2}}\in{\cal Q}}{\cal
R}_{I,N}(P_{V_{1}V_{2}X_{1}X_{2}}).$ (30)

We have the following theorem for the achievable secrecy rate region of the
two-way wiretap channel.

###### Theorem 1

We have the following relations.

$\displaystyle{\cal R}_{J,N}\subset{\cal C}_{J,S,N,2},\quad$
$\displaystyle{\cal R}_{I,N}\subset{\cal C}_{I,S,N,2}.$ (31)

The reference [19, Corollary 1] also showed the same relation as the first
relation in (31).

###### Remark 1

In general, it is a difficult task to establish converse results for the
general TW-WC, undoubtably more difficult than the task to derive converse
results for the TWC (as TWC can be considered as a special case of TW-WC by
taking $Z=\emptyset$). As a well-known fact, it still remains an open problem
to establish the capacity region for the general TWC since TWC was first
introduced by Shannon in [1].

Due to the fact that a TW-WC is a TWC under additional secrecy constraints,
any outer bound on the capacity region for a general TWC, e.g. Shannon’s outer
bound [1]; or capacity region established for a certain TWC, e.g. capacity
region established for the finite field additive TWC [6], could serve as a
trial outer bound on the secrecy capacity regions for the corresponding TW-WC
as well. However, the derivation of a tight outer bound on the secrecy
capacity regions is generally difficult because of the following reasons.

  * •

An adaptive encoder has many random variables to adaptively improve our code.
In general, it is difficult to reduce the analysis into a single-letterized
region.

  * •

We could derive an outer bound only when the channel is additive and the
security criterion is individual, which is stated as (131) in Theorem 4. It is
unclear whether an adaptive encoder outperforms our obtained inner bounds in
the general case.

  * •

Note that a special case individual secrecy capacity region is established in
Corollary 3 for the finite field additive TWC, where the individual secrecy
capacity region for the TW-WC actually coincides with the capacity region for
the TWC.

Analysis is even still difficult for non-adaptive codes due to the following
reasons.

  * •

Wiretap codes employ stochastic encoders. In the one-to-one channel, the
method by Ciszár and Körner [30, Theorem 17.11] converts a joint distribution
on $X^{n}$ and $V^{n}$ on $n$ channel uses to a joint distribution on $X$ and
$V$ on a single channel use. However, in the two-way channel case, we have a
joint distribution $P_{X_{1}^{n}V_{1}^{n}}\times P_{X_{2}^{n}V_{2}^{n}}$ on
$n$ channel uses. In this joint distribution, $X_{1}^{n},V_{1}^{n}$ and
$X_{2}^{n},V_{2}^{n}$ are independent of each other. When we apply the same
method as Ciszár and Körner, $X_{1},V_{1}$ and $X_{2},V_{2}$ might not be
independent of each other under the converted joint distribution
$P_{X_{1}V_{1}X_{2}V_{2}}$ in general.

  * •

Fortunately, we could resolve this problem only for additive channels and
Gaussian additive channels with certain conditions (by exploiting the channel
characteristics instead of using the Ciszár-Körner method). See Theorem 4 and
Theorem 6.

###  II-C Capacity region (with cost constraint)

Next, we introduce cost constraint for our codes. We choose cost functions
$g_{1}$ and $g_{2}$ on ${\cal X}_{1}$ and ${\cal X}_{2}$, respectively. Then,
given two real numbers $c_{1}$ and $c_{2}$, we impose the following cost
constraint for a $(e^{nR_{1}},e^{nR_{2}},n)$ code
$C_{n}=(\phi_{1},\phi_{2},\psi_{1},\psi_{2})$. We define the subset ${\cal
X}_{i}^{n}(c_{i})$ of ${\cal X}_{i}^{n}$ as

$\displaystyle{\cal
X}_{i}^{n}(c_{i}):=\Bigg{\\{}(x_{i,1},\ldots,x_{i,n})\in{\cal
X}_{i}^{n}\Bigg{|}\frac{1}{n}\sum_{j=1}^{n}g_{i}(x_{i,j})\leq c_{i}\Bigg{\\}}$
(32)

for $i=1,2$. We say that the encoder satisfies the peak cost constraint with
$c_{1},c_{2}$ when the input $X_{i}^{n}$ of the encoder $\phi_{i}$ takes
values in the subset ${\cal X}_{i}^{n}(c_{i})$ for $i=1,2$. We say that the
encoder satisfies the average cost constraint with $c_{1},c_{2}$ when the
relation $\frac{1}{n}(\mathbb{E}\sum_{j=1}^{n}g_{i}(X_{i,j}))\leq c_{i}$ holds
for $i=1,2$.

Then, we have the following types of capacity regions.

$\displaystyle{\cal C}_{J,S(W),A(N),1(2)}^{P(A),c_{1},c_{2}}$
$\displaystyle:=\left\\{(R_{1},R_{2})\left|\begin{array}[]{l}(R_{1},R_{2})\hbox{
is achievable under the strong (weak) joint secrecy}\\\ \hbox{constraint by
adaptive (non-adaptive) codes with the 1st (2nd) type decoder}\\\ \hbox{and
with peak (average) cost constraint with
}c_{1},c_{2}.\end{array}\right.\right\\},$ (36) $\displaystyle{\cal
C}_{I,S(W),A(N),1(2)}^{P(A),c_{1},c_{2}}$
$\displaystyle:=\left\\{(R_{1},R_{2})\left|\begin{array}[]{l}(R_{1},R_{2})\hbox{
is achievable under the strong (weak) individual secrecy}\\\ \hbox{constraint
by adaptive (non-adaptive) codes with the 1st (2nd) type decoder}\\\ \hbox{and
with peak (average) cost constraint with
}c_{1},c_{2}.\end{array}\right.\right\\}.$ (40)

We have five indexes in total for characterizing the capacity regions as shown
in Table I. That is, we have $2^{5}=32$ capacity regions.

TABLE I: Summary of problem settings

Indexes | Secrecy criterion | Encoder | Decoder | Cost constraint
---|---|---|---|---
Abbreviations | $J(I)$ | $S(W)$ | $A(N)$ | $1(2)$ | $P(A)$
Definitions |  $J:$ Joint secrecy |  $S:$ Strong secrecy |  $A:$ Adaptive |  $1:$ 1st type decoder |  $P:$ Maximum of cost
is considered. | is imposed. | encoder is used. | is used. | is constrained.
$I:$ Individual secrecy |  $W:$ Weak secrecy |  $N:$ Non-adaptive |  $2:$ 2nd type decoder |  $A:$ Average of cost
is considered. | is imposed. | encoder is used. | is used. | is constrained.

In total, we have $2^{5}=32$ problem settings.

We have the relation

$\displaystyle{\cal C}_{a,b,c,d}^{P,c_{1},c_{2}}\subset{\cal
C}_{a,b,c,d}^{A,c_{1},c_{2}},$ (41)

for $a\in\\{I,J\\}$, $b\in\\{S,W\\}$, $c\in\\{N,A\\}$ and $d\in\\{1,2\\}.$

###### Lemma 1

For $a\in\\{I,J\\}$, $b\in\\{S,W\\}$, $c\in\\{N,A\\}$ and $d\in\\{1,2\\},$ the
regions ${\cal C}_{a,b,c,d}^{P,c_{1},c_{2}}$ and ${\cal
C}_{a,b,c,d}^{A,c_{1},c_{2}}$ are closed sets.

###### Proof:

See Appendix A for a detailed proof. ∎

For a joint distribution $P_{V_{1}V_{2}X_{1}X_{2}}$, we define the condition:

$\displaystyle\sum_{x_{1}\in{\cal X}_{1}}P_{X_{1}}(x_{1})g_{1}(x_{1})\leq
c_{1},\quad\sum_{x_{2}\in{\cal X}_{2}}P_{X_{2}}(x_{2})g_{2}(x_{2})\leq c_{2}.$
(42)

We define the set ${\cal
Q}^{c_{1},c_{2}}:=\Big{\\{}\prod\limits_{i\in\\{1,2\\}}P_{V_{i}}(v_{i})P_{X_{i}|V_{i}}(x_{i}|v_{i})\Big{|}$
the condition (42) holds $\Big{\\}}$. Then, we define the regions

$\displaystyle{\cal R}_{J,N}^{c_{1},c_{2}}:=$ $\displaystyle
cl.\bigcup_{P_{V_{1}V_{2}X_{1}X_{2}}\in{\cal Q}^{c_{1},c_{2}}}{\cal
R}_{J,N}(P_{V_{1}V_{2}X_{1}X_{2}}),$ (43) $\displaystyle{\cal
R}_{I,N}^{c_{1},c_{2}}:=$ $\displaystyle
cl.\bigcup_{P_{V_{1}V_{2}X_{1}X_{2}}\in{\cal Q}^{c_{1},c_{2}}}{\cal
R}_{I,N}(P_{V_{1}V_{2}X_{1}X_{2}}).$ (44)

As a generalization of Theorem 1, we have the following theorem for the
achievable secrecy rate region of the two-way wiretap channel with cost
constraint.

###### Theorem 2

We have the following relations.

$\displaystyle{\cal R}_{J,N}^{c_{1},c_{2}}\subset{\cal
C}_{J,S,N,2}^{P,c_{1},c_{2}},\quad{\cal R}_{I,N}^{c_{1},c_{2}}\subset{\cal
C}_{I,S,N,2}^{P,c_{1},c_{2}}.$ (45)

Further, we extend the regions ${\cal R}_{I,N}^{c_{1},c_{2}}$ and ${\cal
R}_{J,N}^{c_{1},c_{2}}$ as

$\displaystyle\hat{{\cal
R}}_{I(J),N}^{c_{1},c_{2}}:=\bigcup_{P,\vec{c}_{1},\vec{c}_{2}}\Big{\\{}\Big{(}\sum_{j}Q(j)R_{1}^{j},\sum_{j}Q(j)R_{2}^{j}\Big{)}\Big{|}(R_{1}^{j},R_{2}^{j})\in{\cal
R}_{I(J),N}^{c_{1}^{j},c_{2}^{j}}\Big{\\}},$ (46)

where $P,\vec{c}_{1},\vec{c}_{2}$ are chosen as follows. $\vec{c}_{i}$ is a
vector whose components take values in the interval between the infimum of the
range of $g_{i}$ and the supremun of the range of $g_{i}$. The numbers of
components of $\vec{c}_{1}$ and $\vec{c}_{2}$ are arbitrarily chosen as the
same number $n$. $Q$ is a distribution on $\\{1,\ldots,n\\}$ to satisfy
$\sum\limits_{j=1}^{n}Q(j)c_{1}^{j}\leq c_{1}$ and
$\sum\limits_{j=1}^{n}Q(j)c_{2}^{j}\leq c_{2}$. Considering time sharing in
(45), we obtain the following relation.

$\displaystyle\hat{{\cal R}}_{J,N}^{c_{1},c_{2}}\subset{\cal
C}_{J,S,N,2}^{P,c_{1},c_{2}},\quad\hat{{\cal
R}}_{I,N}^{c_{1},c_{2}}\subset{\cal C}_{I,S,N,2}^{P,c_{1},c_{2}}.$ (47)

Here, we stress that the extension (47) is essential because, as pointed in
the reference [28], there exists a one-to-one wiretap channel such that the
inner bound (with time-sharing) in (47) is strictly larger than the inner
bound (without time-sharing) in (45).

##  III Proof of Theorem 1

To prove Theorem 1, we need to find a sequence of non-adaptive codes
${\mathcal{C}_{n}}$ with the 2nd type decoder such that

  * •

the codes have rates pair converge to $(R_{1},R_{2})$ within the region ${\cal
R}_{J,N}$ and ${\cal R}_{I,N}$ for joint secrecy and individual secrecy,
respectively;

  * •

the decoding error is bounded by $\epsilon_{n}$ with
$\lim\limits_{n\to\infty}\epsilon_{n}=0;$

  * •

the information leakage is bounded by $\tau_{n}$ with
$\lim\limits_{n\to\infty}\tau_{n}=0.$

To this end, we propose a coding scheme and evaluate the decoding error (upper
bounded by $\epsilon_{n}$) and respective information leakage (upper bounded
by $\tau_{n}$) for joint secrecy and individual security. Especially, we show
that both $\epsilon_{n}$ and $\tau_{n}$ are decreasing exponentially with
respect to the code length $n$ (thus establishing the strong secrecy). As a
preparation, in Sec. III-A we first give some definitions on the (conditional)
Rényi mutual information, which are used to quantize $\epsilon_{n}$ and
$\tau_{n}$. The single-shot analysis is provided in Sec. III-B, laying
foundations for the analysis of the coding scheme proposed in Sec. III-C. The
reliability and strong secrecy analysis of the coding scheme are provided in
Sec. III-D and Sec. III-E, respectively. Sec. IV-D derives the desired region
in Theorem 1.

###  III-A Conditional Rényi mutual information

We recall the Rényi relative entropy as

$\displaystyle D_{1+s}(P\|Q):=\frac{1}{s}\log\sum_{x}P(x)^{1+s}Q(x)^{-s}.$
(48)

Note that $D_{1+s}(P\|Q)$ is nondecreasing w.r.t. $s$ for $s>0$ and
$\lim\limits_{s\to 0}D_{1+s}(P\|Q)=D(P\|Q),$ i.e., the relative entropy.

We define the Rényi mutual information as

$\displaystyle I_{1+s}^{\downarrow}(Z;X):=D_{1+s}(P_{ZX}\|P_{Z}\times P_{X}).$
(49)

and the conditional Rényi mutual information as

$\displaystyle e^{-sI_{\frac{1}{1+s}}^{\uparrow}(Z;X|Y)}:=$
$\displaystyle\sum_{y}P_{Y}(y)e^{-s\min\limits_{Q_{Z|Y=y}}D_{\frac{1}{1+s}}(P_{ZX|Y=y}\|Q_{Z|Y=y}\times
P_{X|Y=y})}.$ (50)

The minimizer $Q_{Z|Y}^{*}$ is given as

$Q_{Z|Y}^{*}(z|y)=\frac{\sum\limits_{x}P_{X|Y}(x|y)P_{Z|XY}(z|x,y)^{\frac{1}{1+s}}}{\sum\limits_{z^{\prime}}\sum\limits_{x^{\prime}}P_{X|Y}(x^{\prime}|y)P_{Z|XY}(z^{\prime}|x^{\prime},y)^{\frac{1}{1+s}}},$
(51)

and $I_{\frac{1}{1+s}}^{\uparrow}(Z;X|Y)$ is written in the following form
[35, Eq. (54)];

$\displaystyle e^{-sI_{\frac{1}{1+s}}^{\uparrow}(Z;X|Y)}=$
$\displaystyle\sum_{y}P_{Y}(y)\sum_{z}P_{Z|Y}(z|y)\Big{(}\sum_{x}P_{X|Y}(x|y)^{\frac{s}{1+s}}P_{X|ZY}(x|z,y)^{\frac{1}{1+s}}\Big{)}^{1+s}$
$\displaystyle=$
$\displaystyle\sum_{y}P_{Y}(y)\sum_{z}\Big{(}\sum_{x}P_{X|Y}(x|y)\Big{(}\frac{P_{ZX|Y}(z,x|y)}{P_{X|Y}(x|y)}\Big{)}^{\frac{1}{1+s}}\Big{)}^{1+s}$
$\displaystyle=$
$\displaystyle\sum_{y}P_{Y}(y)\sum_{z}\Big{(}\sum_{x}P_{X|Y}(x|y)P_{Z|XY}(z|x,y)^{\frac{1}{1+s}}\Big{)}^{1+s}.$
(52)

When $X$ and $Y$ are independent of each other, it is simplified as

$\displaystyle e^{-sI_{\frac{1}{1+s}}^{\uparrow}(Z;X|Y)}=$
$\displaystyle\sum_{y}P_{Y}(y)\sum_{z}\Big{(}\sum_{x}P_{X}(x)P_{Z|XY}(z|x,y)^{\frac{1}{1+s}}\Big{)}^{1+s}$
$\displaystyle=$
$\displaystyle\sum_{y,z}\Big{(}\sum_{x}P_{X}(x)(P_{Y}(y)P_{Z|XY}(z|x,y))^{\frac{1}{1+s}}\Big{)}^{1+s}=e^{-sI_{\frac{1}{1+s}}^{\uparrow}(ZY;X)}$
(53) $\displaystyle=$ $\displaystyle
e^{-s\min\limits_{Q_{ZY}}D_{\frac{1}{1+s}}(P_{ZYX}\|Q_{ZY}\times P_{X})}.$
(54)

In addition, the minimizer $Q_{ZY}^{*}$ in (54) is given as

$Q_{ZY}^{*}(z,y)=\frac{\sum\limits_{x}P_{X}(x)P_{ZY|X}(z,y|x)^{\frac{1}{1+s}}}{\sum\limits_{z^{\prime},y^{\prime}}\sum\limits_{x^{\prime}}P_{X}(x^{\prime})P_{ZY|X}(z^{\prime},y^{\prime}|x^{\prime})^{\frac{1}{1+s}}}.$
(55)

When we need to identify the joint distribution of $ZY$ and $X$, we denote it
as $I_{\frac{1}{1+s}}^{\uparrow}(ZY;X)[P_{ZYX}]$.

Also, we have the inequality [36, Lemma 1]

$\displaystyle e^{sI_{1+s}^{\downarrow}(Z;X)}\leq
e^{sI_{\frac{1}{1-s}}^{\uparrow}(Z;X)}.$ (56)

When $P_{Z|X}\leq c\cdot Q_{Z|X}$ with a constant $c$, i.e., $P_{Z|X}(z|x)\leq
c\cdot Q_{Z|X}(z|x)$, the expression (53) implies

$\displaystyle e^{sI_{\frac{1}{1-s}}^{\uparrow}(Z;X)[P_{Z|X}\times P_{X}]}\leq
c\cdot e^{sI_{\frac{1}{1-s}}^{\uparrow}(Z;X)[Q_{Z|X}\times P_{X}]}.$ (57)

This quantity coincides with the Gallager function [22].

As another version of Rényi mutual information, the paper [32, (6)] introduced
a modification $\breve{I}_{\frac{1}{1+s}}^{\uparrow}(Z;X)$ of
$I_{\frac{1}{1+s}}^{\uparrow}(Z;X)$ as

$\displaystyle\breve{I}_{\frac{1}{1+s}}^{\uparrow}(Z;X):=$
$\displaystyle\min_{Q_{Z}}\sum_{x}P_{X}(x)D_{\frac{1}{1+s}}(P_{Z|X=x}\|Q_{Z}).$
(58)

Then, $\breve{I}_{\frac{1}{1+s}}^{\uparrow}(Z;X|Y)$ is defined as

$\displaystyle e^{-s\breve{I}_{\frac{1}{1+s}}^{\uparrow}(Z;X|Y)}:=$
$\displaystyle\sum_{y}P_{Y}(y)e^{-s\min_{Q_{Z|Y=y}}\sum_{x}P_{X|Y}(x|y)D_{\frac{1}{1+s}}(P_{Z|X=x,Y=y}\|Q_{Z|Y=y})}.$
(59)

When $X$ and $Y$ are independent of each other, as a generalization of above
quantity, we have
$\breve{I}_{\frac{1}{1+s}}^{\uparrow}(Z;X|Y):=\breve{I}_{\frac{1}{1+s}}^{\uparrow}(ZY;X)$.

###  III-B Single shot analysis

To show Theorem 1, we prepare single shot analysis. We have the following
resolvability lemma.

###### Lemma 2

Let $P_{Z|X_{1}X_{2}}$ be a MAC with two input systems. Assume that
$P_{X_{1}X_{2}}=P_{X_{1}}\times P_{X_{2}}$. Let
$X_{i,1},\ldots,X_{i,\mathsf{L}_{i}}$ be the random variables that are
independently generated subject to $P_{X_{i}},$ for $i=1,2$. Here, we denote
this code selection by ${C}$ and define the distribution $P_{Z|C}$ for the
variable $Z$ dependently of the code selection ${C}$ as
$P_{Z|C}(z):=\frac{1}{\mathsf{L}_{1}\mathsf{L}_{2}}\sum\limits_{j_{1},j_{2}}P_{Z|X_{1}=X_{1,j_{1}},X_{2}=X_{2,j_{2}}}(z).$
Then, for $s\in[0,1]$, we have $D(P_{Z|C}\|P_{Z})\leq D_{1+s}(P_{Z|C}\|P_{Z})$
and

$\displaystyle
s\mathbb{E}_{C}\left[D_{1+s}(P_{Z|C}\|P_{Z})\right]\leq\sum_{\mathcal{S}\neq\emptyset,\mathcal{S}\subset\\{1,2\\}}\frac{1}{\prod_{i\in\mathcal{S}}\mathsf{L}_{i}^{s}}e^{sI^{\downarrow}_{1+s}(Z;X_{\mathcal{S}})}$

for $s\in[0,1]$.

###### Proof:

See Appendix B for a detailed proof. ∎

###### Remark 2

Note that Lemma 2 is a generalization of a special case of [21, Theorem 14].
(Also, the paper [37] uses a method similar to [21, Theorem 14].) Its
extension to $k$-transmitter MAC is given in the reference [20, Lemma 1].
Although the reference [38] discussed a similar type of channel resolvability,
it assumes symmetric structure in the same way as [21, Theorem 17]. Besides,
Lemma 2 can be regarded as an extension of the resolvability for the
2-transmitter MAC [39] by tying the strong secrecy to channel resolvability
and further characterizing the secrecy exponent of the coding mechanism.

For reliability analysis, we have the following lemma as a generalization of
Gallager’s bound [22] for two-transmitter MAC.

###### Lemma 3

Let $P_{Y|X_{1}X_{2}}$ be a MAC with two input systems. Assume that the
$\mathsf{N}$ messages are randomly selected as the random variables
$X_{2,1},\ldots,X_{2,\mathsf{N}}$ that are independently generated subject to
$P_{X_{2}}$. Also, assume that the receiver knows the other random variable
$X_{1}$ and selects it subject to $P_{X_{1}}$ independently of
$X_{2,1},\ldots,X_{2,\mathsf{N}}$. Here, we denote the above selection by
${C^{\prime}}$ and the decoding error probability under the maximum likelihood
(ML) decoder by $e_{C^{\prime}}$. Then, for $s\in[0,1]$, we have

$\displaystyle\mathbb{E}_{C^{\prime}}\left[e_{C^{\prime}}\right]\leq\mathsf{N}^{s}e^{-sI_{\frac{1}{1+s}}^{\uparrow}(YX_{1};X_{2})}=\mathsf{N}^{s}e^{-sI_{\frac{1}{1+s}}^{\uparrow}(Y;X_{2}|X_{1})}.$
(60)

###### Proof:

Lemma 3 can be easily shown from Gallager’s bound [22]. The above situation
can be considered as the channel coding with the channel given by the
conditional distribution $P_{YX_{1}|X_{2}}$. Application of Gallager’s bound
to this channel yields (60). ∎

###  III-C Exponential evaluation

Consider the case where both encoders use non-adaptive codes and both decoders
use the 2nd type decoder.

Codebook Generation: At transmitter $i,$ let
$V_{i,1}^{n},\ldots,V_{i,\mathsf{M}_{i}\cdot\mathsf{L}_{i}}^{n}$ be random
variables that are independently generated subject to
$P_{V_{i}^{n}}=\prod\limits_{j=1}^{n}P_{V_{i,j}},$ where
$V_{i}^{n}=(V_{i,1},\cdots,V_{i,n}),$ $\mathsf{M}_{i}=e^{nR_{i}}$ and
$\mathsf{L}_{i}=e^{nr_{i}}$ for $i=1,2.$

Encoding: When the legitimate user $i$ intends to send the message $m_{i}$,
he(she) chooses one of
${V}^{n}_{i,(m_{i}-1)\mathsf{L}_{i}+1},\ldots,{V}^{n}_{i,m_{i}\mathsf{L}_{i}}$
with equal probability, which is denoted to be ${V}^{n}_{i,(m_{i},m_{i,r})}$
and sends it.

Decoding: At the other legitimate receiver, an ML decoder will be used to
decode ${V}^{n}_{i,(m_{i},m_{i,r})}$ and thus $m_{i}$ by using the receiver’s
information $X_{i\oplus 1}^{n}$.

To show Theorem 1, we prepare the following Lemma, which gives exponential
evaluation on both the decoding error probability and the information leaked
to the eavesdropper.

###### Lemma 4

For a joint distribution $P_{V_{1}V_{2}X_{1}X_{2}}\in{\cal Q}$, a rate pair
$(R_{1},R_{2})$, and two positive numbers $r_{1},r_{2}$, there exists a
sequence of $(e^{nR_{1}},e^{nR_{2}},n)$ non-adaptive codes $C_{n}$ with the
2nd type decoder such that

$\displaystyle P_{e}^{n}(C_{n})$ $\displaystyle\leq\
2\left(e^{ns\left(R_{1}+r_{1}-I_{\frac{1}{1+s}}^{\uparrow}(Y_{2};V_{1}|X_{2})\right)}+e^{ns\left(R_{2}+r_{2}-I_{\frac{1}{1+s}}^{\uparrow}(Y_{1};V_{2}|X_{1})\right)}\right),$
(61) $\displaystyle I(M_{1},M_{2};Z^{n}|C_{n})$ $\displaystyle\leq
2\left(\sum_{\mathcal{S}\neq\emptyset,\mathcal{S}\subset\\{1,2\\}}e^{ns\left(I^{\downarrow}_{1+s}(Z;V_{\mathcal{S}})-\sum\limits_{j\in\mathcal{S}}r_{j}\right)}\right).$
(62)

Also, there exists a sequence of $(e^{nR_{1}},e^{nR_{2}},n)$ non-adaptive
codes $C_{n}$ with the 2nd type decoder such that

$\displaystyle P_{e}^{n}(C_{n})$ $\displaystyle\leq
3\left(e^{ns\left(R_{1}+r_{1}-I_{\frac{1}{1+s}}^{\uparrow}(Y_{2};V_{1}|X_{2})\right)}+e^{ns\left(R_{2}+r_{2}-I_{\frac{1}{1+s}}^{\uparrow}(Y_{1};V_{2}|X_{1})\right)}\right),$
(63) $\displaystyle I(M_{1};Z^{n}|C_{n})$ $\displaystyle\leq
3\left(e^{ns\left(I^{\downarrow}_{1+s}(Z;V_{1},V_{2})-(r_{1}+r_{2}+R_{2})\right)}+e^{ns\left(I^{\downarrow}_{1+s}(Z;V_{1})-r_{1}\right)}+e^{ns\left(I^{\downarrow}_{1+s}(Z;V_{2})-(r_{2}+R_{2})\right)}\right),$
(64) $\displaystyle I(M_{2};Z^{n}|C_{n})$ $\displaystyle\leq
3\left(e^{ns\left(I^{\downarrow}_{1+s}(Z;V_{1},V_{2})-(r_{1}+r_{2}+R_{1})\right)}+e^{ns\left(I^{\downarrow}_{1+s}(Z;V_{2})-r_{2}\right)}+e^{ns\left(I^{\downarrow}_{1+s}(Z;V_{1})-(r_{1}+R_{1})\right)}\right).$
(65)

###### Proof:

According to the reliability analysis in Section III-D and the strong secrecy
analysis based on resolvability in Section III-E, we have

$\displaystyle\mathbb{E}_{C_{n}}\left[P_{e}^{n}(C_{n})\right]$
$\displaystyle\leq
e^{ns\left(R_{1}+r_{1}-I_{\frac{1}{1+s}}^{\uparrow}(Y_{2};V_{1}|X_{2})\right)}+e^{ns\left(R_{2}+r_{2}-I_{\frac{1}{1+s}}^{\uparrow}(Y_{1};V_{2}|X_{1})\right)},$
(66) $\displaystyle\mathbb{E}_{C_{n}}\left[I(M_{1},M_{2};Z^{n}|C_{n})\right]$
$\displaystyle\leq\sum_{\mathcal{S}\neq\emptyset,\mathcal{S}\subset\\{1,2\\}}e^{ns\left(I^{\downarrow}_{1+s}(Z;V_{\mathcal{S}})-\sum\limits_{j\in\mathcal{S}}r_{j}\right)},$
(67) $\displaystyle\mathbb{E}_{C_{n}}\left[I(M_{1};Z^{n}|C_{n})\right]$
$\displaystyle\leq
e^{ns\left(I^{\downarrow}_{1+s}(Z;V_{1},V_{2})-(r_{1}+r_{2}+R_{2})\right)}+e^{ns\left(I^{\downarrow}_{1+s}(Z;V_{1})-r_{1}\right)}+e^{ns\left(I^{\downarrow}_{1+s}(Z;V_{2})-(r_{2}+R_{2})\right)},$
(68) $\displaystyle\mathbb{E}_{C_{n}}\left[I(M_{2};Z^{n}|C_{n})\right]$
$\displaystyle\leq
e^{ns\left(I^{\downarrow}_{1+s}(Z;V_{1},V_{2})-(r_{1}+r_{2}+R_{1})\right)}+e^{ns\left(I^{\downarrow}_{1+s}(Z;V_{2})-r_{2}\right)}+e^{ns\left(I^{\downarrow}_{1+s}(Z;V_{1})-(r_{1}+R_{1})\right)}.$
(69)

With Markov inequality, the relations (66) and (67) imply the first statement
of Lemma 4. In the same way, the relations (66), (68), and (69) imply the
second statement of Lemma 4. ∎

###  III-D Reliability Analysis Based on Gallager’s Bound

We show (66). Consider the case when User 1 decodes the message $M_{2},$ while
holding $X_{1}^{n}=(X_{i,1},\cdots,X_{i,n})$ which was chosen according to the
distribution $P_{X_{1}^{n}}=\prod\limits_{j=1}^{n}P_{X_{1,j}}$. User 2 encodes
the message into one of
$V_{2,1}^{n},\ldots,V_{2,\mathsf{M}_{2}\cdot\mathsf{L}_{2}}^{n}$ that are
independently generated subject to
$P_{V_{2}^{n}}=\prod\limits_{j=1}^{n}P_{V_{2,j}}$.

We apply Lemma 3 to the $n$-fold asymptotic case with
$\mathsf{N}_{i}=\mathsf{M}_{i}\mathsf{L}_{i}=e^{n(R_{i}+r_{i})}$. In this
case, the relation (52) guarantees the additivity property

$\displaystyle
I_{\frac{1}{1+s}}^{\uparrow}(Y_{1}^{n};V_{2}^{n}|X_{1}^{n})=nI_{\frac{1}{1+s}}^{\uparrow}(Y_{1};V_{2}|X_{1}),\quad$
(70)

Then, we found that the expectation of the decoding error probability of User
1 is upper bounded by
$e^{ns\left(R_{2}+r_{2}-I_{\frac{1}{1+s}}^{\uparrow}(Y_{1};V_{2}|X_{1})\right)}$.
In the same way, we can show that the expectation of the decoding error
probability of User 2 is upper bounded by
$e^{ns\left(R_{1}+r_{1}-I_{\frac{1}{1+s}}^{\uparrow}(Y_{2};V_{1}|X_{2})\right)}$.
Hence, we obtain (66).

###  III-E Strong secrecy analysis based on resolvability

For the strong secrecy analysis, we consider both joint secrecy and individual
secrecy.

First, we consider the joint secrecy, i.e., to show (67). We have

$\displaystyle\mathbb{E}_{C_{n}}\left[I(M_{1},M_{2};Z^{n}|C_{n})\right]=$
$\displaystyle\mathbb{E}_{C_{n}}\left[\sum\limits_{m_{1},m_{2}}P_{M_{1}M_{2}}(m_{1},m_{2})D(P_{Z^{n}|M_{1}=m_{1},M_{2}=m_{2}}\|P_{Z^{n}|C_{n}})\right]$
$\displaystyle=$
$\displaystyle\mathbb{E}_{C_{n}}\left[\sum\limits_{m_{1},m_{2}}P_{M_{1}M_{2}}(m_{1},m_{2})D(P_{Z^{n}|M_{1}=m_{1},M_{2}=m_{2}}\|P_{Z^{n}})-D(P_{Z^{n}|C_{n}}\|P_{Z^{n}})\right]$
$\displaystyle\leq$
$\displaystyle\mathbb{E}_{C_{n}}\left[\sum\limits_{m_{1},m_{2}}P_{M_{1}M_{2}}(m_{1},m_{2})D(P_{Z^{n}|M_{1}=m_{1},M_{2}=m_{2}}\|P_{Z^{n}})\right]$
$\displaystyle=$ $\displaystyle\sum\limits_{m_{1},m_{2}}P_{M_{1}M_{2}}\
(m_{1},m_{2})\mathbb{E}_{C_{n}}\left[D(P_{Z^{n}|M_{1}=m_{1},M_{2}=m_{2}}\|P_{Z^{n}})\right],$
(71)

where $\mathcal{M}_{1},\mathcal{M}_{2}$ are fixed sets of messages, and
correspondingly

$\displaystyle P_{Z^{n}|M_{1}=m_{1},M_{2}=m_{2}}(z^{n})$
$\displaystyle:=\frac{1}{\mathsf{L}_{1}\mathsf{L}_{2}}\sum_{j_{1},j_{2}}P_{Z^{n}|V_{1}^{n}=V^{n}_{1,j_{1}},V^{n}_{2}=V^{n}_{2,j_{2}}}(z^{n}),$

with $(m_{i}-1)\mathsf{L}_{i}+1\leq j_{i}\leq m_{i}\mathsf{L}_{i},$ for $1\leq
i\leq k.$

We apply Lemma 2 to the $n$-fold asymptotic case with inputs
$(V_{1}^{n},V_{2}^{n})$ and $\mathsf{L}_{i}=e^{nr_{i}}$. In this case, the
relation (49) guarantees the additivity property

$\displaystyle
I^{\downarrow}_{1+s}(Z^{n};V^{n}_{\mathcal{S}})=nI^{\downarrow}_{1+s}(Z;V_{\mathcal{S}}),\quad\mbox{for
}\mathcal{S}\neq\emptyset,\mathcal{S}\subset\\{1,2\\}.$ (72)

Lemma 2 and (72) together imply that

$\displaystyle\mathbb{E}_{C_{n}}\left[D(P_{Z^{n}|M_{1}=m_{1},M_{2}=m_{2}}\|P_{Z^{n}})\right]\leq\sum_{\mathcal{S}\neq\emptyset,\mathcal{S}\subset\\{1,2\\}}e^{ns\left(I^{\downarrow}_{1+s}(Z;V_{\mathcal{S}})-\sum\limits_{j\in\mathcal{S}}r_{j}\right)}.$
(73)

The combination of (71) and (73) implies (67).

Next, we consider the individual secrecy, i.e., to show (68). We apply Lemma 2
to the $n$-fold asymptotic case with
$\mathsf{N}_{i}=\mathsf{M}_{i}\mathsf{L}_{i}=e^{n(R_{i}+r_{i})}$.

$\displaystyle\mathbb{E}_{C_{n}}\left[I(M_{i};Z^{n}|C_{n})\right]=$
$\displaystyle\mathbb{E}_{C_{n}}\left[\sum\limits_{m_{i}}P_{M_{i}}(m_{i})D(P_{Z^{n}|M_{i}=m_{i}}\|P_{Z^{n}|C_{n}})\right]$
$\displaystyle\leq$
$\displaystyle\sum\limits_{m_{i}}P_{M_{i}}(m_{i})\mathbb{E}_{C_{n}}\left[D(P_{Z^{n}|M_{i}=m_{i}}\|P_{Z^{n}})\right],$
(74)

Then, we have

$\displaystyle P_{Z^{n}|M_{1}=m_{1}}(z^{n})$
$\displaystyle:=\frac{1}{\mathsf{L}_{1}\mathsf{N}_{2}}\sum_{j_{1}=(m_{i}-1)\mathsf{L}_{i}+1}^{m_{i}\mathsf{L}_{i}}\sum_{j_{2}=1}^{\mathsf{N}_{2}}P_{Z^{n}|V_{1}^{n}=V^{n}_{1,j_{1}},V^{n}_{2}=V^{n}_{2,j_{2}}}(z^{n}),$

where $\mathsf{L}_{1}=e^{nr_{1}}$ and $\mathsf{N}_{2}=e^{n(R_{2}+r_{2})}.$

Lemma 2 and (72) together implies that

$\displaystyle\mathbb{E}_{C_{n}}\left[D(P_{Z^{n}|M_{1}=m_{1}}\|P_{Z^{n}})\right]\leq
e^{ns\left(I^{\downarrow}_{1+s}(Z;V_{1},V_{2})-(r_{1}+r_{2}+R_{2})\right)}+e^{ns\left(I^{\downarrow}_{1+s}(Z;V_{1})-r_{1}\right)}+e^{ns\left(I^{\downarrow}_{1+s}(Z;V_{2})-(r_{2}+R_{2})\right)}.$
(75)

The relations (74) and (75) imply (68). In the same way, we can show (69).

###  III-F Deriving the achievable region under strong secrecy

In order to show the relations (31), it is sufficient to show the relations

$\displaystyle{\cal R}_{J,N}(P_{V_{1}V_{2}X_{1}X_{2}})\subset{\cal
C}_{J,S,N,2},\quad$ $\displaystyle{\cal
R}_{I,N}(P_{V_{1}V_{2}X_{1}X_{2}})\subset{\cal C}_{I,S,N,2}$ (76)

for $P_{V_{1}V_{2}X_{1}X_{2}}\in{\cal Q}$, because the application of time
sharing to the obtained region in (76) yields the relations (31). Therefore,
in the following, we show the relations (76).

Using Lemma 4, we prove Theorem 1. When

$\displaystyle R_{2}+r_{2}$ $\displaystyle<I(Y_{1};V_{2}|X_{1}),$ (77)
$\displaystyle R_{1}+r_{1}$ $\displaystyle<I(Y_{2};V_{1}|X_{2}),$ (78)

the decoding error probability $P_{e}^{n}(C_{n})$ goes to zero as
$n\to\infty$.

First, we address the strong joint secrecy. When

$\displaystyle\sum_{i\in\mathcal{S}}r_{i}>I(Z;V_{\mathcal{S}}),\quad\forall\mathcal{S}\neq\emptyset,\mathcal{S}\subseteq\\{1,2\\},$
(79)

the mutual information $I(M_{1},M_{2};Z^{n}|C_{n})$ goes to zero as
$n\to\infty$. For strong joint secrecy, we have the following constraints on
rates $r_{i}$ for the two-way wiretap channel:

$\displaystyle r_{1}$ $\displaystyle>I(Z;V_{1});$ (80) $\displaystyle r_{2}$
$\displaystyle>I(Z;V_{2});$ (81) $\displaystyle r_{1}+r_{2}$
$\displaystyle>I(Z;V_{1},V_{2}).$ (82)

For the joint secrecy, applying Fourier-Motzkin elimination to remove
$r_{1},r_{2}$ in (77) – (82), we obtain the rate region of $(R_{1},R_{2})$
defined by

$\displaystyle\begin{split}R_{1}&\leq I(Y_{2};V_{1}|X_{2})-I(Z;V_{1})\\\
R_{2}&\leq I(Y_{1};V_{2}|X_{1})-I(Z;V_{2})\\\ R_{1}+R_{2}&\leq
I(Y_{2};V_{1}|X_{2})+I(Y_{1};V_{2}|X_{1})-I(V_{1},V_{2};Z),\end{split}$ (83)

which implies the first relation in (31).

Next, we address the strong individual secrecy. When

$\displaystyle r_{1}$ $\displaystyle>I(Z;V_{1}),$ $\displaystyle r_{2}+R_{2}$
$\displaystyle>I(Z;V_{2}),$ $\displaystyle r_{1}+r_{2}+R_{2}$
$\displaystyle>I(Z;V_{1},V_{2}),$

the mutual information $I(M_{1};Z^{n}|C_{n})$ goes to zero as $n\to\infty$.
Since the same discussion holds for the mutual information
$I(M_{2};Z^{n}|C_{n})$ for strong individual secrecy, we have the following
constraints on rates $r_{i}$ and $R_{i}$ for the two-way wiretap channel:

$\displaystyle r_{1}$ $\displaystyle>I(Z;V_{1});$ (84) $\displaystyle r_{2}$
$\displaystyle>I(Z;V_{2});$ (85) $\displaystyle r_{1}+r_{2}+\min(R_{1},R_{2})$
$\displaystyle>I(Z;V_{1},V_{2}).$ (86)

For the individual secrecy, applying Fourier-Motzkin elimination to remove
$r_{1},r_{2}$ in (77), (78), and (84) – (86), we obtain the rate region of
$(R_{1},R_{2})$ defined by

$\displaystyle\begin{split}R_{1}&\leq I(Y_{2};V_{1}|X_{2})-I(Z;V_{1})\\\
R_{2}&\leq I(Y_{1};V_{2}|X_{1})-I(Z;V_{2})\\\ \max\\{R_{1},R_{2}\\}&\leq
I(Y_{2};V_{1}|X_{2})+I(Y_{1};V_{2}|X_{1})-I(V_{1},V_{2};Z),\end{split}$ (87)

which implies the second relation in (31). The detail of Fourier-Motzkin
elimination is explained in Appendix C.

##  IV Proof of Theorem 2

To prove Theorem 2, we need to find a sequence of non-adaptive codes
${\mathcal{C}_{n}}$ with peak cost constraint $c_{1},c_{2}$ such that

  * •

the codes have rates pair converge to $(R_{1},R_{2})$ within the region ${\cal
R}_{J,N}^{c_{1},c_{2}}$ and ${\cal R}_{I,N}^{c_{1},c_{2}}$ for joint secrecy
and individual secrecy, respectively;

  * •

the decoding error is bounded by $\epsilon_{n}$ with
$\lim\limits_{n\to\infty}\epsilon_{n}=0;$

  * •

the information leakage is bounded by $\tau_{n}$ with
$\lim\limits_{n\to\infty}\tau_{n}=0.$

To this end, we use constant composition codes in our coding scheme to satisfy
the cost constraint and evaluate the decoding error (upper bounded by
$\epsilon_{n}$) and respective information leakage (upper bounded by
$\tau_{n}$) for joint secrecy and individual security (similar to the proof of
Theorem 1). As a preparation, we first introduce the basics of type method in
Sec. IV-A, which is used to establish the constant composition version of the
resolvability lemma and generalization of Gallager’s bound in Sec. IV-B. The
coding scheme using constant composition codes and its analysis are provided
in Sec. IV-C. Sec. IV-D derives the desired region in Theorem 2.

###  IV-A Basics of type method

First, we give notations for the method of types for a single finite system
${\cal X}$ with $d_{X}=|{\cal X}|$. Given an element $x^{n}\in{\cal X}^{n}$
and an element $x\in{\cal X}$, we define the subset ${\cal
N}(x^{n},x):=\\{i|x_{i}=x\\}$, the integer $n_{x}:=|{\cal N}(x^{n},x)|$, and
the empirical distribution
$T_{Y}(x^{n}):=(\frac{n_{1}}{n},\ldots,\frac{n_{d_{X}}}{n})$, which is called
a type. The set of types is denoted by $T_{n}({\cal X})$. For ${P}\in
T_{n}({\cal X})$, a subset of ${\cal X}^{n}$ is defined by:

$\displaystyle T_{P}^{n}({\cal X}):=\\{x^{n}\in{\cal
X}^{n}|T_{Y}(x^{n})=P\\}.$

We call $T_{P}^{n}({\cal X})$ the fixed-type subset of $P$. We simplify
$T_{P}^{n}({\cal X})$ to $T_{P}^{n}$ when we do not need to identify ${\cal
X}.$ In this paper, the value $\nu_{n}(d_{X}):=\max\limits_{P\in T_{n}({\cal
X})}\frac{e^{nH(P)}}{|T_{P}^{n}|}\leq(1+n)^{d_{X}}$ plays a key role. The
uniform distribution $P_{\mathop{\rm Unif},T_{P}^{n}}$ on the subset
$T_{P}^{n}$ satisfies

$\displaystyle P_{\mathop{\rm
Unif},T_{P}^{n}}\leq\nu_{n}(d_{X})P^{n}\leq(1+n)^{d_{X}}P^{n}.$ (88)

Next, we discuss the method of types for two finite systems ${\cal X}$ and
${\cal V}$. For $P_{XV}\in T_{n}({\cal X}\times{\cal V})$ and $v^{n}\in{\cal
V}^{n}$, we define a subset of ${\cal X}^{n}$ as

$\displaystyle T_{P_{X|V}}^{n}({\cal X},v^{n}):=\\{x^{n}\in{\cal
X}^{n}|(x^{n},v^{n})\in T_{P_{XV}}^{n}\\}.$

We simplify $T_{P_{X|V}}^{n}({\cal X},v^{n})$ to $T_{P_{X|V}}^{n}(v^{n})$.

Now, we introduce notations over the finite sets ${\cal X}_{1}$, ${\cal
X}_{2}$, ${\cal V}_{1}$ and ${\cal V}_{2}.$ We fix a positive integer $n$. Let
$P_{Z|X_{1}X_{2}}$ be a MAC with two inputs system. Let $P_{V_{1}X_{1}}$ and
$P_{V_{2}X_{2}}$ be joint types with length $n$. Define
$P_{Z|V_{1}V_{2}}(z|v_{1},v_{2}):=\sum\limits_{x_{1},x_{2}}P_{Z|X_{1}X_{2}}(z|x_{1},x_{2})P_{X_{1}|V_{1}}(x_{1}|v_{1})P_{X_{2}|V_{2}}(x_{2}|v_{2})$
and $P_{V_{1}V_{2}}:=P_{V_{1}}\times P_{V_{2}}$. Then, we have the joint
distribution $P_{ZV_{1}V_{2}}$. Based on this joint distribution, we define
the information quantity
$\breve{I}^{\uparrow}_{\frac{1}{1-s}}(Z;V_{\mathcal{S}})$ for
$\mathcal{S}\neq\emptyset,\mathcal{S}\subset\\{1,2\\}$.

Define the uniform distribution $\breve{P}_{X_{i}^{n}V_{i}^{n}}$ on the subset
$T_{P_{V_{i}X_{i}}}^{n}$ of $({\cal V}_{i}\times{\cal X}_{i})^{n}$ for
$i=1,2$. Then, we define its marginal distributions $\breve{P}_{V_{i}^{n}}$,
$\breve{P}_{X_{i}^{n}}$ and its conditional distribution
$\breve{P}_{X_{i}^{n}|V_{i}^{n}}$. Notice that $\breve{P}_{V_{i}^{n}}$ and
$\breve{P}_{X_{i}^{n}}$ are the uniform distributions on the subsets
$T_{P_{V_{i}}}^{n}$ and $T_{P_{X_{i}}}^{n}$, respectively. Also, the
conditional distribution $\breve{P}_{X_{i}^{n}|V_{i}^{n}=v_{i}^{n}}$ is the
uniform distribution on $T^{n}_{P_{X_{i}|V_{i}}}(v^{n})$.

###  IV-B Constant composition codes

We define two-transmitter MAC as the conditional distribution
$P_{Z^{n}|V_{1}^{n}V_{2}^{n}}:=P_{Z^{n}|X_{1}^{n}X_{2}^{n}}\cdot\breve{P}_{X_{1}^{n}|V_{1}^{n}}\cdot\breve{P}_{X_{2}^{n}|V_{2}^{n}}$.
Then, we have a constant composition version of the following resolvability
lemma.

###### Lemma 5

Let $V_{i,1}^{n},\ldots,V_{i,\mathsf{L}_{i}}^{n}$ be the random variables that
are independently generated subject to $\breve{P}_{V_{i}^{n}}$ for $i=1,2$.
Here, we denote this code selection by ${C}$ and define the distribution
$P_{Z^{n}|C}$ for the variable $Z^{n}$ dependently of the code selection ${C}$
as
$P_{Z^{n}|C}(z^{n}):=\frac{1}{\mathsf{L}_{1}\mathsf{L}_{2}}\sum\limits_{j_{1},j_{2}}P_{Z^{n}|V_{1}^{n}=V_{1,j_{1}}^{n},V_{2}^{n}=V_{2,j_{2}}^{n}}(z^{n})$.

Then, for $s\in[0,1]$, we have $D(P_{Z^{n}|C}\|P_{Z^{n}})\leq
D_{1+s}(P_{Z^{n}|C}\|P_{Z^{n}})$ and

$\displaystyle
s\mathbb{E}_{C}\left[D_{1+s}(P_{Z^{n}|C}\|P_{Z^{n}})\right]\leq\sum_{\mathcal{S}\neq\emptyset,\mathcal{S}\subset\\{1,2\\}}\frac{\nu_{n}(|{\cal
X}_{1}|)\nu_{n}(|{\cal
X}_{2}|)}{\prod_{i\in\mathcal{S}}\mathsf{L}_{i}^{s}}e^{s\breve{I}_{\frac{1}{1-s}}^{\uparrow}(Z;V_{\mathcal{S}})}$
(89)

for $s\in[0,1]$.

###### Proof:

See Appendix D for a detailed proof. ∎

To state a generalization of Gallager’s bound [22] for constant composition
codes, we assume that $|{\cal Y}|$, $|{\cal X}_{1}|$, $|{\cal X}_{2}|$, and
$|{\cal V}_{2}|$ are finite. We fix a positive integer $n$. Let
$P_{Y|X_{1}X_{2}}$ be a MAC with two inputs system. Let $P_{X_{1}}$ and
$P_{V_{2}X_{2}}$ be (joint) types with length $n$. Define
$P_{Y|X_{1}V_{2}}(y|x_{1},v_{2}):=\sum\limits_{x_{2}}P_{Y|X_{1}X_{2}}(y|x_{1},x_{2})P_{X_{2}|V_{2}}(x_{2}|v_{2})$
and $P_{X_{1}V_{2}}:=P_{X_{1}}\times P_{V_{2}}$. Then, we have the joint
distribution $P_{YX_{1}V_{2}}$. Based on this joint distribution, we define
the information quantity
$\breve{I}_{\frac{1}{1+s}}^{\uparrow}(Y;V_{2}|X_{1})$.

Define the uniform distribution $\breve{P}_{X_{2}^{n}V_{2}^{n}}$ on the subset
$T_{P_{V_{2}X_{2}}}^{n}$ of $({\cal V}_{2}\times{\cal X}_{2})^{n}.$ Then, we
define its marginal distribution $\breve{P}_{V_{2}^{n}}$ and its conditional
distribution $\breve{P}_{X_{2}^{n}|V_{2}^{n}}$. We define the uniform
distribution $\breve{P}_{X_{1}^{n}}$ on the subset $T_{P_{X_{1}}}^{n}$ of
${\cal X}_{1}^{n}$. We define two-transmitter MAC
$P_{Y^{n}|X_{1}^{n}V_{2}^{n}}(y^{n}|x_{1}^{n},v_{2}^{n}):=\sum\limits_{x_{2}^{n}}P_{Y^{n}|X_{1}^{n}X_{2}^{n}}(y^{n}|x_{1}^{n},x_{2}^{n})\breve{P}_{X_{2}^{n}|V_{2}^{n}}(x_{2}^{n}|v_{2}^{n})$.

Then, we have the following as a generalization of Gallager’s bound [22].

###### Lemma 6

Assume that the $\mathsf{N}$ messages are randomly selected as the random
variables $V_{2,1}^{n},\ldots,V_{2,\mathsf{N}}^{n}$ that are independently
generated subject to $\breve{P}_{V_{2}^{n}}$. Also, assume that the receiver
knows the other random variable $X_{1}^{n}$ and selects it subject to
$\breve{P}_{X_{1}^{n}}$ independently of
$V_{2,1}^{n},\ldots,V_{2,\mathsf{N}}^{n}$. Here, we denote the above selection
by ${C^{\prime}}$ and the decoding error probability under the maximum
likelihood (ML) decoder by $e_{C^{\prime}}$.

Then, for $s\in[0,1]$, we have

$\displaystyle\mathbb{E}_{C^{\prime}}\left[e_{C^{\prime}}\right]\leq|T_{n}({\cal
Y}\times{\cal X}_{1})|^{1+s}\nu_{n}(|{\cal Y}||{\cal
X}_{1}|)^{s}\nu_{n}(|{\cal X}_{1}|)\nu_{n}(|{\cal
X}_{2}|)\mathsf{N}^{s}e^{-s\breve{I}_{\frac{1}{1+s}}^{\uparrow}(Y;V_{2}|X_{1})}.$
(90)

###### Proof:

See Appendix E for a detailed proof. ∎

###  IV-C Exponential evaluation

To show Theorem 2, we prepare the following Lemma, which gives exponential
evaluation.

###### Lemma 7

Assume that $|{\cal Y}_{1}|$, $|{\cal Y}_{2}|$, $|{\cal X}_{1}|$, $|{\cal
X}_{2}|$, $|{\cal V}_{1}|$, and $|{\cal V}_{2}|$ are finite. We fix a positive
integer $n$. Let $P_{ZY_{1}Y_{2}|X_{1}X_{2}}$ be a MAC with two inputs system.
Let $P_{V_{1}X_{1}}$ and $P_{V_{2}X_{2}}$ be joint types with length $n$.
Define

$P_{ZY_{1}Y_{2}V_{1}V_{2}X_{1}X_{2}}(z,v_{1},v_{2},x_{1},x_{2}):=P_{ZY_{1}Y_{2}|X_{1}X_{2}}(z,y_{1},y_{2}|x_{1},x_{2})P_{X_{1}V_{1}}(x_{1},v_{1})P_{X_{2}V_{2}}(x_{2},v_{2}).$

Then, based on this joint distribution, we define the information quantities
$\breve{I}_{\frac{1}{1+s}}^{\uparrow}(Y_{2};V_{1}|X_{2})$,
$\breve{I}_{\frac{1}{1+s}}^{\uparrow}(Y_{1};V_{2}|X_{1})$, and
$\breve{I}_{\frac{1}{1-s}}^{\uparrow}(Z;V_{\mathcal{S}})$ for
$\mathcal{S}\neq\emptyset,\mathcal{S}\subset\\{1,2\\}$. We choose $\nu_{n}$
and $\nu_{n}^{\prime}$ as

$\displaystyle\nu_{n}:=$ $\displaystyle\nu_{n}(|{\cal X}_{1}||{\cal
X}_{2}|\max(|{\cal V}_{1}|,|{\cal V}_{2}|)),$ (91)
$\displaystyle\nu_{n}^{\prime}:=$ $\displaystyle\nu_{n}(|{\cal
X}_{1}|)\nu_{n}(|{\cal X}_{2}|)\max(|T_{n}({\cal Y}\times{\cal
X}_{1})|^{1+s}\nu_{n}(|{\cal Y}||{\cal X}_{1}|)^{s},|T_{n}({\cal Y}\times{\cal
X}_{2})|^{1+s}\nu_{n}(|{\cal Y}||{\cal X}_{2}|)^{s}).$ (92)

For a rate pair $(R_{1},R_{2})$, and two positive numbers $r_{1},r_{2}$, there
exists a sequence of $(e^{nR_{1}},e^{nR_{2}},n)$ non-adaptive codes $C_{n}$
with the 2nd type decoder in $T_{P_{X_{1}}}^{n}\times T_{P_{X_{2}}}^{n}$ such
that

$\displaystyle P_{e}^{n}(C_{n})$ $\displaystyle\leq\
2\nu_{n}^{\prime}\left(e^{ns\left(R_{1}+r_{1}-\breve{I}_{\frac{1}{1+s}}^{\uparrow}(Y_{2};V_{1}|X_{2})\right)}+e^{ns\left(R_{2}+r_{2}-\breve{I}_{\frac{1}{1+s}}^{\uparrow}(Y_{1};V_{2}|X_{1})\right)}\right),$
(93) $\displaystyle I(M_{1},M_{2};Z^{n}|C_{n})$ $\displaystyle\leq
2\nu_{n}\left(\sum_{\mathcal{S}\neq\emptyset,\mathcal{S}\subset\\{1,2\\}}e^{ns\left(\breve{I}_{\frac{1}{1-s}}^{\uparrow}(Z;V_{\mathcal{S}})-\sum\limits_{j\in\mathcal{S}}r_{j}\right)}\right).$
(94)

Also, there exists a sequence of $(e^{nR_{1}},e^{nR_{2}},n)$ non-adaptive
codes $C_{n}$ in $T_{P_{X_{1}}}^{n}\times T_{P_{X_{2}}}^{n}$ such that

$\displaystyle P_{e}^{n}(C_{n})$ $\displaystyle\leq
3\nu_{n}^{\prime}\left(e^{ns\left(R_{1}+r_{1}-\breve{I}_{\frac{1}{1+s}}^{\uparrow}(Y_{2};V_{1}|X_{2})\right)}+e^{ns\left(R_{2}+r_{2}-\breve{I}_{\frac{1}{1+s}}^{\uparrow}(Y_{1};V_{2}|X_{1})\right)}\right),$
(95) $\displaystyle I(M_{1};Z^{n}|C_{n})$ $\displaystyle\leq
3\nu_{n}\left(e^{ns\left(\breve{I}_{\frac{1}{1-s}}^{\uparrow}(Z;V_{1},V_{2})-(r_{1}+r_{2}+R_{2})\right)}+e^{ns\left(\breve{I}_{\frac{1}{1-s}}^{\uparrow}(Z;V_{1})-r_{1}\right)}+e^{ns\left(\breve{I}_{\frac{1}{1-s}}^{\uparrow}(Z;V_{2})-(r_{2}+R_{2})\right)}\right),$
(96) $\displaystyle I(M_{2};Z^{n}|C_{n})$ $\displaystyle\leq
3\nu_{n}\left(e^{ns\left(\breve{I}_{\frac{1}{1-s}}^{\uparrow}(Z;V_{1},V_{2})-(r_{1}+r_{2}+R_{1})\right)}+e^{ns\left(\breve{I}_{\frac{1}{1-s}}^{\uparrow}(Z;V_{2})-r_{2}\right)}+e^{ns\left(\breve{I}_{\frac{1}{1-s}}^{\uparrow}(Z;V_{1})-(r_{1}+R_{1})\right)}\right).$
(97)

###### Proof:

The transmitters encode their messages independently. Before transmission,
they generate the random variables
$V_{i,1}^{n},\ldots,V_{i,\mathsf{M}_{i}\cdot\mathsf{L}_{i}}^{n}$ independently
subject to $\breve{P}_{V_{i}^{n}},$ where $\mathsf{M}_{i}=e^{nR_{i}}$ and
$\mathsf{L}_{i}=e^{nr_{i}}$ for $i=1,2$.

Encoding: When the legitimate user $i$ intends to send the message $m_{i}$,
he(she) chooses one of
${V}^{n}_{i,(m_{i}-1)\mathsf{L}_{i}+1},\ldots,{V}^{n}_{i,m_{i}\mathsf{L}_{i}}$.
When the selected one element is $v_{i}^{n}$, user $i$ generates $x_{i}^{n}$
according to $\breve{P}_{X_{i}^{n}|V_{i}^{n}=v_{i}^{n}}$, and sends
$x_{i}^{n}$ to the channel.

Decoding: At the other legitimate receiver, a ML decoder will be used to
decode $M_{i}$ by using the receiver’s information $X_{i\oplus 1}^{n}$.

Then, in the same way as the proof of Lemma 4, replacing the roles of Lemmas 2
and 3 by the roles of Lemmas 5 and 6, we have

$\displaystyle\mathbb{E}_{C_{n}}\left[P_{e}^{n}(C_{n})\right]$
$\displaystyle\leq\nu_{n}^{\prime}\left(e^{ns\left(R_{1}+r_{1}-\breve{I}_{\frac{1}{1+s}}^{\uparrow}(Y_{2};V_{1}|X_{2})\right)}+e^{ns\left(R_{2}+r_{2}-\breve{I}_{\frac{1}{1+s}}^{\uparrow}(Y_{1};V_{2}|X_{1})\right)}\right),$
(98) $\displaystyle\mathbb{E}_{C_{n}}\left[I(M_{1},M_{2};Z^{n}|C_{n})\right]$
$\displaystyle\leq\nu_{n}\left(\sum_{\mathcal{S}\neq\emptyset,\mathcal{S}\subset\\{1,2\\}}e^{ns\left(\breve{I}_{\frac{1}{1-s}}^{\uparrow}(Z;V_{\mathcal{S}})-\sum\limits_{j\in\mathcal{S}}r_{j}\right)}\right),$
(99) $\displaystyle\mathbb{E}_{C_{n}}\left[I(M_{1};Z^{n}|C_{n})\right]$
$\displaystyle\leq\nu_{n}\left(e^{ns\left(\breve{I}_{\frac{1}{1-s}}^{\uparrow}(Z;V_{1},V_{2})-(r_{1}+r_{2}+R_{2})\right)}+e^{ns\left(\breve{I}_{\frac{1}{1-s}}^{\uparrow}(Z;V_{1})-r_{1}\right)}+e^{ns\left(\breve{I}_{\frac{1}{1-s}}^{\uparrow}(Z;V_{2})-(r_{2}+R_{2})\right)}\right),$
(100) $\displaystyle\mathbb{E}_{C_{n}}\left[I(M_{2};Z^{n}|C_{n})\right]$
$\displaystyle\leq\nu_{n}\left(e^{ns\left(\breve{I}_{\frac{1}{1-s}}^{\uparrow}(Z;V_{1},V_{2})-(r_{1}+r_{2}+R_{1})\right)}+e^{ns\left(\breve{I}_{\frac{1}{1-s}}^{\uparrow}(Z;V_{2})-r_{2}\right)}+e^{ns\left(\breve{I}_{\frac{1}{1-s}}^{\uparrow}(Z;V_{1})-(r_{1}+R_{1})\right)}\right).$
(101)

In the same way as Lemma 4, the above relations yield the statement of Lemma
7. ∎

###  IV-D Deriving the achievable region under strong secrecy

Assume that $|{\cal Y}_{1}|$, $|{\cal Y}_{2}|$, $|{\cal X}_{1}|$, $|{\cal
X}_{2}|$, $|{\cal V}_{1}|$, and $|{\cal V}_{2}|$ are finite. Given
$P_{V_{1}V_{2}X_{1}X_{2}}\in{\cal Q}^{c_{1},c_{2}}$, we have
$T^{n}_{P_{X_{1}}}\subset{\cal X}_{1}^{n}(c_{1})$ and
$T^{n}_{P_{X_{2}}}\subset{\cal X}_{2}^{n}(c_{2})$. Hence, in the same way as
the discussion in Section III-F, Lemma 7 yields

$\displaystyle{\cal R}_{J,N}(P_{V_{1}V_{2}X_{1}X_{2}})\subset{\cal
C}_{J,S,N,2}^{P,c_{1},c_{2}},\quad{\cal
R}_{I,N}(P_{V_{1}V_{2}X_{1}X_{2}})\subset{\cal C}_{I,S,N,2}^{P,c_{1},c_{2}}.$
(102)

Applying the time sharing in the same way as Subsection III-F, we obtain the
statement of Theorem 2.

Next, we consider the case when $|{\cal Y}_{1}|$, $|{\cal Y}_{2}|$, $|{\cal
X}_{1}|$, $|{\cal X}_{2}|$, $|{\cal V}_{1}|$, and $|{\cal V}_{2}|$ are not
finite. In the following, we consider the finite-discretization of ${\cal
Y}_{1}$ and ${\cal Y}_{2}$. User $i$ divides the set ${\cal Y}_{i}$ into
finite disjoint subsets ${\cal R}_{i}:=\\{{\cal T}_{i,1},\ldots,{\cal
T}_{i,l_{i}}\\}$. Let $\tilde{Y}_{i}$ be the information for which subset
contains $Y_{i}$. Let ${\cal R}_{J,N}(P_{V_{1}V_{2}X_{1}X_{2}})[{\cal
R}_{1},{\cal R}_{2}]$ and ${\cal R}_{I,N}(P_{V_{1}V_{2}X_{1}X_{2}})[{\cal
R}_{1},{\cal R}_{2}]$ be the regions defined in (24) and (29) with replacing
$Y_{i}$ by $\tilde{Y}_{i}$, respectively. The above regions shows the regions
when User $i$ uses the information $\tilde{Y}_{i}$ instead of $Y_{i}$. When
$P_{V_{1}V_{2}X_{1}X_{2}}^{\prime}\in{\cal Q}^{c_{1},c_{2}}$ has a finite
support, the relation (102) shows that

${\cal R}_{J,N}(P_{V_{1}V_{2}X_{1}X_{2}}^{\prime})[{\cal R}_{1},{\cal
R}_{2}]\subset{\cal C}_{J,S,N,2}^{P,c_{1},c_{2}}\quad\mbox{and}\quad{\cal
R}_{I,N}(P_{V_{1}V_{2}X_{1}X_{2}}^{\prime})[{\cal R}_{1},{\cal
R}_{2}]\subset{\cal C}_{I,S,N,2}^{P,c_{1},c_{2}}.$ (103)

Given $P_{V_{1}V_{2}X_{1}X_{2}}\in{\cal Q}^{c_{1},c_{2}}$, we can choose
finite-discretizations ${\cal R}_{1},{\cal R}_{2}$ such that $\allowbreak{\cal
R}_{J,N}(P_{V_{1}V_{2}X_{1}X_{2}})[{\cal R}_{1},{\cal R}_{2}]$ and ${\cal
R}_{I,N}(P_{V_{1}V_{2}X_{1}X_{2}})[{\cal R}_{1},{\cal R}_{2}]$ are
sufficiently close to ${\cal R}_{J,N}(P_{V_{1}V_{2}X_{1}X_{2}})$ and
$\allowbreak{\cal R}_{I,N}(P_{V_{1}V_{2}X_{1}X_{2}}),$ respectively. Also, we
can choose a distribution $P_{V_{1}V_{2}X_{1}X_{2}}^{\prime}\in{\cal
Q}^{c_{1},c_{2}}$ with finite support such that $\allowbreak{\cal
R}_{J,N}(P_{V_{1}V_{2}X_{1}X_{2}}^{\prime})[{\cal R}_{1},{\cal R}_{2}]$ and
${\cal R}_{I,N}(P_{V_{1}V_{2}X_{1}X_{2}}^{\prime})[{\cal R}_{1},{\cal R}_{2}]$
are sufficiently close to $\allowbreak{\cal
R}_{J,N}(P_{V_{1}V_{2}X_{1}X_{2}})[{\cal R}_{1},{\cal R}_{2}]$ and ${\cal
R}_{I,N}(P_{V_{1}V_{2}X_{1}X_{2}})[{\cal R}_{1},{\cal R}_{2}]$, respectively.
Therefore, given $P_{V_{1}V_{2}X_{1}X_{2}}\in{\cal Q}^{c_{1},c_{2}}$, we can
choose a pair of a distribution $P_{V_{1}V_{2}X_{1}X_{2}}^{\prime}\in{\cal
Q}^{c_{1},c_{2}}$ with finite support and finite-discretizations ${\cal
R}_{1},{\cal R}_{2}$ such that $\allowbreak{\cal
R}_{J,N}(P_{V_{1}V_{2}X_{1}X_{2}}^{\prime})[{\cal R}_{1},{\cal R}_{2}]$ and
${\cal R}_{I,N}(P_{V_{1}V_{2}X_{1}X_{2}}^{\prime})[{\cal R}_{1},{\cal R}_{2}]$
are sufficiently close to ${\cal R}_{J,N}(P_{V_{1}V_{2}X_{1}X_{2}})$ and
$\allowbreak{\cal R}_{I,N}(P_{V_{1}V_{2}X_{1}X_{2}}),$ respectively.

We denote the inner of a set $S$ by $\mathsf{inn}(S)$. Due to the above
discussion, for any element $(R_{1},R_{2})$ of $\mathsf{inn}({\cal
R}_{J,N}(P_{V_{1}V_{2}X_{1}X_{2}}))$, there exists a pair of a distribution
$P_{V_{1}V_{2}X_{1}X_{2}}^{\prime}\in{\cal Q}^{c_{1},c_{2}}$ with finite
support and finite-discretizations ${\cal R}_{1},{\cal R}_{2}$ such that
$(R_{1},R_{2})$ is included in $\allowbreak{\cal
R}_{J,N}(P_{V_{1}V_{2}X_{1}X_{2}}^{\prime})[{\cal R}_{1},{\cal R}_{2}]$. The
combinaiton of this discussion and (103) implies that $\mathsf{inn}({\cal
R}_{J,N}(P_{V_{1}V_{2}X_{1}X_{2}}))\subset{\cal C}_{J,S,N,2}^{P,c_{1},c_{2}}$.
In the same way, we can show the relation $\mathsf{inn}({\cal
R}_{I,N}(P_{V_{1}V_{2}X_{1}X_{2}}))\subset{\cal C}_{I,S,N,2}^{P,c_{1},c_{2}}$.
Since Lemma 1 guarantees that ${\cal C}_{J,S,N,2}^{P,c_{1},c_{2}}$ and ${\cal
C}_{I,S,N,2}^{P,c_{1},c_{2}}$ are closed sets, we have ${\cal
R}_{J,N}(P_{V_{1}V_{2}X_{1}X_{2}})\subset{\cal C}_{J,S,N,2}^{P,c_{1},c_{2}}$
and ${\cal R}_{I,N}(P_{V_{1}V_{2}X_{1}X_{2}})\subset{\cal
C}_{I,S,N,2}^{P,c_{1},c_{2}}$, respectively. Therefore, we obtain the
statement of Theorem 2 in the general case.

##  V Additive two-way wiretap channel over a finite field

In additive two-way wiretap channel, $X_{1},X_{2},Y_{1},Y_{2},Z$ are variables
in the finite field $\mathbb{F}_{q}$. Then, we have

$\displaystyle Y_{1}$ $\displaystyle=a_{1}X_{1}+b_{1}X_{2}+N_{1},$ (104)
$\displaystyle Y_{2}$ $\displaystyle=a_{2}X_{1}+b_{2}X_{2}+N_{2},$ (105)
$\displaystyle Z$ $\displaystyle=a_{3}X_{1}+b_{3}X_{2}+N_{3},$ (106)

where $a_{1},b_{1},a_{2},b_{2},a_{3},b_{3}$ are non-zero elements of
$\mathbb{F}_{q}$. $N_{1}$, $N_{2}$, $N_{3}$ are the independent random
variables in $\mathbb{F}_{q},$ which are independent of $X_{1},X_{2}$.

When $X_{1}$ and $X_{2}$ are independent uniform random variables, we have

$\displaystyle I(Y_{2};X_{1}|X_{2})$ $\displaystyle=\log q-H(N_{2})$ (107)
$\displaystyle I(Y_{1};X_{2}|X_{1})$ $\displaystyle=\log q-H(N_{1})$ (108)
$\displaystyle I(Z;X_{1})$ $\displaystyle=I(Z;X_{2})=0$ (109) $\displaystyle
I(X_{1},X_{2};Z)$ $\displaystyle=\log q-H(N_{3}),$ (110)

where $H(X)$ expresses the Shannon entropy of a random variable $X$ defined as
$-\sum\limits_{x}P_{X}(x)\log P_{X}(x)$.

Therefore, the joint secrecy achievable rate region (83) is evaluated to

$\displaystyle{\cal R}_{J,N}\supseteq{\cal
A}_{J}:=\left\\{(R_{1},R_{2})\left|\begin{array}[]{l}R_{1}\leq\log
q-H(N_{2}),\\\ R_{2}\leq\log q-H(N_{1}),\\\ R_{1}+R_{2}\leq\log
q+H(N_{3})-H(N_{1})-H(N_{2})\end{array}\right.\right\\}.$ (114)

Also, the individual secrecy achievable rate region (87) is evaluated to

$\displaystyle{\cal R}_{I,N}\supseteq{\cal
A}_{I}:=\left\\{(R_{1},R_{2})\left|\begin{array}[]{l}R_{1}\leq\log
q-H(N_{2}),\\\ R_{2}\leq\log q-H(N_{1}),\\\ \max(R_{1},R_{2})\leq\log
q+H(N_{3})-H(N_{1})-H(N_{2})\end{array}\right.\right\\}.$ (118)

We define the outer region

$\displaystyle{\cal
A}_{O}:=\left\\{(R_{1},R_{2})\left|\begin{array}[]{l}R_{1}\leq\log
q-H(N_{2}),\\\ R_{2}\leq\log q-H(N_{1})\end{array}\right.\right\\}.$ (121)

Considering the case when $V_{i}=X_{i}$ is the uniform random number for
$i=1,2$, we have the following corollary of Theorem 1.

###### Corollary 1

Under the channel model (104) – (106), we have the following relations.

$\displaystyle{\cal A}_{J}\subset{\cal C}_{J,S,N,2},\quad{\cal
A}_{I}\subset{\cal C}_{I,S,N,2}.$ (122)

The reference [15, Corollary 1] showed the achievability of the rate region
${\cal A}_{J}$ under the weak secrecy for the binary additive TW-WC, i.e.,

$\displaystyle{\cal A}_{J}\subset{\cal C}_{J,W,N,2}.$ (123)

Corollary 1 extended the above relation to additive TW-WC over any finite
field and to the case with strong secrecy.

Next, to apply Lemma 4 with the above choice of $X_{i}$ and $V_{i}$ for
$i=1,2$, we prepare the Rényi entropy $H_{1+s}(X)$ of a random variable $X$
which is defined as $-\frac{1}{s}\log\sum\limits_{x}P_{X}(x)^{1+s}$. Then, for
the additive TW-WC, the quantities
$e^{-sI_{\frac{1}{1+s}}^{\uparrow}(Y_{2};X_{1}|X_{2})}$ and
$e^{-sI_{\frac{1}{1+s}}^{\uparrow}(Y_{1};X_{2}|X_{1})}$ are calculated as

$\displaystyle e^{-sI_{\frac{1}{1+s}}^{\uparrow}(Y_{2};X_{1}|X_{2})}=e^{-s\log
q+sH_{\frac{1}{1+s}}(N_{2})},\quad
e^{-sI_{\frac{1}{1+s}}^{\uparrow}(Y_{1};X_{2}|X_{1})}=e^{-s\log
q+sH_{\frac{1}{1+s}}(N_{1})}.$ (124)

Also, $e^{sI^{\downarrow}_{1+s}(Z;X_{1})}$,
$e^{sI^{\downarrow}_{1+s}(Z;X_{2})}$, and
$e^{sI^{\downarrow}_{1+s}(Z;X_{1},X_{2})}$ are calculated as

$\displaystyle e^{sI^{\downarrow}_{1+s}(Z;X_{1})}=1,\quad
e^{sI^{\downarrow}_{1+s}(Z;X_{2})}=1,\quad
e^{sI^{\downarrow}_{1+s}(Z;X_{1},X_{2})}=e^{s\log q-sH_{1+s}(N_{3})}.$ (125)

Hence, Lemma 4 implies the following corollary.

###### Corollary 2

For a rate pair $(R_{1},R_{2})$ and two positive numbers $r_{1},r_{2}$, there
exists a sequence of $(e^{nR_{1}},e^{nR_{2}},n)$ non-adaptive codes
$\\{C_{n}\\}$ with the 2nd type decoder such that

$\displaystyle P_{e}^{n}(C_{n})$ $\displaystyle\leq
2\left(e^{ns\left(R_{1}+r_{1}-\log
q+H_{\frac{1}{1+s}}(N_{2})\right)}+e^{ns\left(R_{2}+r_{2}-\log
q+H_{\frac{1}{1+s}}(N_{1})\right)}\right),$ (126) $\displaystyle
I(M_{1},M_{2};Z^{n}|C_{n})$ $\displaystyle\leq 2\left(e^{ns\left(\log
q-H_{1+s}(N_{3})-r_{1}-r_{2}\right)}+e^{-nsr_{1}}+e^{-nsr_{2}}\right).$ (127)

Also, there exists a sequence of $(e^{nR_{1}},e^{nR_{2}},n)$ non-adaptive
codes $\\{C_{n}\\}$ such that

$\displaystyle P_{e}^{n}(C_{n})$ $\displaystyle\leq
3\left(e^{ns\left(R_{1}+r_{1}-\log
q+H_{\frac{1}{1+s}}(N_{2})\right)}+e^{ns\left(R_{2}+r_{2}-\log
q+H_{\frac{1}{1+s}}(N_{1})\right)}\right),$ (128) $\displaystyle
I(M_{1};Z^{n}|C_{n})$ $\displaystyle\leq 3\left(e^{ns\left(\log
q-H_{1+s}(N_{3})-r_{1}-r_{2}-R_{2}\right)}+e^{-nsr_{1}}+e^{-ns(r_{2}+R_{2})}\right),$
(129) $\displaystyle I(M_{2};Z^{n}|C_{n})$ $\displaystyle\leq
3\left(e^{ns\left(\log
q-H_{1+s}(N_{3})-r_{1}-r_{2}-R_{1}\right)}+e^{-nsr_{2}}+e^{-ns(r_{1}+R_{1})}\right).$
(130)

###### Theorem 3

When there exists an independent random variable $N_{1}^{\prime}$ or
$N_{2}^{\prime}$ from $N_{1}$ and $N_{2}$ such that
$Z=a_{3}a_{1}^{-1}Y_{1}+N_{1}^{\prime}$ or
$Z=a_{3}a_{2}^{-1}Y_{2}+N_{2}^{\prime}$, the equalities in (114) and (118)
hold, i.e., ${\cal R}_{J,N}={\cal A}_{J}$ and ${\cal R}_{I,N}={\cal A}_{I}.$

###### Proof:

See Appendix F for a detailed proof. ∎

Notice that the condition $Z=a_{3}a_{1}^{-1}Y_{1}+N_{1}^{\prime}$ is a
stronger condition than the condition $a_{3}a_{1}^{-1}=b_{3}b_{1}^{-1}$. Also,
the condition $Z=a_{3}a_{2}^{-1}Y_{2}+N_{2}^{\prime}$ is a stronger condition
than the condition $a_{3}a_{2}^{-1}=b_{3}b_{2}^{-1}$.

###### Theorem 4

We have the following relation.

$\displaystyle{\cal C}_{I,W,A,1}$ $\displaystyle\subseteq{\cal A}_{O}$ (131)

When there exists an independent random variable $N_{3}^{\prime}$ from $N_{1}$
and $N_{2}$ such that
$N_{3}=\frac{b_{3}}{b_{1}}{N}_{1}+\frac{a_{3}}{a_{2}}{N}_{2}+N_{3}^{\prime}$,
we have

$\displaystyle{\cal C}_{J,W,N,1}\subseteq{\cal A}_{J}.$ (132)

Since ${\cal C}_{J,W,N,2}\subset{\cal C}_{J,W,N,1}$, with Corollary 1, the
relation (132) implies the relation

$\displaystyle{\cal C}_{J,b,N,d}={\cal R}_{J,N}={\cal A}_{J}$ (133)

for $b=S,W$ and $d=1,2$.

###### Proof:

See Appendix G for a detailed proof. ∎

We notice that Theorem 4 implies the equality in (114) with a condition
different from Theorem 3.

When $\max\\{H(N_{1}),H(N_{2})\\}\leq H(N_{3})$, we have
$H(N_{1})+H(N_{2})-H(N_{3})\leq\min\\{H(N_{1}),H(N_{2})\\}$, which indicates
that the first and second condition in (118) implies the third condition in
(118). Hence, under this condition, the region ${\cal A}_{I}$ is simplified to
${\cal A}_{O}$. Hence, we have the following corollary.

###### Corollary 3

When $\max\\{H(N_{1}),H(N_{2})\\}\leq H(N_{3})$, we have the following
relation.

$\displaystyle{\cal C}_{I,b,c,d}={\cal A}_{O}$ (134)

for $b=S,W$, $c=N,A$, and $d=1,2$.

##  VI Gaussian two-way wiretap channel

In Gaussian two-way wiretap channel, $X_{1},X_{2},Y_{1},Y_{2},Z$ are variables
in $\mathbb{R}$. Then, we have

$\displaystyle Y_{1}$ $\displaystyle=a_{1}X_{1}+b_{1}X_{2}+N_{1},$ (135)
$\displaystyle Y_{2}$ $\displaystyle=a_{2}X_{1}+b_{2}X_{2}+N_{2},$ (136)
$\displaystyle Z$ $\displaystyle=a_{3}X_{1}+b_{3}X_{2}+N_{3},$ (137)

where $N_{1}$, $N_{2}$, $N_{3}$ are Gaussian random variables with mean 0 and
variance $v_{1},$ $v_{2},$ $v_{3}$. Also,
$a_{1},b_{1},a_{2},b_{2},a_{3},b_{3}$ are constants. Choosing $g_{i}$ as
$g_{i}(X_{i}):=X_{i}^{2}$, we consider the cost constraint.

Assume that $X_{1}$($X_{2}$) is a Gaussian random variable with mean $0$ and
variance $p_{1}$ ($p_{2}$). Then, we have

$\displaystyle I(Y_{2};X_{1}|X_{2})$
$\displaystyle=\frac{1}{2}(\log(a_{2}^{2}p_{1}+v_{2})-\log v_{2})$ (138)
$\displaystyle I(Y_{1};X_{2}|X_{1})$
$\displaystyle=\frac{1}{2}(\log(b_{1}^{2}p_{2}+v_{1})-\log v_{1})$ (139)
$\displaystyle I(Z;X_{1})$
$\displaystyle=\frac{1}{2}(\log(a_{3}^{2}p_{1}+b_{3}^{2}p_{2}+v_{3})-\log(b_{3}^{2}p_{2}+v_{3}))$
(140) $\displaystyle I(Z;X_{2})$
$\displaystyle=\frac{1}{2}(\log(a_{3}^{2}p_{1}+b_{3}^{2}p_{2}+v_{3})-\log(a_{3}^{2}p_{1}+v_{3}))$
(141) $\displaystyle I(X_{1},X_{2};Z)$
$\displaystyle=\frac{1}{2}(\log(a_{3}^{2}p_{1}+b_{3}^{2}p_{2}+v_{3})-\log
v_{3}).$ (142)

Therefore, the joint secrecy achievable rate region (24) is converted to

$\displaystyle{\cal
R}_{J}^{p_{1},p_{2}}:=\left\\{(R_{1},R_{2})\left|\begin{array}[]{l}R_{1}\leq\frac{1}{2}\Big{(}\log(a_{2}^{2}p_{1}+v_{2})-\log
v_{2}-\log(a_{3}^{2}p_{1}+b_{3}^{2}p_{2}+v_{3})+\log(b_{3}^{2}p_{2}+v_{3})\Big{)}\\\
R_{2}\leq\frac{1}{2}\Big{(}\log(b_{1}^{2}p_{2}+v_{1})-\log
v_{1}-\log(a_{3}^{2}p_{1}+b_{3}^{2}p_{2}+v_{3})+\log(a_{3}^{2}p_{1}+v_{3})\Big{)}\\\
R_{1}+R_{2}\leq\frac{1}{2}\Big{(}\log(b_{1}^{2}p_{2}+v_{1})+\log(a_{2}^{2}p_{1}+v_{2})-\log(a_{3}^{2}p_{1}+b_{3}^{2}p_{2}+v_{3})\\\
\hskip 51.6665pt+\log v_{3}-\log v_{1}-\log
v_{2})\Big{)}.\end{array}\right.\right\\}.$ (147)

An inner bound on the strong joint secrecy capacity region is obtained as

${\cal A}_{J}^{c_{1},c_{2}}:=cl.\bigcup_{0\leq p_{1}\leq c_{1};0\leq p_{2}\leq
c_{2}}{\cal R}_{J}^{p_{1},p_{2}}.$ (148)

Also, the individual secrecy achievable rate region (29) is converted to

$\displaystyle{\cal
R}_{I}^{p_{1},p_{2}}:=\left\\{(R_{1},R_{2})\left|\begin{array}[]{l}R_{1}\leq\frac{1}{2}\Big{(}\log(a_{2}^{2}p_{1}+v_{2})-\log
v_{2}-\log(a_{3}^{2}p_{1}+b_{3}^{2}p_{2}+v_{3})+\log(b_{3}^{2}p_{2}+v_{3})\Big{)}\\\
R_{2}\leq\frac{1}{2}\Big{(}\log(b_{1}^{2}p_{2}+v_{1})-\log
v_{1}-\log(a_{3}^{2}p_{1}+b_{3}^{2}p_{2}+v_{3})+\log(a_{3}^{2}p_{1}+v_{3})\Big{)}\\\
\max(R_{1},R_{2})\leq\frac{1}{2}\Big{(}\log(b_{1}^{2}p_{2}+v_{1})+\log(a_{2}^{2}p_{1}+v_{2})-\log(a_{3}^{2}p_{1}+b_{3}^{2}p_{2}+v_{3})\\\
\hskip 68.88867pt+\log v_{3}-\log v_{1}-\log
v_{2})\Big{)}.\end{array}\right.\right\\}.$ (153)

An inner bound on the strong individual secrecy capacity region is obtained as

${\cal A}_{I}^{c_{1},c_{2}}:=cl.\bigcup_{0\leq p_{1}\leq c_{1};0\leq p_{2}\leq
c_{2}}{\cal R}_{I}^{p_{1},p_{2}}.$ (154)

For the direct part, we have the following theorem.

###### Theorem 5

we have the following relations.

$\displaystyle{\cal A}_{J}^{c_{1},c_{2}}\subset{\cal
C}_{J,S,N,2}^{P,c_{1},c_{2}},\quad{\cal A}_{I}^{c_{1},c_{2}}\subset{\cal
C}_{I,S,N,2}^{P,c_{1},c_{2}}.$ (155)

As observed in [19] that the reference [12, 13] has showed the achievability
of the rate region ${\cal A}_{J}^{c_{1},c_{2}}$ under the weak secrecy and the
average cost constraint, i.e.,

$\displaystyle{\cal A}_{J}^{c_{1},c_{2}}\subset{\cal
C}_{J,W,N,2}^{A,c_{1},c_{2}}.$ (156)

Theorem 5 extended the above relation to the case with strong secrecy and the
peak cost constraint. It is worth mentioning that this region (under the weak
secrecy and the average cost constraint) is extended in [15] by applying
adaptive coding. Here, we restrict our focus to the non-adaptive coding.

For the converse part, we have the following theorem.

###### Theorem 6

When $\frac{b_{3}^{2}}{b_{1}^{2}}v_{1}+\frac{a_{3}^{2}}{a_{2}^{2}}v_{2}\leq
v_{3}$, we have the following relations.

$\displaystyle{\cal C}_{J,W,N,1}^{A,c_{1},c_{2}}$ $\displaystyle\subset{\cal
A}_{J,N}^{c_{1},c_{2}}:=\left\\{(R_{1},R_{2})\left|\begin{array}[]{l}R_{1}\leq\frac{1}{2}\Big{(}\log(a_{2}^{2}c_{1}+v_{2})-\log
v_{2}\Big{)}\\\ R_{2}\leq\frac{1}{2}\Big{(}\log(b_{1}^{2}c_{2}+v_{1})-\log
v_{1}\Big{)}\\\
R_{1}+R_{2}\leq\frac{1}{2}\Big{(}\log(b_{1}^{2}c_{2}+v_{1})+\log(a_{2}^{2}c_{1}+v_{2})-\log(a_{3}^{2}c_{1}+b_{3}^{2}c_{2}+v_{3})\\\
\hskip 51.6665pt+\log v_{3}-\log v_{1}-\log
v_{2})\Big{)}\end{array}\right.\right\\}.$ (161)

###### Proof:

See Appendix I for a detailed proof. ∎

##  VII Conclusion

In this paper, we have studied the TW-WC under a strong joint or individual
secrecy constraint. Restricting our focus to non-adaptive coding, we have
derived inner bounds on the strong secrecy capacity regions for the TW-WC and
characterized the secrecy and error exponents by the conditional Rényi mutual
information. Moreover, we have introduced the peak cost constraint and
extended our results by using the constant composition codes, where both the
secrecy and error exponents are accordingly characterized by a modification of
Rényi mutual information. Our method works even when a pre-noisy processing is
employed based on a conditional distribution in the encoder and the results
are valid for the general discrete memoryless TW-WC without/with cost
constraint. In addition, we have applied our results to the case with additive
noise channel over a finite field and the case with the Gaussian noise. In
these examples, we have derived outer bounds in various settings in addition
to inner bounds.

Our results can naturally be extended in several directions. First, our
results can be generalized to other multi-user communication channels. For the
TW-WC, we did not include the scenario with one-sided secrecy, since the
analysis is straightforward by following our approaches for the scenarios with
joint/individual secrecy. Besides, it is not difficult to generalize our
results to channels with more than two legitimate users under diverse secrecy
constraints. In fact, our technique for strong secrecy analysis reflects the
channel resolvability for the $k$-transmitter MAC [20]; our error exponent
could be regarded as a generalization of Gallager’s error exponent [22]; and
our method covers also the channels with cost constraints by using random
coding on constant composition codes, all together laying a solid foundation
for yielding inner bounds and the corresponding secrecy and error exponents
for various settings of multi-user communication channels with strong secrecy
constraint that could be requested differently depending on the actual
applications. Besides, in this paper we have focused on the non-adaptive codes
only. As already witnessed in [15] for the Gaussian TW-WC under joint weak
secrecy, the adaptive coding could enlarge the inner bounds that were
established based on the non-adaptive coding. A further notice is that the
adaptive coding scheme proposed in [15] could be extended to the scenario with
joint strong secrecy as shown in [19]. Clearly, this extension is not limited
to the joint secrecy only. It can be extended to the TW-WCs with individual
secrecy or one-sided secrecy as well, to potentially improve the existing
inner bounds, while the analysis for secrecy and reliability needs to be
revised since the input is adapted to the previously received signals in the
adaptive coding. A more challenging task could be to find an adaptive coding
scheme, differently from [15], that actually could improve the secrecy sum-
rate that is achieved by non-adaptive coding; or to provide th e converse
proof if adaption is useless to improve the secrecy sum-rate.

## Acknowledgments

MH is very grateful to Prof. Te Sun Han for explaining the relation between
the references [25, 28].

##  Appendix A Proof of Lemma 1

We show the closedness of ${\cal C}_{J,S(W),A(N),1(2)}^{P(A),c_{1},c_{2}}$. It
is sufficient to show the following. When we choose any sequence
$\\{\kappa_{n}\\}$ in ${\cal C}_{J,S(W),A(N),1(2)}^{P(A),c_{1},c_{2}}$ such
that $\kappa_{m}\to\kappa_{0}$ as $m\to\infty$, $\kappa_{0}$ belongs to ${\cal
C}_{J,S(W),A(N),1(2)}^{P(A),c_{1},c_{2}}$. We choose a sequence codes
$\\{C_{n,m}\\}_{n=1}^{\infty}$ such that $\kappa(C_{n,m})\to\kappa_{m}$,
$P_{e}^{n}(C_{n,m})\to 0$, and $I(M_{1},M_{2};Z^{n}|C_{n,m})\to 0$ as
$n\to\infty$. For any positive integer $l>0$, we choose $m_{l}$ such that
$\|\kappa_{m}-\kappa_{0}\|\leq\frac{1}{l}$. We choose $n_{l}$ satisfying the
following conditions; $n_{l+1}>n_{l}$. For any $n\geq n_{l}$,we have
$\|\kappa(C_{n,m_{l}})-\kappa_{m_{l}}\|\leq\frac{1}{l}$,
$P_{e}^{n}(C_{n,m_{l}})\leq\frac{1}{l}$, and
$I(M_{1},M_{2};Z^{n}|C_{n,m_{l}})\leq\frac{1}{l}$. We choose $C_{n}$ as
$C_{n,m_{l}}$ for $n_{l}\leq n<n_{l+1}$. Then, we have
$\kappa(C_{n})\to\kappa_{0}$, $P_{e}^{n}(C_{n})\to 0$, and
$I(M_{1},M_{2};Z^{n}|C_{n})\to 0$ as $n\to\infty$. Therefore, $\kappa_{0}$
belongs to ${\cal C}_{J,S(W),A(N),1(2)}^{I(II),c_{1},c_{2}}$.

The closedness of ${\cal C}_{I,S(W),A(N),1(2)}^{P(A),c_{1},c_{2}}$ can be
shown in the same way.

##  Appendix B Proof of Lemma 2

To prove Lemma 2, we consider

$\displaystyle\mathbb{E}_{C}\left[e^{sD_{1+s}(P_{Z|C}\|P_{Z})}\right]$
$\displaystyle\stackrel{{\scriptstyle(a)}}{{=}}$
$\displaystyle\sum_{z}\mathbb{E}_{C}\left[\left(\frac{1}{\mathsf{L}_{1}\mathsf{L}_{2}}\sum_{j_{1}^{\prime},j_{2}^{\prime}}P_{Z|X_{1}=X_{1,j_{1}^{\prime}},X_{2}=X_{2,j_{2}^{\prime}}}(z)\right)\left(\frac{1}{\mathsf{L}_{1}\mathsf{L}_{2}}\sum_{j_{1},j_{2}}P_{Z|X_{1}=X_{1,j_{1}},X_{2}=X_{2,j_{2}}}(z)\right)^{s}P_{Z}(z)^{-s}\right]$
$\displaystyle\stackrel{{\scriptstyle(b)}}{{=}}$
$\displaystyle\sum_{z}\mathbb{E}_{C}\left[\frac{1}{\mathsf{L}_{1}\mathsf{L}_{2}}\sum_{j_{1}^{\prime},j_{2}^{\prime}}P_{Z|X_{1}=X_{1,j_{1}^{\prime}},X_{2}=X_{2,j_{2}^{\prime}}}(z)\right.$
$\displaystyle\left.\cdot\mathbb{E}_{C|X_{1}=X_{1,j_{1}^{\prime}},X_{2}=X_{2,j_{2}^{\prime}}}\left[\left(\frac{1}{\mathsf{L}_{1}\mathsf{L}_{2}}\sum_{j_{1},j_{2}}P_{Z|X_{1}=X_{1,j_{1}},X_{2}=X_{2,j_{2}}}(z)\right)^{s}\right]P_{Z}(z)^{-s}\right]$
$\displaystyle\stackrel{{\scriptstyle(c)}}{{\leq}}$
$\displaystyle\sum_{z}\mathbb{E}_{C}\left[\frac{1}{\mathsf{L}_{1}\mathsf{L}_{2}}\sum_{j_{1}^{\prime},j_{2}^{\prime}}P_{Z|X_{1}=X_{1,j_{1}^{\prime}},X_{2}=X_{2,j_{2}^{\prime}}}(z)\right.$
$\displaystyle\left.\cdot\left(\mathbb{E}_{C|X_{1}=X_{1,j_{1}^{\prime}},X_{2}=X_{2,j_{2}^{\prime}}}\left[\frac{1}{\mathsf{L}_{1}\mathsf{L}_{2}}\sum_{j_{1},j_{2}}P_{Z|X_{1}=X_{1,j_{1}},X_{2}=X_{2,j_{2}}}(z)\right]\right)^{s}P_{Z}(z)^{-s}\right]$
$\displaystyle=$
$\displaystyle\sum_{z}\mathbb{E}_{C}\left[\frac{1}{\mathsf{L}_{1}\mathsf{L}_{2}}\sum_{j_{1}^{\prime},j_{2}^{\prime}}P_{Z|X_{1}=X_{1,j_{1}^{\prime}},X_{2}=X_{2,j_{2}^{\prime}}}(z)\right.$
$\displaystyle\left.\cdot\left(\mathbb{E}_{C|X_{1}=X_{1,j_{1}^{\prime}},X_{2}=X_{2,j_{2}^{\prime}}}\left[\frac{1}{\mathsf{L}_{1}\mathsf{L}_{2}}\sum_{\mathcal{S}\subseteq\\{1,2\\}}\sum_{\scriptsize\begin{array}[]{c}j_{i}\neq
j_{i}^{\prime}\\\
i\in\mathcal{S}\end{array}}P_{Z|(X_{i}=X_{i,j_{i}})_{i\in\mathcal{S}},(X_{i}=X_{i,j_{i}^{\prime}})_{i\in{\mathcal{S}}^{c}}}(z)\right]\right)^{s}P_{Z}(z)^{-s}\right]$
(164) $\displaystyle=$
$\displaystyle\sum_{z}\mathbb{E}_{C}\left[\frac{1}{\mathsf{L}_{1}\mathsf{L}_{2}}\sum_{j_{1}^{\prime},j_{2}^{\prime}}P_{Z|X_{1}=X_{1,j_{1}^{\prime}},X_{2}=X_{2,j_{2}^{\prime}}}(z)\left(\frac{1}{\mathsf{L}_{1}\mathsf{L}_{2}}\sum_{\mathcal{S}\subseteq\\{1,2\\}}\left(\prod_{i\in\mathcal{S}}(\mathsf{L}_{i}-1)\right)P_{Z|(X_{i}=X_{i,j_{i}^{\prime}})_{i\in{\mathcal{S}}^{c}}}(z)\right)^{s}P_{Z}(z)^{-s}\right]$
$\displaystyle\stackrel{{\scriptstyle(d)}}{{\leq}}$
$\displaystyle\sum_{z}\mathbb{E}_{C}\left[\frac{1}{\mathsf{L}_{1}\mathsf{L}_{2}}\sum_{j_{1}^{\prime},j_{2}^{\prime}}P_{Z|X_{1}=X_{1,j_{1}^{\prime}},X_{2}=X_{2,j_{2}^{\prime}}}(z)\left(\frac{1}{\mathsf{L}_{1}^{s}\mathsf{L}_{2}^{s}}\sum_{\mathcal{S}\subseteq\\{1,2\\}}\left(\prod_{i\in{\mathcal{S}}}(\mathsf{L}_{i}-1)^{s}\right)P_{Z|(X_{i}=X_{i,j_{i}^{\prime}})_{i\in{\mathcal{S}}^{c}}}(z)^{s}\right)P_{Z}(z)^{-s}\right]$
$\displaystyle\stackrel{{\scriptstyle(e)}}{{\leq}}$
$\displaystyle\sum_{z}\mathbb{E}_{C}\left[\frac{1}{\mathsf{L}_{1}\mathsf{L}_{2}}\sum_{j_{1}^{\prime},j_{2}^{\prime}}P_{Z|X_{1}=X_{1,j_{1}^{\prime}},X_{2}=X_{2,j_{2}^{\prime}}}(z)\left(\sum_{\mathcal{S}\subseteq\\{1,2\\}}\frac{1}{\prod_{i\in{\mathcal{S}}^{c}}\mathsf{L}_{i}^{s}}P_{Z|(X_{i}=X_{i,j_{i}^{\prime}})_{i\in{\mathcal{S}}^{c}}}(z)^{s}\right)P_{Z}(z)^{-s}\right]$
$\displaystyle=$
$\displaystyle\sum_{z}\frac{1}{\mathsf{L}_{1}\mathsf{L}_{2}}\sum_{j_{1}^{\prime},j_{2}^{\prime}}\left(\sum_{{\mathcal{S}}\subseteq\\{1,2\\}}\frac{1}{\prod_{i\in{\mathcal{S}}^{c}}\mathsf{L}_{i}^{s}}\mathbb{E}_{C}\left[P_{Z|X_{1}=X_{1,j_{1}^{\prime}},X_{2}=X_{2,j_{2}^{\prime}}}(z)P_{Z|(X_{i}=X_{i,j_{i}^{\prime}})_{i\in{\mathcal{S}}^{c}}}(z)^{s}P_{Z}(z)^{-s}\right]\right)$
$\displaystyle=$
$\displaystyle\frac{1}{\mathsf{L}_{1}\mathsf{L}_{2}}\sum_{j_{1}^{\prime},j_{2}^{\prime}}\sum_{{\mathcal{S}}\subseteq\\{1,2\\}}\frac{1}{\prod_{i\in{\mathcal{S}}^{c}}\mathsf{L}_{i}^{s}}\sum_{(x_{i})_{i\in{\mathcal{S}}^{c}}}\left(\prod_{i\in{\mathcal{S}}^{c}}P_{X_{i}}(x_{i})\right)\sum_{z}P_{Z|(X_{i}=x_{i})_{i\in{\mathcal{S}}^{c}}}(z)^{1+s}P_{Z}(z)^{-s}$
$\displaystyle\stackrel{{\scriptstyle(f)}}{{=}}$
$\displaystyle\sum_{\mathcal{S}\subseteq\\{1,2\\}}\frac{1}{\prod_{i\in\mathcal{S}^{c}}\mathsf{L}_{i}^{s}}e^{sI_{1+s}^{\downarrow}(Z;X_{\mathcal{S}^{c}})}=\sum_{\mathcal{S}\subseteq\\{1,2\\}}\frac{1}{\prod_{i\in\mathcal{S}}\mathsf{L}_{i}^{s}}e^{sI_{1+s}^{\downarrow}(Z;X_{\mathcal{S}})}$
$\displaystyle=$ $\displaystyle
1+\sum_{\mathcal{S}\neq\emptyset,\mathcal{S}\subseteq\\{1,2\\}}\frac{1}{\prod_{i\in\mathcal{S}}\mathsf{L}_{i}^{s}}e^{sI_{1+s}^{\downarrow}(Z;X_{\mathcal{S}})},$
(165)

where $X_{\mathcal{S}}=\\{X_{i}|i\in\mathcal{S}\\}$ with
$P_{X_{\mathcal{S}}}=\prod_{i\in\mathcal{S}}P_{X_{i}}.$ In more details, $(a)$
is by the definition of Rényi relative entropy in (48); $(b)$ is by the law of
total expectation; $(c)$ is by the concavity of $x^{s}$ for $0\leq s\leq 1$;
$(d)$ is due to the fact that
$(\sum_{i}a_{i})^{s}\leq\sum\limits_{i}a_{i}^{s}$ for $a_{i}\geq 0$ and $0\leq
s\leq 1$; $(e)$ follows from $\mathsf{L}_{i}-1\leq\mathsf{L}_{i}$; and $(f)$
is by the definition of Rényi mutual information in (49). Taking the logarithm
to (165), we have

$s\mathbb{E}_{C}\left[D_{1+s}(P_{Z|C}\|P_{Z})\right]\leq\log\Big{(}1+\sum_{\mathcal{S}\neq\emptyset,\mathcal{S}\subseteq\\{1,2\\}}\frac{1}{\prod_{i\in\mathcal{S}}\mathsf{L}_{i}^{s}}e^{sI_{1+s}^{\downarrow}(Z;X_{\mathcal{S}})}\Big{)}\leq\sum_{{\mathcal{S}}\neq\emptyset,\mathcal{S}\subseteq\\{1,2\\}}\frac{1}{\prod_{i\in\mathcal{S}}\mathsf{L}_{i}^{s}}e^{sI_{1+s}^{\downarrow}(Z;X_{\mathcal{S}})}.$
(166)

##  Appendix C Fourier-Motzkin elimination to derive individual secrecy region

To derive individual secrecy region, recall that we have the following rate
constraints:

$\displaystyle r_{1}$ $\displaystyle>I(Z;X_{1});$ (167) $\displaystyle r_{2}$
$\displaystyle>I(Z;X_{2});$ (168) $\displaystyle
r_{1}+r_{2}+\min(R_{1},R_{2})$ $\displaystyle>I(Z;X_{1},X_{2});$ (169)
$\displaystyle R_{1}+r_{1}$ $\displaystyle<I(Y_{2};X_{1}|X_{2});$ (170)
$\displaystyle R_{2}+r_{2}$ $\displaystyle<I(Y_{1};X_{2}|X_{1}).$ (171)

First consider (167), (169) and (170) to remove $r_{1}.$ We obtain

$\displaystyle R_{1}$ $\displaystyle<I(Y_{2};X_{1}|X_{2})-I(Z;X_{1});$ (172)
$\displaystyle r_{2}$
$\displaystyle>R_{1}-I(Y_{2};X_{1}|X_{2})+I(Z;X_{1},X_{2})-\min(R_{1},R_{2}).$
(173)

Now consider (168), (171) and (173) to remove $r_{2},$ We obtain

$\displaystyle R_{2}$ $\displaystyle\leq I(Y_{1};X_{2}|X_{1})-I(Z;X_{2});$
(174) $\displaystyle\max\\{R_{1},R_{2}\\}$ $\displaystyle\leq
I(Y_{2};X_{1}|X_{2})+I(Y_{1};X_{2}|X_{1})-I(X_{1},X_{2};Z).$ (175)

The achievable region is thus established by (172),(174) and (175).

##  Appendix D Proof of Lemma 5

Using (88), we have

$\displaystyle\breve{P}_{X_{i}^{n}|V_{i}^{n}}\leq\nu_{n}(|{\cal
X}_{i}|)P_{X_{i}|V_{i}}^{n},\quad\breve{P}_{V_{i}^{n}}\leq\nu_{n}(|{\cal
V}_{i}|)P_{V_{i}}^{n}.$ (176)

Thus, we have

$\displaystyle
P_{Z|X_{1}X_{2}}^{n}(z^{n}|x_{1}^{n},x_{2}^{n})\breve{P}_{X^{n}_{2}}(x_{2}^{n})\breve{P}_{X^{n}_{1}|V^{n}_{1}}(x_{1}^{n}|v_{1}^{n})\leq\nu_{n}(|{\cal
X}_{1}|)\nu_{n}(|{\cal
X}_{2}|)P_{Z|X_{1}X_{2}}^{n}(z^{n}|x_{1}^{n},x_{2}^{n}){P}_{X_{2}}^{n}(x_{2}^{n}){P}_{X_{1}|V_{1}}^{n}(x_{1}^{n}|v_{1}^{n}).$
(177)

Taking the sum for $x_{1}^{n},x_{2}^{n}$, we have

$\displaystyle
P_{Z|X_{1}X_{2}}^{n}\cdot\breve{P}_{X^{n}_{2}}\cdot\breve{P}_{X^{n}_{1}|V^{n}_{1}}\leq\nu_{n}(|{\cal
X}_{1}|)\nu_{n}(|{\cal
X}_{2}|)P_{Z|X_{1}X_{2}}^{n}\cdot{P}_{X_{2}}^{n}\cdot{P}_{X_{1}|V_{1}}^{n}.$
(178)

Thus, we have

$\displaystyle
P_{Z|X_{1}X_{2}}^{n}(z^{n}|x_{1}^{n},x_{2}^{n})\breve{P}_{X^{n}_{\\{1,2\\}}|V^{n}_{\\{1,2\\}}}(x_{\\{1,2\\}}^{n}|v_{\\{1,2\\}}^{n})=P_{Z|X_{1}X_{2}}^{n}(z^{n}|x_{1}^{n},x_{2}^{n})\breve{P}_{X^{n}_{1}|V^{n}_{1}}(x_{1}^{n}|v_{1}^{n})\breve{P}_{X^{n}_{2}|V^{n}_{2}}(x_{2}^{n}|v_{2}^{n})$
$\displaystyle\leq$ $\displaystyle\nu_{n}(|{\cal X}_{1}|)\nu_{n}(|{\cal
X}_{2}|)P_{Z|X_{1}X_{2}}^{n}(z^{n}|x_{1}^{n},x_{2}^{n}){P}_{X_{1}|V_{1}}^{n}(x_{1}^{n}|v_{1}^{n}){P}_{X_{2}|V_{2}}^{n}(x_{2}^{n}|v_{2}^{n}).$
(179)

Taking the sum for $x_{1}^{n},x_{2}^{n}$, we have

$\displaystyle
P_{Z|X_{1}X_{2}}^{n}\cdot\breve{P}_{X^{n}_{\\{1,2\\}}|V^{n}_{\\{1,2\\}}}\leq\nu_{n}(|{\cal
X}_{1}|)\nu_{n}(|{\cal
X}_{2}|)P_{Z|X_{1}X_{2}}^{n}\cdot{P}_{X_{\\{1,2\\}}|V_{\\{1,2\\}}}^{n}.$ (180)

In the same way, we have

$\displaystyle
P_{Z|X_{1}X_{2}}^{n}\cdot\breve{P}_{X^{n}_{\\{1,2\\}}}\leq\nu_{n}(|{\cal
X}_{1}|)\nu_{n}(|{\cal
X}_{2}|)P_{Z|X_{1}X_{2}}^{n}\cdot{P}_{X_{\\{1,2\\}}}^{n}.$ (181)

In summary, we have

$\displaystyle
P_{Z|X_{1}X_{2}}^{n}\cdot\breve{P}_{X^{n}_{\mathcal{S}^{c}}}\cdot\breve{P}_{X^{n}_{\mathcal{S}}|V^{n}_{\mathcal{S}}}\leq\nu_{n}(|{\cal
X}_{1}|)\nu_{n}(|{\cal
X}_{2}|)P_{Z|X_{1}X_{2}}^{n}\cdot{P}_{X_{\mathcal{S}^{c}}}^{n}\cdot{P}_{X_{\mathcal{S}}|V_{\mathcal{S}}}^{n}.$
(182)

Using (57), we have

$\displaystyle
e^{sI_{\frac{1}{1-s}}^{\uparrow}(Z^{n};V^{n}_{\mathcal{S}})[P_{Z|X_{1}X_{2}}^{n}\cdot\breve{P}_{X^{n}_{\mathcal{S}^{c}}}\cdot\breve{P}_{X^{n}_{\mathcal{S}}|V^{n}_{\mathcal{S}}}\times\breve{P}_{V^{n}_{\mathcal{S}}}]}\leq\nu_{n}(|{\cal
X}_{1}|)\nu_{n}(|{\cal
X}_{2}|)e^{sI_{\frac{1}{1-s}}^{\uparrow}(Z^{n};V^{n}_{\mathcal{S}})[P_{Z|X_{1}X_{2}}^{n}\cdot{P}_{X_{\mathcal{S}^{c}}}^{n}\cdot{P}_{X_{\mathcal{S}}|V_{\mathcal{S}}}^{n}\times\breve{P}_{V^{n}_{\mathcal{S}}}]}$
(183)

Therefore, for $s\in[0,1]$, we have

$\displaystyle\mathbb{E}_{C}\left[e^{sD_{1+s}(P_{Z^{n}|C}\|P_{Z^{n}})}\right]-1$
$\displaystyle\stackrel{{\scriptstyle(a)}}{{\leq}}$
$\displaystyle\sum_{\mathcal{S}\neq\emptyset,\mathcal{S}\subseteq\\{1,2\\}}\frac{1}{\prod_{i\in\mathcal{S}}\mathsf{L}_{i}^{s}}e^{sI_{1+s}^{\downarrow}(Z^{n};V^{n}_{\mathcal{S}})[P_{Z|X_{1}X_{2}}^{n}\cdot\breve{P}_{X^{n}_{\mathcal{S}^{c}}}\cdot\breve{P}_{X^{n}_{\mathcal{S}}|V^{n}_{\mathcal{S}}}\times\breve{P}_{V^{n}_{\mathcal{S}}}]}$
$\displaystyle\stackrel{{\scriptstyle(b)}}{{\leq}}$
$\displaystyle\sum_{\mathcal{S}\neq\emptyset,\mathcal{S}\subseteq\\{1,2\\}}\frac{1}{\prod_{i\in\mathcal{S}}\mathsf{L}_{i}^{s}}e^{sI_{\frac{1}{1-s}}^{\uparrow}(Z^{n};V^{n}_{\mathcal{S}})[P_{Z|X_{1}X_{2}}^{n}\cdot\breve{P}_{X^{n}_{\mathcal{S}^{c}}}\cdot\breve{P}_{X^{n}_{\mathcal{S}}|V^{n}_{\mathcal{S}}}\times\breve{P}_{V^{n}_{\mathcal{S}}}]}$
$\displaystyle\stackrel{{\scriptstyle(c)}}{{\leq}}$
$\displaystyle\sum_{\mathcal{S}\neq\emptyset,\mathcal{S}\subseteq\\{1,2\\}}\frac{\nu_{n}(|{\cal
X}_{1}|)\nu_{n}(|{\cal
X}_{2}|)}{\prod_{i\in\mathcal{S}}\mathsf{L}_{i}^{s}}e^{sI_{\frac{1}{1-s}}^{\uparrow}(Z^{n};V^{n}_{\mathcal{S}})[P_{Z|X_{1}X_{2}}^{n}\cdot{P}_{X_{\mathcal{S}^{c}}}^{n}\cdot{P}_{X_{\mathcal{S}}|V_{\mathcal{S}}}^{n}\times\breve{P}_{V^{n}_{\mathcal{S}}}]}$
$\displaystyle=$
$\displaystyle\sum_{\mathcal{S}\neq\emptyset,\mathcal{S}\subseteq\\{1,2\\}}\frac{\nu_{n}(|{\cal
X}_{1}|)\nu_{n}(|{\cal
X}_{2}|)}{\prod_{i\in\mathcal{S}}\mathsf{L}_{i}^{s}}e^{sI_{\frac{1}{1-s}}^{\uparrow}(Z^{n};V^{n}_{\mathcal{S}})[P_{Z|V_{\mathcal{S}}}^{n}\times\breve{P}_{V^{n}_{\mathcal{S}}}]}$
$\displaystyle\stackrel{{\scriptstyle(d)}}{{=}}$
$\displaystyle\sum_{\mathcal{S}\neq\emptyset,\mathcal{S}\subseteq\\{1,2\\}}\frac{\nu_{n}(|{\cal
X}_{1}|)\nu_{n}(|{\cal
X}_{2}|)}{\prod_{i\in\mathcal{S}}\mathsf{L}_{i}^{s}}e^{s\min_{Q_{Z^{n}}}D_{\frac{1}{1-s}}(P_{Z|V_{\mathcal{S}}}^{n}\times\breve{P}_{V^{n}_{\mathcal{S}}}\|Q_{Z^{n}}\times\breve{P}_{V^{n}_{\mathcal{S}}})}$
$\displaystyle\stackrel{{\scriptstyle(e)}}{{\leq}}$
$\displaystyle\sum_{\mathcal{S}\neq\emptyset,\mathcal{S}\subseteq\\{1,2\\}}\frac{\nu_{n}(|{\cal
X}_{1}|)\nu_{n}(|{\cal
X}_{2}|)}{\prod_{i\in\mathcal{S}}\mathsf{L}_{i}^{s}}e^{s\min_{Q_{Z}}D_{\frac{1}{1-s}}(P_{Z|V_{\mathcal{S}}}^{n}\times\breve{P}_{V^{n}_{\mathcal{S}}}\|Q_{Z}^{n}\times\breve{P}_{V^{n}_{\mathcal{S}}})}$
$\displaystyle=$
$\displaystyle\sum_{\mathcal{S}\neq\emptyset,\mathcal{S}\subseteq\\{1,2\\}}\frac{\nu_{n}(|{\cal
X}_{1}|)\nu_{n}(|{\cal
X}_{2}|)}{\prod_{i\in\mathcal{S}}\mathsf{L}_{i}^{s}}\min_{Q_{Z}}\sum_{v^{n}_{\mathcal{S}}\in
T_{P_{V_{\mathcal{S}}}}^{n}}\breve{P}_{V^{n}_{\mathcal{S}}}(v^{n}_{\mathcal{S}})e^{sD_{\frac{1}{1-s}}(P_{Z|V_{\mathcal{S}}}^{n}(\cdot|v_{\mathcal{S}}^{n})\|Q_{Z}^{n})}$
$\displaystyle\stackrel{{\scriptstyle(f)}}{{=}}$
$\displaystyle\sum_{\mathcal{S}\neq\emptyset,\mathcal{S}\subseteq\\{1,2\\}}\frac{\nu_{n}(|{\cal
X}_{1}|)\nu_{n}(|{\cal
X}_{2}|)}{\prod_{i\in\mathcal{S}}\mathsf{L}_{i}^{s}}\min_{Q_{Z}}e^{ns\sum_{v_{\mathcal{S}}}P_{V_{\mathcal{S}}}(v_{\mathcal{S}})D_{\frac{1}{1-s}}(P_{Z|V_{\mathcal{S}}=v_{\mathcal{S}}}\|Q_{Z})}$
$\displaystyle=$
$\displaystyle\sum_{\mathcal{S}\neq\emptyset,\mathcal{S}\subseteq\\{1,2\\}}\frac{\nu_{n}(|{\cal
X}_{1}|)\nu_{n}(|{\cal
X}_{2}|)}{\prod_{i\in\mathcal{S}}\mathsf{L}_{i}^{s}}e^{ns\min_{Q_{Z}}\sum_{v_{\mathcal{S}}}P_{V_{\mathcal{S}}}(v_{\mathcal{S}})D_{\frac{1}{1-s}}(P_{Z|V_{\mathcal{S}}=v_{\mathcal{S}}}\|Q_{Z})}$
$\displaystyle=$
$\displaystyle\sum_{\mathcal{S}\neq\emptyset,\mathcal{S}\subset\\{1,2\\}}\frac{\nu_{n}(|{\cal
X}_{1}|)\nu_{n}(|{\cal
X}_{2}|)}{\prod_{i\in\mathcal{S}}\mathsf{L}_{i}^{s}}e^{ns\breve{I}_{\frac{1}{1-s}}^{\uparrow}(Z;V_{\mathcal{S}})},$
(184)

where steps $(a)$, $(b)$, and $(c)$, follow from (165) in the proof of Lemma
2, (56), and (183), respectively. Also, $(d)$ and $(e)$ follow from (52) and
the restriction of the range of the minimization from $Q_{Z^{n}}$ to
$Q_{Z}^{n}$, respectively. Also, $(f)$ follows from the equations

$\displaystyle
D_{\frac{1}{1-s}}(P_{Z|V_{\mathcal{S}}}^{n}(\cdot|v_{\mathcal{S}}^{n})\|Q_{Z}^{n})=\sum_{j=1}^{n}D_{\frac{1}{1+s}}(P_{Z|V_{\mathcal{S}}=v_{\mathcal{S},j}}\|Q_{Z})=n\sum_{v_{\mathcal{S}}}P_{V_{\mathcal{S}}}(v_{\mathcal{S}})D_{\frac{1}{1+s}}(P_{Z|V_{\mathcal{S}}=v_{\mathcal{S}}}\|Q_{Z}),$
(185)

which follow from the fact that
$v_{\mathcal{S}}^{n}=(v_{\mathcal{S},1},\ldots,v_{\mathcal{S},n})\in
T_{P_{V_{\mathcal{S}}}}^{n}$. Notice that the RHS does not depend on the
choice of $v_{\mathcal{S}}^{n}\in T_{P_{V_{\mathcal{S}}}}^{n}$. Therefore, we
obtain (89).

###### Remark 3

When the conditional distribution $P_{X_{i}|V_{i}}$ is deterministic, the
first inequality in (176) can be simplified to
$\breve{P}_{X_{i}^{n}|V_{i}^{n}}=P_{X_{i}|V_{i}}^{n}$. Therefore, in this
case, (89) is simplified as

$\displaystyle
s\mathbb{E}_{C}\left[D_{1+s}(P_{Z^{n}|C}\|P_{Z^{n}})\right]\leq\sum_{\mathcal{S}\neq\emptyset,\mathcal{S}\subset\\{1,2\\}}\frac{\prod_{i^{\prime}\in\mathcal{S}}\nu_{n}(|{\cal
X}_{i^{\prime}}|)}{\prod_{i\in\mathcal{S}}\mathsf{L}_{i}^{s}}e^{s\breve{I}_{\frac{1}{1-s}}^{\uparrow}(Z;V_{\mathcal{S}})}.$
(186)

##  Appendix E Proof of Lemma 6

We choose $Q_{Y^{n}X_{1}^{n}}^{*}$ as
$Q_{Y^{n}X_{1}^{n}}^{*}(y^{n},x_{1}^{n}):=\frac{\sum_{v_{2}^{n}}P_{YX_{1}|V_{2}}^{n}(y^{n},x_{1}^{n}|v_{2}^{n})^{\frac{1}{1+s}}\breve{P}_{V_{2}^{n}}(v_{2}^{n})}{\sum_{y^{n},x_{1}^{n},v_{2}^{n}}P_{YX_{1}|V_{2}}^{n}(y^{n},x_{1}^{n}|v_{2}^{n})^{\frac{1}{1+s}}\breve{P}_{V_{2}^{n}}(v_{2}^{n})}$.
Then, we have

$\displaystyle
e^{-sI_{\frac{1}{1+s}}^{\uparrow}(Y^{n}X_{1}^{n};V_{2}^{n})[P_{YX_{1}|V_{2}}^{n}\times\breve{P}_{V_{2}^{n}}]}=e^{-sD_{\frac{1}{1+s}}(P_{YX_{1}|V_{2}}^{n}\times\breve{P}_{V_{2}^{n}}\|Q_{Y^{n}X_{1}^{n}}^{*}\times\breve{P}_{V_{2}^{n}})}.$
(187)

Since $Q_{Y^{n}X_{1}^{n}}^{*}$ is permutation invariant,
$Q_{Y^{n}X_{1}^{n}}^{*}$ is written as a probabilistic mixture of the uniform
distribution on a fixed-type subset. Hence, there exists a distribution $q$ on
$T_{n}({\cal Y}\times{\cal X}_{1})$ such that
$Q_{Y^{n}X_{1}^{n}}^{*}=\sum_{Q_{Y,X_{1}}\in T_{n}({\cal Y}\times{\cal
X}_{1})}q(Q_{Y,X_{1}})\breve{Q}_{Y^{n},X_{1}^{n}}$. Then, we have

$\displaystyle
Q_{Y^{n}X_{1}^{n}}^{*}(y^{n}x_{1}^{n})^{\frac{s}{1+s}}=\Big{(}\sum_{Q_{Y,X_{1}}\in
T_{n}({\cal Y}\times{\cal
X}_{1})}q(Q_{Y,X_{1}})\breve{Q}_{Y^{n},X_{1}^{n}}(y^{n}x_{1}^{n})\Big{)}^{\frac{s}{1+s}}$
$\displaystyle\leq$ $\displaystyle\sum_{Q_{Y,X_{1}}\in T_{n}({\cal
Y}\times{\cal
X}_{1})}\Big{(}q(Q_{Y,X_{1}})\breve{Q}_{Y^{n},X_{1}^{n}}(y^{n}x_{1}^{n})\Big{)}^{\frac{s}{1+s}}\leq\sum_{Q_{Y,X_{1}}\in
T_{n}({\cal Y}\times{\cal
X}_{1})}\breve{Q}_{Y^{n},X_{1}^{n}}(y^{n}x_{1}^{n})^{\frac{s}{1+s}}.$ (188)

Thus, we have

$\displaystyle
e^{-\frac{s}{1+s}D_{\frac{1}{1+s}}(P_{YX_{1}|V_{2}}^{n}\times\breve{P}_{V_{2}^{n}}\|Q_{Y^{n}X_{1}^{n}}^{*}\times\breve{P}_{V_{2}^{n}})}=\sum_{v_{2}^{n}\in
T_{P_{V_{2}}}^{n}}\breve{P}_{V_{2}^{n}}(v_{2}^{n})\sum_{y^{n}x_{1}^{n}}P_{YX_{1}|V_{2}}^{n}(y^{n}x_{1}^{n}|v_{2}^{n})^{\frac{1}{1+s}}Q_{Y^{n}X_{1}^{n}}^{*}(y^{n}x_{1}^{n})^{\frac{s}{1+s}}$
$\displaystyle\stackrel{{\scriptstyle(a)}}{{\leq}}$
$\displaystyle\sum_{v_{2}^{n}\in
T_{P_{V_{2}}}^{n}}\breve{P}_{V_{2}^{n}}(v_{2}^{n})\sum_{y^{n}x_{1}^{n}}P_{YX_{1}|V_{2}}^{n}(y^{n}x_{1}^{n}|v_{2}^{n})^{\frac{1}{1+s}}\sum_{Q_{Y,X_{1}}\in
T_{n}({\cal Y}\times{\cal
X}_{1})}\breve{Q}_{Y^{n},X_{1}^{n}}(y^{n}x_{1}^{n})^{\frac{s}{1+s}}$
$\displaystyle=$ $\displaystyle\sum_{Q_{Y,X_{1}}\in T_{n}({\cal Y}\times{\cal
X}_{1})}\sum_{v_{2}^{n}\in
T_{P_{V_{2}}}^{n}}\breve{P}_{V_{2}^{n}}(v_{2}^{n})\sum_{y^{n}x_{1}^{n}}P_{YX_{1}|V_{2}}^{n}(y^{n}x_{1}^{n}|v_{2}^{n})^{\frac{1}{1+s}}\breve{Q}_{Y^{n},X_{1}^{n}}(y^{n}x_{1}^{n})^{\frac{s}{1+s}}$
$\displaystyle\stackrel{{\scriptstyle(b)}}{{\leq}}$
$\displaystyle\nu_{n}(|{\cal Y}||{\cal
X}_{1}|)^{\frac{s}{1+s}}\sum_{Q_{Y,X_{1}}\in T_{n}({\cal Y}\times{\cal
X}_{1})}\sum_{v_{2}^{n}\in
T_{P_{V_{2}}}^{n}}\breve{P}_{V_{2}^{n}}(v_{2}^{n})\sum_{y^{n}x_{1}^{n}}P_{YX_{1}|V_{2}}^{n}(y^{n}x_{1}^{n}|v_{2}^{n})^{\frac{1}{1+s}}Q_{Y,X_{1}}^{n}(y^{n}x_{1}^{n})^{\frac{s}{1+s}}$
$\displaystyle=$ $\displaystyle\nu_{n}(|{\cal Y}||{\cal
X}_{1}|)^{\frac{s}{1+s}}\sum_{Q_{Y,X_{1}}\in T_{n}({\cal Y}\times{\cal
X}_{1})}\sum_{v_{2}^{n}\in
T_{P_{V_{2}}}^{n}}\breve{P}_{V_{2}^{n}}(v_{2}^{n})e^{-\frac{s}{1+s}D_{\frac{1}{1+s}}(P_{YX_{1}|V_{2}}^{n}(\cdot|v_{2}^{n})\|Q_{YX_{1}}^{n})}$
$\displaystyle\leq$ $\displaystyle\nu_{n}(|{\cal Y}||{\cal
X}_{1}|)^{\frac{s}{1+s}}|T_{n}({\cal Y}\times{\cal
X}_{1})|\max_{Q_{Y,X_{1}}}\sum_{v_{2}^{n}\in
T_{P_{V_{2}}}^{n}}\breve{P}_{V_{2}^{n}}(v_{2}^{n})e^{-\frac{s}{1+s}D_{\frac{1}{1+s}}(P_{YX_{1}|V_{2}}^{n}(\cdot|v_{2}^{n})\|Q_{YX_{1}}^{n})}$
$\displaystyle\stackrel{{\scriptstyle(c)}}{{=}}$ $\displaystyle|T_{n}({\cal
Y}\times{\cal X}_{1})|\nu_{n}(|{\cal Y}||{\cal
X}_{1}|)^{\frac{s}{1+s}}\max_{Q_{YX_{1}}}e^{-\frac{s}{1+s}n\sum_{v_{2}}P_{V_{2}}(v_{2})D_{\frac{1}{1+s}}(P_{YX_{1}|V_{2}=v_{2}}\|Q_{YX_{1}})}$
$\displaystyle=$ $\displaystyle|T_{n}({\cal Y}\times{\cal
X}_{1})|\nu_{n}(|{\cal Y}||{\cal
X}_{1}|)^{\frac{s}{1+s}}e^{-\frac{s}{1+s}\breve{I}_{\frac{1}{1+s}}^{\uparrow}(YX_{1};V_{2})}$
$\displaystyle=$ $\displaystyle|T_{n}({\cal Y}\times{\cal
X}_{1})|\nu_{n}(|{\cal Y}||{\cal
X}_{1}|)^{\frac{s}{1+s}}e^{-\frac{s}{1+s}\breve{I}_{\frac{1}{1+s}}^{\uparrow}(Y;V_{2}|X_{1})},$
(189)

where steps $(a)$ and $(b)$ follow from (188) and (88), respectively. Also,
$(c)$ follows from the equations

$\displaystyle
D_{\frac{1}{1+s}}(P_{YX_{1}|V_{2}}^{n}(\cdot|v_{2}^{n})\|Q_{YX_{1}}^{n})=\sum_{j=1}^{n}D_{\frac{1}{1+s}}(P_{YX_{1}|V_{2}=v_{2,j}}\|Q_{YX_{1}})=n\sum_{v_{2}}P_{V_{2}}(v_{2})D_{\frac{1}{1+s}}(P_{YX_{1}|V_{2}=v_{2}}\|Q_{YX_{1}}),$
(190)

which follow from the fact that $v_{2}^{n}=(v_{2,1},\ldots,v_{2,n})\in
T_{P_{V_{2}}}^{n}$.

In the same way as (178), we have

$\displaystyle
P_{Y|X_{1}X_{2}}^{n}\cdot\breve{P}_{X_{2}^{n}|V_{2}^{n}}\times\breve{P}_{X_{1}^{n}}\leq$
$\displaystyle\nu_{n}(|{\cal X}_{1}|)\nu_{n}(|{\cal
X}_{2}|)P_{Y|X_{1}X_{2}}^{n}\cdot P_{X_{2}|V_{2}}^{n}\times P_{X_{1}}^{n}$
$\displaystyle=$ $\displaystyle\nu_{n}(|{\cal X}_{1}|)\nu_{n}(|{\cal
X}_{2}|)P_{YX_{1}|V_{2}}^{n}.$ (191)

Therefore, we have

$\displaystyle\mathbb{E}_{C^{\prime}}\left[e_{C^{\prime}}\right]\stackrel{{\scriptstyle(a)}}{{\leq}}$
$\displaystyle(\mathsf{N}-1)^{s}e^{-sI_{\frac{1}{1+s}}^{\uparrow}(Y^{n}X_{1}^{n};V_{2}^{n})[P_{Y|X_{1}X_{2}}^{n}\cdot\breve{P}_{X_{2}^{n}|V_{2}^{n}}\times\breve{P}_{X_{1}^{n}}\times\breve{P}_{V_{2}^{n}}]}$
$\displaystyle\stackrel{{\scriptstyle(b)}}{{\leq}}$
$\displaystyle(\mathsf{N}-1)^{s}\nu_{n}(|{\cal X}_{1}|)\nu_{n}(|{\cal
X}_{2}|)e^{-sI_{\frac{1}{1+s}}^{\uparrow}(Y^{n}X_{1}^{n};V_{2}^{n})[P_{YX_{1}|V_{2}}^{n}\times\breve{P}_{V_{2}^{n}}]}$
$\displaystyle\stackrel{{\scriptstyle(c)}}{{\leq}}$ $\displaystyle|T_{n}({\cal
Y}\times{\cal X}_{1})|^{1+s}\nu_{n}(|{\cal Y}||{\cal
X}_{1}|)^{s}\nu_{n}(|{\cal X}_{1}|)\nu_{n}(|{\cal
X}_{2}|)\mathsf{N}^{s}e^{-s\breve{I}_{\frac{1}{1+s}}^{\uparrow}(Y;V_{2}|X_{1})},$
(192)

where steps $(a)$, $(b)$, and $(c)$ follow from (60) of Lemma 3, the
combination of (57) and (191), and the combination of (187) and (189),
respectively. Therefore, we obtain (90).

###### Remark 4

When the conditional distribution $P_{X_{i}|V_{i}}$ is deterministic, (191) is
simplified as

$\displaystyle
P_{Y|X_{1}X_{2}}^{n}\cdot\breve{P}_{X_{2}^{n}|V_{2}^{n}}\times\breve{P}_{X_{1}^{n}}\leq$
$\displaystyle\nu_{n}(|{\cal X}_{1}|)P_{YX_{1}|V_{2}}^{n}.$ (193)

Thus, (90) is simplified as

$\displaystyle\mathbb{E}_{C^{\prime}}\left[e_{C^{\prime}}\right]\leq|T_{n}({\cal
Y}\times{\cal X}_{1})|^{1+s}\nu_{n}(|{\cal Y}||{\cal
X}_{1}|)^{s}\nu_{n}(|{\cal
X}_{1}|)\mathsf{N}^{s}e^{-s\breve{I}_{\frac{1}{1+s}}^{\uparrow}(Y;V_{2}|X_{1})}.$
(194)

##  Appendix F Proof of Theorem 3

Since $H(Y_{2}|V_{1},V_{2})\geq
H(Y_{2}|V_{1},V_{2},X_{1},X_{2})=H(Y_{2}|X_{1},X_{2})=H(N_{2})$, we have

$\displaystyle I(Y_{2};V_{1}|V_{2})-I(Z;V_{1})\leq
I(Y_{2};V_{1}|V_{2})\leq\log q-H(N_{2}).$ (195)

In the same way, we have

$\displaystyle I(Y_{1};V_{2}|V_{1})-I(Z;V_{2})\leq\log q-H(N_{1}).$ (196)

We assume that there exists an independent random variable $N_{1}^{\prime}$
from $N_{1}$ such that $Z=a_{3}a_{1}^{-1}Y_{1}+N_{1}^{\prime}$. Then, we have

$\displaystyle I(Y_{2};V_{1}|V_{2})+I(Y_{1};V_{2}|V_{1})-I(V_{1},V_{2};Z)$
$\displaystyle=$ $\displaystyle
I(Y_{2};V_{1}|V_{2})+I(V_{1},V_{2};Y_{1})-I(Y_{1};V_{1})-I(V_{1},V_{2};Z)$
$\displaystyle\stackrel{{\scriptstyle(a)}}{{=}}$ $\displaystyle
I(Y_{2};V_{1}|V_{2})+I(Y_{1};V_{1},V_{2}|Z)-I(Y_{1};V_{1})$
$\displaystyle\stackrel{{\scriptstyle(b)}}{{\leq}}$ $\displaystyle
I(Y_{2};V_{1}|V_{2})+I(Y_{1};X_{1},X_{2}|Z)-I(Y_{1};V_{1})$ $\displaystyle=$
$\displaystyle
H(Y_{2}|V_{2})-H(Y_{2}|V_{1},V_{2})+I(Y_{1};X_{1},X_{2}|Z)-I(Y_{1};V_{1})$
$\displaystyle\stackrel{{\scriptstyle(c)}}{{\leq}}$ $\displaystyle
H(Y_{2}|V_{2})-H(Y_{2}|X_{1},X_{2})+I(Y_{1};X_{1},X_{2}|Z)-I(Y_{1};V_{1})$
$\displaystyle\stackrel{{\scriptstyle(d)}}{{=}}$ $\displaystyle
H(Y_{2}|V_{2})-H(Y_{2}|X_{1},X_{2})+I(Y_{1};X_{1},X_{2})-I(Z;X_{1},X_{2})-I(Y_{1};V_{1})$
$\displaystyle=$ $\displaystyle
H(Y_{2}|V_{2})-H(Y_{2}|X_{1},X_{2})+H(Y_{1})-H(Z)-H(Y_{1}|X_{1},X_{2})+H(Z|X_{1},X_{2})-I(Y_{1};V_{1})$
$\displaystyle\stackrel{{\scriptstyle(e)}}{{\leq}}$ $\displaystyle\log
q-H(Y_{2}|X_{1},X_{2})-H(Y_{1}|X_{1},X_{2})+H(Z|X_{1},X_{2})$ $\displaystyle=$
$\displaystyle\log q-H(N_{2})+H(N_{3})-H(N_{1}),$ (197)

where $(a)$, $(b)$ and $(d)$ follows by the Markov chain
$(V_{1},V_{2})-(X_{1},X_{2})-Y_{1}-Z$ which holds since
$Z=a_{3}a_{1}^{-1}Y_{1}+N_{1}^{\prime};$ $(c)$ follows by the Markov chain
$(V_{1},V_{2})-(X_{1},X_{2})-Y_{2};$ and $(e)$ is due to the fact that
$H(Y_{1})\leq H(Z)$ as $Z=a_{3}a_{1}^{-1}Y_{1}+N_{1}^{\prime}.$

Therefore, all conditions in ${\cal R}_{J,N}$ and ${\cal R}_{I,N}$ are upper
bounded by the conditions in the set in the RHS of (114) and (118). Hence, the
equalities in (114) and (118) hold.

##  Appendix G Proof of Theorem 4

To prove Theorem 4, we need to show the validity of (131) and (132), the
proofs of which are given as follows.

###  G-A Proof of (131)

First we show that $h(q)-H(N_{2})$ serves as an outer bound on $R_{1}$ even
with adaptive codes with the 1st type decoder. Fano’s inequality implies

$\displaystyle nR_{1}-I(M_{1};M_{2},X_{2}^{n})=$ $\displaystyle
H(M_{1})-I(M_{1};M_{2},X_{2}^{n})$ $\displaystyle=$ $\displaystyle
H(M_{1}|M_{2},X_{2}^{n})$ $\displaystyle=$ $\displaystyle
H(M_{1}|M_{2},X_{2}^{n},Y_{2}^{n})+I(M_{1};Y_{2}^{n}|M_{2},X_{2}^{n})$
$\displaystyle\leq$ $\displaystyle\log 2+\epsilon\log
M_{1}+I(M_{1};Y_{2}^{n}|M_{2},X_{2}^{n}).$ (198)

Thus, we have

$\displaystyle nR_{1}$ $\displaystyle\leq\log 2+\epsilon\log
M_{1}+I(M_{1};Y_{2}^{n}|M_{2},X_{2}^{n})+I(M_{1};M_{2},X_{2}^{n})$
$\displaystyle=\log 2+\epsilon\log M_{1}+I(M_{1};M_{2},X_{2}^{n},Y_{2}^{n}).$
(199)

Consider

$\displaystyle I(M_{1};M_{2},X_{2}^{n},Y_{2}^{n})=$ $\displaystyle
I(M_{1};X_{2}^{n},Y_{2}^{n}|M_{2})$ $\displaystyle=$ $\displaystyle
I(M_{1};Y_{2}^{n}|M_{2})+I(M_{1};X_{2}^{n}|M_{2},Y_{2}^{n})$
$\displaystyle\stackrel{{\scriptstyle(a)}}{{=}}$ $\displaystyle
I(M_{1};Y_{2}^{n}|M_{2})$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{n}I(M_{1};Y_{2,j}|M_{2},Y_{2}^{j-1})$
$\displaystyle\stackrel{{\scriptstyle(b)}}{{\leq}}$
$\displaystyle\sum_{j=1}^{n}H(Y_{2,j})-H(Y_{2,j}|M_{1},M_{2},X_{1,j},X_{2,j},Y_{2}^{j-1})$
$\displaystyle\stackrel{{\scriptstyle(c)}}{{=}}$
$\displaystyle\sum_{j=1}^{n}H(Y_{2,j})-H(Y_{2,j}|X_{1,j},X_{2,j})$
$\displaystyle\leq$ $\displaystyle n(\log q-H(N_{2})),$ (200)

where $(a)$ follows by the Markov chain $M_{1}-(M_{2},Y_{2}^{n})-X_{2}^{n}$,
which is shown by an inductive use of the Markov chain
$M_{1}-(M_{2},X_{2}^{j-1},Y_{2}^{j-1})-X_{2,j};$ $(b)$ follows by the fact
that conditioning reduces entropy; and $(c)$ follows from the Markov chain
$(M_{1},M_{2},X_{1}^{j-1},X_{2}^{j-1},Y_{1}^{j-1},Y_{2}^{j-1},Z^{j-1})-(X_{1,j},X_{2,j})-(Y_{1,j},Y_{2,j},Z_{j})$.
Combining (199) and (200), we have

$\displaystyle nR_{1}\leq n(\log q-H(N_{2}))+\log 2+\epsilon\log M_{1},$ (201)

which implies $R_{1}\leq\log q-H(N_{2})$.

A similar proof can be applied to show that $\log q-H(N_{1})$ serves as an
outer bound on $R_{2}.$

###  G-B Proof of (132)

Consider the sum-rate $R_{1}+R_{2}$ for non-adaptive codes with the 1st type
decoder under joint secrecy constraint. We have

$\displaystyle n(R_{1}+R_{2})=$ $\displaystyle H(M_{1}|M_{2})+H(M_{2}|M_{1})$
$\displaystyle=$ $\displaystyle
H(M_{1}|X_{2}^{n},Y_{2}^{n},M_{2})+I(M_{1};X_{2}^{n},Y_{2}^{n}|M_{2})+H(M_{2}|X_{1}^{n},Y_{1}^{n},M_{1})+I(M_{2};X_{1}^{n},Y_{1}^{n}|M_{1})$
$\displaystyle=$ $\displaystyle
H(M_{1}|X_{2}^{n},Y_{2}^{n},M_{2})+I(M_{1};Y_{2}^{n}|X_{2}^{n},M_{2})+I(M_{1};X_{2}^{n}|M_{2})$
$\displaystyle+H(M_{2}|X_{1}^{n},Y_{1}^{n},M_{1})+I(M_{2};Y_{1}^{n}|X_{1}^{n},M_{1})+I(M_{2};X_{1}^{n}|M_{1})$
$\displaystyle\stackrel{{\scriptstyle(a)}}{{=}}$ $\displaystyle
H(M_{1}|X_{2}^{n},Y_{2}^{n},M_{2})+I(M_{1};Y_{2}^{n}|X_{2}^{n},M_{2})+H(M_{2}|X_{1}^{n},Y_{1}^{n},M_{1})+I(M_{2};Y_{1}^{n}|X_{1}^{n},M_{1})$
$\displaystyle\stackrel{{\scriptstyle(b)}}{{\leq}}$ $\displaystyle
I(M_{1};Y_{2}^{n}|X_{2}^{n},M_{2})+I(M_{2};Y_{1}^{n}|X_{1}^{n},M_{1})+2n\mathcal{O}(\epsilon_{n})$
$\displaystyle\stackrel{{\scriptstyle(c)}}{{=}}$ $\displaystyle
I(M_{1};Y_{2}^{n}|X_{2}^{n})+I(M_{2};Y_{1}^{n}|X_{1}^{n})+2n\mathcal{O}(\epsilon_{n})$
$\displaystyle\stackrel{{\scriptstyle(d)}}{{\leq}}$ $\displaystyle
I(M_{1};Y_{2}^{n}|X_{2}^{n})+I(M_{2};Y_{1}^{n}|X_{1}^{n})-I(M_{1},M_{2};Z^{n})+2n\mathcal{O}(\epsilon_{n})+n\tau_{n},$
(202)

where $(a)$ from the fact that for non-adaptive codes, $M_{1}$ and $X_{1}^{n}$
are independent of $M_{2}$ and $X_{2}^{n};$ $(b)$ follows from Fano’s
inequality; $(c)$ follows from the Markovian chains
$M_{2}-X_{2}^{n}-(M_{1},Y_{2}^{n})$ and $M_{1}-X_{1}^{n}-(M_{2},Y_{1}^{n})$;
$(d)$ follows from the joint secrecy constraint as defined in (9).

Since
$N_{3}=\frac{b_{3}}{b_{1}}{N}_{1}+\frac{a_{3}}{a_{2}}{N}_{2}+N_{3}^{\prime}$,
we have
$a_{3}{X}_{1}^{n}+b_{3}{X}_{2}^{n}+{N}_{3}^{n}=b_{3}{X}_{2}^{n}+\frac{b_{3}}{b_{1}}{N}_{1}^{n}+(a_{3}{X}_{1}^{n}+\frac{a_{3}}{a_{2}}{N}_{2}^{n}+N_{3}^{\prime})$.
Hence, we have $H(a_{3}{X}_{1}^{n}+b_{3}{X}_{2}^{n}+{N}_{3}^{n})\geq
H(b_{3}{X}_{2}^{n}+\frac{b_{3}}{b_{1}}{N}_{1}^{n})$. Thus,

$\displaystyle
H\Big{(}a_{3}{X}_{1}^{n}+\frac{a_{3}}{a_{2}}{N}_{2}^{n}\Big{)}+H\Big{(}b_{3}{X}_{2}^{n}+\frac{b_{3}}{b_{1}}{N}_{1}^{n}\Big{)}-H(a_{3}{X}_{1}^{n}+b_{3}{X}_{2}^{n}+{N}_{3}^{n})\leq
n\log q.$ (203)

Thus, using Lemma 8 in Appendix H, we have

$\displaystyle
I(M_{1};Y_{2}^{n}|X_{2}^{n})+I(M_{2};Y_{1}^{n}|X_{1}^{n})-I(M_{1},M_{2};Z^{n})$
$\displaystyle\leq$ $\displaystyle
H\Big{(}a_{3}{X}_{1}^{n}+\frac{a_{3}}{a_{2}}{N}_{2}^{n}\Big{)}-H\Big{(}\frac{a_{3}}{a_{2}}{N}_{2}^{n}\Big{)}+H\Big{(}b_{3}{X}_{2}^{n}+\frac{b_{3}}{b_{1}}{N}_{1}^{n}\Big{)}-H\Big{(}\frac{b_{3}}{b_{1}}{N}_{1}^{n}\Big{)}-H(a_{3}{X}_{1}^{n}+b_{3}{X}_{2}^{n}+{N}_{3}^{n})+H({N}_{3}^{n})$
$\displaystyle\leq$ $\displaystyle n(\log q+H(N_{3})-H(N_{1})-H(N_{2})).$
(204)

The combination of (202) and (204) implies

$\displaystyle R_{1}+R_{2}\leq\log q+H(N_{3})-H(N_{1})-H(N_{2}).$ (205)

##  Appendix H Useful lemmas for additive channels

First we prepare a useful lemma for additive channels. Assume that ${\cal
X}_{1}$, ${\cal X}_{2}$, ${\cal Y}_{1}$, ${\cal Y}_{2}$, and ${\cal Z}$ are
over a general field $\mathbb{K}$, which is not necessarily a finite field. We
consider the additive channels given in (104) – (106) or (135) – (137) over
the field $\mathbb{K}$, where all coefficients are elements of $\mathbb{K}$.
Then, we have the following lemma.

###### Lemma 8

When there exists independent random variable $N_{3}^{\prime}$ from $N_{1}$
and $N_{2}$ such that
$N_{3}=\frac{b_{3}}{b_{1}}{N}_{1}+\frac{a_{3}}{a_{2}}{N}_{2}+N_{3}^{\prime}$,
we have

$\displaystyle
I(M_{1};Y_{2}^{n}|X_{2}^{n})+I(M_{2};Y_{1}^{n}|X_{1}^{n})-I(M_{1},M_{2};Z^{n})$
$\displaystyle\leq$ $\displaystyle
H\Big{(}a_{3}X_{1}^{n}+\frac{a_{3}}{a_{2}}N_{2}^{n}\Big{)}-H\Big{(}\frac{a_{3}}{a_{2}}N_{2}^{n}\Big{)}+H\Big{(}b_{3}X_{2}^{n}+\frac{b_{3}}{b_{1}}N_{1}^{n}\Big{)}-H\Big{(}\frac{b_{3}}{b_{1}}N_{1}^{n}\Big{)}-H(a_{3}X_{1}^{n}+b_{3}X_{2}^{n}+N_{3}^{n})+H(N_{3}^{n}).$

Proof of Lemma 8:  We have

$\displaystyle
I(M_{1};Y_{2}^{n}|X_{2}^{n})+I(M_{2};Y_{1}^{n}|X_{1}^{n})-I(M_{1},M_{2};Z^{n})$
$\displaystyle=$ $\displaystyle
H(a_{2}X_{1}^{n}+N_{2}^{n})-H(a_{2}X_{1}^{n}+N_{2}^{n}|M_{1})+H(b_{1}X_{2}^{n}+N_{1}^{n})-H(b_{1}X_{2}^{n}+N_{1}^{n}|M_{2})$
$\displaystyle-(H(a_{3}X_{1}^{n}+b_{2}X_{2}^{n}+N_{3}^{n})-H(a_{3}X_{1}^{n}+b_{3}X_{2}^{n}+N_{3}^{n}|M_{1},M_{2}))$
$\displaystyle\stackrel{{\scriptstyle(a)}}{{\leq}}$ $\displaystyle
H(a_{2}X_{1}^{n}+N_{2}^{n})-H(a_{2}X_{1}^{n}+N_{2}^{n}|M_{1}M_{2})+H(b_{1}X_{2}^{n}+N_{1}^{n})-H(b_{1}X_{2}^{n}+N_{1}^{n}|M_{1},M_{2})$
$\displaystyle-(H(a_{3}X_{1}^{n}+b_{3}X_{2}^{n}+N_{3}^{n})-H(a_{3}X_{1}^{n}+b_{3}X_{2}^{n}+N_{3}^{n}|M_{1},M_{2}))$
$\displaystyle=$ $\displaystyle
H\Big{(}a_{3}X_{1}^{n}+\frac{a_{3}}{a_{2}}N_{2}^{n}\Big{)}-H\Big{(}\frac{a_{3}}{a_{2}}N_{2}^{n}\Big{)}+H\Big{(}b_{3}X_{2}^{n}+\frac{b_{3}}{b_{1}}N_{1}^{n}\Big{)}-H\Big{(}\frac{b_{3}}{b_{1}}N_{1}^{n}\Big{)}-H(a_{3}X_{1}^{n}+b_{3}X_{2}^{n}+N_{3}^{n})+H(N_{3}^{n})$
$\displaystyle-\Big{(}H\Big{(}a_{3}X_{1}^{n}+\frac{a_{3}}{a_{2}}N_{2}^{n}\Big{|}M_{1},M_{2}\Big{)}-H\Big{(}\frac{a_{3}}{a_{2}}N_{2}^{n}\Big{)}+H\Big{(}b_{3}X_{2}^{n}+\frac{b_{3}}{b_{1}}N_{1}^{n}\Big{|}M_{1},M_{2}\Big{)}-H\Big{(}\frac{b_{3}}{b_{1}}N_{1}^{n}\Big{)}$
$\displaystyle-H(a_{3}X_{1}^{n}+b_{3}X_{2}^{n}+N_{3}^{n}|M_{1},M_{2})+H(N_{3}^{n})\Big{)}$
$\displaystyle\stackrel{{\scriptstyle(b)}}{{\leq}}$ $\displaystyle
H\Big{(}a_{3}X_{1}^{n}+\frac{a_{3}}{a_{2}}N_{2}^{n}\Big{)}-H\Big{(}\frac{a_{3}}{a_{2}}N_{2}^{n}\Big{)}+H\Big{(}b_{3}X_{2}^{n}+\frac{b_{3}}{b_{1}}N_{1}^{n}\Big{)}-H\Big{(}\frac{b_{3}}{b_{1}}N_{1}^{n}\Big{)}-H(a_{3}X_{1}^{n}+b_{3}X_{2}^{n}+N_{3}^{n})+H(N_{3}^{n}),$

where $(a)$ follows from the fact that conditioning reduces entropy; $(b)$
follows from the proof below.

$\displaystyle
H\Big{(}a_{3}X_{1}^{n}+\frac{a_{3}}{a_{2}}N_{2}^{n}\Big{|}M_{1},M_{2}\Big{)}-H\Big{(}\frac{a_{3}}{a_{2}}N_{2}^{n}\Big{)}+H\Big{(}b_{3}X_{2}^{n}+\frac{b_{3}}{b_{1}}N_{1}^{n}\Big{|}M_{1},M_{2}\Big{)}-H\Big{(}\frac{b_{3}}{b_{1}}N_{1}^{n}\Big{)}$
$\displaystyle-H(a_{3}X_{1}^{n}+b_{3}X_{2}^{n}+N_{3}^{n}|M_{1},M_{2})+H(N_{3}^{n})$
$\displaystyle=$ $\displaystyle
I\Big{(}a_{3}X_{1}^{n}+\frac{a_{3}}{a_{2}}N_{2}^{n};a_{3}X_{1}^{n}\Big{|}M_{1},M_{2}\Big{)}+I\Big{(}b_{3}X_{2}^{n}+\frac{b_{3}}{b_{1}}N_{1}^{n};b_{3}X_{2}^{n}\Big{|}M_{1},M_{2}\Big{)}$
$\displaystyle-I\Big{(}a_{3}X_{1}^{n}+b_{3}X_{2}^{n}+N_{3}^{n};a_{3}X_{1}^{n}+b_{3}X_{2}^{n}\Big{|}M_{1},M_{2}\Big{)}$
$\displaystyle\stackrel{{\scriptstyle(c)}}{{=}}$ $\displaystyle
I\Big{(}a_{3}X_{1}^{n}+\frac{a_{3}}{a_{2}}N_{2}^{n},b_{3}X_{2}^{n}+\frac{b_{3}}{b_{1}}N_{1}^{n};a_{3}X_{1}^{n},b_{3}X_{2}^{n}\Big{|}M_{1},M_{2}\Big{)}-I\Big{(}a_{3}X_{1}^{n}+b_{3}X_{2}^{n}+N_{3}^{n};a_{3}X_{1}^{n}+b_{3}X_{2}^{n}\Big{|}M_{1},M_{2}\Big{)}$
$\displaystyle\stackrel{{\scriptstyle(d)}}{{\geq}}$ $\displaystyle
I\Big{(}a_{3}X_{1}^{n}+b_{3}X_{2}^{n}+\frac{b_{3}}{b_{1}}N_{1}^{n}+\frac{a_{3}}{a_{2}}N_{2}^{n};a_{3}X_{1}^{n}+b_{3}X_{2}^{n}\Big{|}M_{1},M_{2}\Big{)}-I\Big{(}a_{3}X_{1}^{n}+b_{3}X_{2}^{n}+N_{3}^{n};a_{3}X_{1}^{n}+b_{3}X_{2}^{n}\Big{|}M_{1},M_{2}\Big{)}$
$\displaystyle\stackrel{{\scriptstyle(e)}}{{\geq}}$ $\displaystyle 0,$

where $(c)$ holds because $X_{1}^{n}$ and $X_{2}^{n}$ are independent and they
are also independent of $N_{1}^{n},N_{2}^{n}$; $(d)$ follows from the non-
negativity of the mutual information; and $(e)$ follows from the assumption
$N_{3}=\frac{b_{3}}{b_{1}}{N}_{1}+\frac{a_{3}}{a_{2}}{N}_{2}+N_{3}^{\prime}$.

Moreover, for the additive Gaussian TWC as defined in (135) – (137). We have
the following lemma:

###### Lemma 9

Assume the condition

$\displaystyle\mathbb{E}[\|X_{1}^{n}\|^{2}]\leq
nc_{1},\quad\mathbb{E}[\|X_{2}^{n}\|^{2}]\leq nc_{2}.$ (206)

For non-adaptive codes, when
$\frac{b_{3}^{2}}{b_{1}^{2}}v_{1}+\frac{a_{3}^{2}}{a_{2}^{2}}v_{2}\leq v_{3}$,
we have

$\displaystyle
H\Big{(}a_{3}X_{1}^{n}+\frac{a_{3}}{a_{2}}N_{2}^{n}\Big{)}+H\Big{(}b_{3}X_{2}^{n}+\frac{b_{3}}{b_{1}}N_{1}^{n}\Big{)}-H(a_{3}X_{1}^{n}+b_{3}X_{2}^{n}+N_{3}^{n})$
$\displaystyle\leq$ $\displaystyle\frac{n}{2}(\log 2\pi
e(b_{3}^{2}c_{2}+\frac{b_{3}^{2}}{b_{1}^{2}}v_{1})+\log 2\pi
e(a_{3}^{2}c_{1}+\frac{a_{3}^{2}}{a_{2}^{2}}v_{2})-\log 2\pi
e(a_{3}^{2}c_{1}+b_{3}^{2}c_{2}+v_{3})).$ (207)

Proof of Lemma 9:  We define the functions

$\displaystyle g(x):=$ $\displaystyle\frac{n}{2}\log(2\pi e\frac{x}{n})$ (208)
$\displaystyle h(y,z):=$ $\displaystyle g(z+g^{-1}(y))-y,$ (209)

where $g^{-1}$ is the inverse function of $g$. Hence, we have

$\displaystyle g^{-1}(y)=\frac{ne^{2\frac{y}{n}}}{2\pi e}.$ (210)

Under the average power constraints, the fact that $N_{1}^{n},N_{2}^{n}$ are
independent of $X_{1}^{n},X_{2}^{n}$, and the maximum differential entropy
lemma, we have

$\displaystyle H(b_{3}X_{2}^{n}+\frac{b_{3}}{b_{1}}N_{1}^{n})$
$\displaystyle\leq g(nb_{3}^{2}c_{2}+\frac{b_{3}^{2}}{b_{1}^{2}}nv_{1})$ (211)
$\displaystyle H(a_{3}X_{1}^{n}+\frac{a_{3}}{a_{2}}N_{2}^{n})$
$\displaystyle\leq g(na_{3}^{2}c_{1}+\frac{a_{3}^{2}}{a_{2}^{2}}nv_{2}).$
(212)

As a simple extension of [23, Lemma 8], we have the following lemma.

###### Lemma 10

For a fixed $z$, $h(y,z)$ is monotonically decreasing for $y$.

######  Lemma 11 ([24, Eq. (5)])

When $X_{1}^{n}$ and $X_{2}^{n}$ are independent $n$-dimensional vectors

$\displaystyle e^{\frac{2}{n}H(X_{1}^{n}+X_{2}^{n})}\geq
e^{\frac{2}{n}H(X_{1}^{n})}+e^{\frac{2}{n}H(X_{2}^{n})}.$ (213)

The equality holds when $X_{1}^{n}$ and $X_{2}^{n}$ are Gaussian random
variables with proportional covariance matrices.

We define a Gaussian random variable $N_{3}^{\prime}$ with mean 0 and variance
$v_{3}^{\prime}:=v_{3}-\frac{b_{3}^{2}}{b_{1}^{2}}v_{1}-\frac{a_{3}^{2}}{a_{2}^{2}}v_{2}$
that is independent of other random variables $X_{1},X_{2},N_{1},N_{2},N_{3}$.
Since $X_{1}^{n}$ and $X_{2}^{n}$ are independent of each other, using Lemma
11, we have

$\displaystyle H(a_{3}X_{1}^{n}+b_{3}X_{2}^{n}+N_{3})=$ $\displaystyle
H(a_{3}X_{1}^{n}+b_{3}X_{2}^{n}+\frac{b_{3}}{b_{1}}N_{1}^{n}+\frac{a_{3}}{a_{2}}N_{2}^{n}+N_{3}^{\prime})$
$\displaystyle\geq$
$\displaystyle\frac{n}{2}\log(e^{\frac{2}{n}H(b_{3}X_{2}^{n}+\frac{b_{3}}{b_{1}}N_{1}^{n})}+e^{\frac{2}{n}H(a_{3}X_{1}^{n}+\frac{a_{3}}{a_{2}}N_{2}^{n}+N_{3}^{\prime})})$
$\displaystyle\geq$
$\displaystyle\frac{1}{2}\log(e^{\frac{2}{n}H(b_{3}X_{2}^{n}+\frac{b_{3}}{b_{1}}N_{1}^{n})}+e^{\frac{2}{n}H(a_{3}X_{1}^{n}+\frac{a_{3}}{a_{2}}N_{2}^{n})}+e^{\frac{2}{n}H(N_{3}^{\prime})})$
$\displaystyle=$