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arXiv:0809.4804v1 [cs.IT] 27 Sep 2008

On the Secure Degrees of Freedom of Wireless X

Networks

Tiangao Gou, Syed A. Jafar

Electrical Engineering and Computer Science

University of California Irvine, Irvine, California, 92697

Email: {tgou,syed}@uci.edu

Abstract—Previous work showed that the X network with M

transmitters, N receivers has

work we study the degrees of freedom of the X network with

secrecy constraints, i.e. the X network where some/all messages

are confidential. We consider the M × N network where all

messages are secured and show that

can be achieved. Secondly, we show that if messages from only

M − 1 transmitters are confidential, then

freedom can be achieved meaning that there is no loss of degrees

of freedom because of secrecy constraints. We also consider the

achievable secure degrees of freedom under a more conservative

secrecy constraint. We require that messages from any subset

of transmitters are secure even if other transmitters are com-

promised, i.e., messages from the compromised transmitter are

revealed to the unintended receivers. We also study the achievable

secure degrees of freedom of the K user Gaussian interference

channel under two different secrecy constraints where

degrees of freedom per message can be achieved. The achievable

scheme in all cases is based on random binning combined with

interference alignment.

MN

M+N−1degrees of freedom. In this

N(M−1)

M+N−1degrees of freedom

MN

M+N−1degrees of

1

2secure

I. INTRODUCTION

Security is an important issue if the transmitted informa-

tion is confidential. Researchers have studied the information

theoretic secrecy for different channel models. In [1], Wyner

first proposed the wiretap channel model to characterize single

user secure communication problem, i.e., a sender transmits

a confidential message to its receiver while keeping a wire-

tapper totally ignorant of the message. The secrecy level is

measured by the equivocation rate, i.e., the entropy rate of

the confidential message conditioned on the received signal

at the wire-tapper. More recent information-theoretic research

on secure communication focuses on multi-user scenarios. In

[2], the authors study the compound wire-tap channel where

the sender multicasts its messages to multiple receivers while

ensuring the confidentiality of the messages at multiple wire-

tappers. Multiple access channel with confidential messages

has been studied in [3]–[5]. Broadcast channel with confiden-

tial messages has been studied in [6], [7]. The two user discrete

memoryless interference channel with confidential messages is

studied in [7].

It is well known that the secrecy capacity of the Gaussian

wiretap channel is the difference between the capacities of

the main and the wiretap channels [8]. In other words, there

The work of S. Jafar was supported by ONR Young Investigator Award

N00014-08-1-0872.

is a rate penalty for ensuring the secrecy. From the degrees

of freedom perspective, this is pessimistic since the channel

loses all its degrees of freedom. Further, even the two user

Gaussian interference channel loses all its degrees of freedom

if we have to ensure that messages from both transmitters are

confidential, i.e., a message should remain secure from the

undesired receiver. The results of the Gaussian wiretap channel

and the 2 user Gaussian interference channel prompt one to ask

whether it is possible for a network to have positive number of

degrees of freedom if the messages in the network are secure.

The answer to this question lies in the study of the K user

Gaussian interference channel with secure messages [9] which

indeed has positive number of degrees of freedom if K > 2.

It is shown that the network has

freedom. The key to increase the secure degrees of freedom is

interference alignment. Interference signals associated with the

messages needed to be secured are aligned to occupy smaller

dimension so that the secrecy penalty rate is minimized. At

the same time, the degrees of the freedom for the legitimate

channel is maximized by interference alignment. Thus, the

tool of interference alignment serves the dual purpose of

minimizing the secrecy penalty rate and maximizing the rate

of the legitimate messages, thus improving the secure degrees

of freedom of the network.

In this paper, we generalize the result of [9] to the X

network. We study the achievable secure degrees of freedom

of the M×N user wireless X network, i.e., a network with M

transmitters and N receivers where independent confidential

messages need to be conveyed from each transmitter to each

receiver. X networks are interesting since they encompasses

different communication scenarios. For example, each trans-

mitter is associated with a broadcast channel, each receiver is

associated with a multiple access channel and every pair of

transmitters and receivers comprises an interference channel.

In other words, broadcast channel, multiple access channel

and interference channel are special cases of X networks.

In addition, interference alignment is also feasible on X net-

works. In [11], interference alignment schemes are constructed

to achieve

slot for each message without secrecy constraint. In this

paper, we exploit alignment of interference to assist secrecy

in the network. We study the achievable secure degrees of

freedom under four different secrecy constraints. We show

K(K−2)

2K−2

secure degrees of

1

M+N−1degrees of freedom per frequency/time

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that if the set of all unintended messages is secured at each

receiver, then each message can achieveM−1

degrees of freedom for a total of

of freedom. In other words, only a fraction

freedom is lost under this secrecy constraint. Interestingly,

if we only secure the set of unintended messages from any

M − 1 transmitters at each receiver, then each message can

achieve

as what one can achieve without secrecy constraint. This

corresponds to a scenario where one transmitter’s messages

need not be secure, perhaps because their confidentiality is

ensured cryptographically, by some higher layer. In this case,

the other messages increase their degrees of freedom by

exploiting this. Next, we consider a more conservative secrecy

constraint. Transmitters do not trust each other, so we require

that even if any subset of transmitters S is compromised, i.e.,

the messages from the compromised transmitter are revealed

to the unintended receivers (through a genie), the remaining

transmitters’ messages are still secure. For this case, we show

that if the set of all unintended messages is secured then

N(M−|S|−1)

M+N−1

secure degrees of freedom can be achieved for

the remaining (M −|S|)×N users. If we only need to secure

the set of unintended messages from M −|S|−1 transmitters,

then

each message. The achievable scheme for all cases is based

on random binning combined with interference alignment.

M

1

M+N−1secure

N(M−1)

M+N−1secure degrees

1

Mdegrees of

1

M+N−1secure degrees of freedom which is the same

1

M+N−1secure degrees of freedom can be achieved for

II. SYSTEM MODEL AND SECRECY CONSTRAINTS

A. System Model

The M×N user X network is comprised of M transmitters

and N receivers. Each transmitter has an independent message

for each receiver. The channel output at the jthreceiver over

the fthfrequency slot and the tthtime slot is described as

follows:

Yj(f,t) =

M

?

i=1

Hji(f)Xi(f,t) + Zj(f,t), j = 1,2,...,N

where Xi(f,t) is the input signal at Transmitter i, Hji(f) is

the channel coefficient from Transmitter i to Receiver j and

Zj(f,t) represents the additive white Gaussian noise (AWGN)

at Receiver j. We assume the channel coefficients vary across

frequencyslots but remain constant in time and are drawn from

a continuous distribution. We assume all channel coefficients

are known to all transmitters and receivers. Using the symbol

extension channel in [11], the input-output relationship is

characterized as follows:

¯Yj(t) =

M

?

i=1

¯Hji¯Xi(t) +¯Zj(t)

(1)

where¯Xi(t) is the F × 1 column vector representing the F

symbol extension of the transmitted symbol Xi, i.e.,¯Xi(t) =

[Xi(1,t) Xi(2,t) ··· Xi(F,t)]T. Similarly,¯Yj(t) and¯Zj(t)

represent the symbol extension of Yjand Zj, respectively.¯Hji

is the F ×F diagonal matrix representing the extension of the

channel, i.e.,

¯ Hji=

Hji(1)

0

...

0

0

···

···

...

···

0

0

...

Hji(2)

...

0

Hji(F)

Transmitter i has message Wji

Receiver j, for each i ∈ {1,2,...,M}, j ∈ {1,2,...,N},

resulting in a total of MN independent messages. An

(M11,...,MNM,n,F,Pe) code for the X channel consists

of the following:

∈{1,2,...,Mji} for

• MN independent message sets: Wji= {1,2,...,Mji}

• M encoding functions, fi: W1i× W2i× ··· × WNi→

¯Xn

i, where¯Xn

the message tuple (w1i,w2i,··· ,wNi) ∈ W1i× W2i×

··· × WNito transmitted symbols. Each transmitter has

a power constraint, i.e.

i= [¯Xi(1)¯Xi(2) ···¯Xi(n)], which map

1

nF

F

?

f=1

n

?

t=1

|Xi(t,f)|2≤ P, i ∈ {1,2,...,M}.

• N decoding functions, gj:¯Yn

WjM, where¯Yn

map the received sequence¯Yn

tuple ( ˆ wj1, ˆ wj2,··· , ˆ wjM) ∈ Wj1× Wj2× ··· × WjM.

The maximal average probability of error Pe

(M11,...,MNM,n,F,Pe) code is defined as

j→ Wj1× Wj2× ··· ×

j= [¯Yj(1)¯Yj(2) ··· ¯Yj(n)], which

jto the decoded message

for an

Pe? max{Pe,11,Pe,21,··· ,Pe,NM}

where

Pe,ji=

1

Mji

?

wji∈Wji

P{g(¯Yn

j) ?= wji|wjisent}

We use the equivocation rate

measure.

A rate tuple (R11,R21,...,RNM) is said to be achievable

for the M × N user X network with confidential messages

if for any ǫ > 0, there exists an (M11,...,MNM,n,F,Pe)

code such that

1

nFlog2(Mji) ≥ Rji

and the reliability requirement

1

nFH(W|¯Yn

j) as the secrecy

Pe≤ ǫ

and the security constraints which will be defined shortly are

satisfied. The secure degrees of freedom tuple (η11,...,ηNM)

is achievable if the rate tuple (R11,...,RNM) is achievable

and

ηji= lim

P→∞

Rji(P)

log(P)

∀(j,i) ∈ J × I,

I = {1,2,...,M}, J = {1,2,...,N}

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B. Secrecy Constraints

We will define four different secrecy constraints as follows:

1) Secrecy Constraint 1: The secrecy constraint is defined

as

1

nFH(W(J−j)×I|¯Yn

j) ≥

?

(r,i)∈(J−j)×I

Rri− ǫ

where

W(J−j)×I? {Wri: ∀(r,i) ∈ (J − j) × I}

This ensures the perfect secrecy of the set of all unintended

messages at each receiver. Moreover, it can be shown that

perfect secrecy for a set of messages guarantees perfect secrecy

for any subset of that message set, i.e.,

1

nFH(WS|¯Yn

j) ≥

?

(r,i)∈S

Rri− ǫ ∀S ⊆ (J − j) × I (2)

To see this, consider

H(W(J−j)×I|¯Yn

j)=

H(WS|¯Yn

j) + H(WSc|WS,¯Yn

j)

(3)

(4)

≤

H(WS|¯Yn

j) + H(WSc)

where Scdenotes the complement of S and (3) follows from

the chain rule and (4) follows from the fact that conditioning

reduces the entropy. If the message set satisfies the secrecy

constraint, i.e.,

H(W(J−j)×I|¯Yn

j) ≥ H(WS) + H(WSc) − ǫ

then from (4) we have

H(WS|¯Yn

j) + H(WSc) ≥ H(WS) + H(WSc) − ǫ

⇒ H(WS|¯Yn

j) ≥ H(WS) − ǫ

Thus, the confidentiality of the subset WSis preserved.

2) Secrecy Constraint 2: Instead of ensuring the confiden-

tiality of the set of unintended messages of all transmitters, we

only secure the set of unintended messages from any M − 1

transmitters. Secrecy constraint 2 is defined as

1

nFH(W(J−j)×(I−l)|¯Yn

j) ≥

?

(r,i)∈(J−j)×(I−l)

Rri−ǫ ∀l ∈ I

where

W(J−j)×(I−l)= {Wri: ∀(r,i) ∈ (J − j) × (I − l)}

Again, the perfect secrecy of a message set guarantees perfect

secrecy for any subset of that message set, i.e.,

1

nFH(WSJ×SI|¯Yn

j) ≥

?

(r,i)∈SJ×SI

Rri− ǫ,

∀SJ⊆ J − j,∀SI⊆ I − l

Note that satisfying secrecy constraint 1 ensures satisfying

secrecy constraint 2.

3) Secrecy Constraint 3: Let us define SI⊂ I to be the set

of transmitters that are compromised, i.e., the messages from

the compromised transmitter are revealed to the unintended

receivers and Sc

We define secrecy constraint 3 as

Ito be the set of the remaining transmitters.

1

nFH(W(J−j)×Sc

I|¯Yn

j,W(J−j)×SI) ≥

?

(r,i)∈(J−j)×Sc

I

∀SI⊂ I

Rri− ǫ

This constraint ensures that secrecy of any subset of transmit-

ters even if all other transmitters are compromised. Also, this

secrecy constraint guarantees that

1

nFH(WSJ×Sc

I|¯Yn

j,W(J−j)×SI) ≥

?

(r,i)∈SJ×Sc

∀SI⊂ I,∀SJ⊆ J − j

I

Rri− ǫ

4) Secrecy Constraint 4: Even if any subset of transmitters

SI ⊂ I is compromised, we require secrecy of the set of

messages from Sc

secrecy constraint 4 as

I− l transmitters for any l ∈ Sc

I. We define

1

nFH(W(J−j)×(Sc

?

(r,i)∈(J−j)×(Sc

I−l)|¯Yn

j,W(J−j)×SI)

≥

I−l)

Rri− ǫ

∀SI⊂ I, ∀l ∈ Sc

I

III. THE M × N USER X NETWORK WITH CONFIDENTIAL

MESSAGES

In this section, we consider the achievable secure degrees

of freedom of the M × N user X channel under different

secrecy constraints. In order to satisfy the secrecy constraints,

we use the random binning coding scheme to generate the

codebook. This is a natural extension of the coding scheme

used in [7] to achieve the inner bound of the capacity region

of the two user discrete memoryless interference channel with

confidential messages. To maximize the achievable degrees of

freedom, we adopt the interference alignment scheme used

in [11]. The main results of this section are presented in the

following theorems:

Theorem 1: For the M × N user X network with single

antenna nodes,

be achieved for each message Wji, ∀j ∈ {1,...,N},∀i ∈

{1,...,M} and hence a total of

freedom can be achieved under secrecy constraint 1.

Proof: We provide a detailed proof in the Appendix. A

sketch of the proof is provided here. Consider the F symbol

extension channel where F = N(m+1)Γ+(M−1)mΓ,∀m ∈

N and Γ = (N − 1)(M − 1). Over the F symbol extension

channel, message Wj1is encoded at Transmitter 1 into m1=

(m+1)Γindependent streams Xj1(t) which is an (m+1)Γ×1

vector and message Wji,i ?= 1 is encoded at Transmitter

i into mi = mΓindependent streams Xji(t) which is an

mΓ×1 vector based on random binning coding scheme. Note

that such coding scheme introduces randomness to ensure the

secrecy. Then transmitter i employs the interference alignment

M−1

M(M+N−1)secure degrees of freedom can

N(M−1)

M+N−1secure degrees of

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scheme mapping Xji(t) to Vji(t)Xji(t) where Vji is the

F × mi matrix. At last, Transmitter i sends signal¯Xi(t) =

?N

matrices Vji(t) are chosen as given in [11] so that at each

receiver, the desired signal vectors span a signal space which

is disjoint with the space spanned by the interference vectors.

Therefore, each receiver can decode its desired data streams

by zero forcing the interference. Note that at Receiver j, the

signal vectors associated with M desired messages Wji,∀i =

1,...,M span a (m+1)Γ+(M−1)mΓdimensional subspace

in the F = N(m + 1)Γ+ (M − 1)mΓdimensional signal

space. Thus, to get an interference-free signal subspace, the

dimension of the subspace spanned by all interference vectors

has to be less than or equal to (N−1)(m+1)Γ. Notice that the

interference vectors from Transmitter 1 span a (N−1)(m+1)Γ

dimensional subspace. Therefore, we can align the interference

vectors from all other transmitters within this subspace so

that each receiver can decode its desired data streams by zero

forcing the interference in this subspace. Next, it can be shown

that the following secrecy rate is achievable:

j=1Vji(t)Xji(t) into the channel. Note that the precoding

Rji

=

1

FI(Xji;¯ Yj)

1

F

−

1

M(N − 1)max

k∈JI(X(J−k)×I;¯Yk|Xk×I)

∀(j,i) ∈ J × I (5)

From [11], we have

I(Xji;¯Yj) = (m + 1)Γlog(P) + o(log(P)) i = 1

and

I(Xji;¯ Yj) = mΓlog(P) + o(log(P)) i = 2,...,M

Next, consider the term I(X(J−k)×I;¯Yk|Xk×I) which de-

notes the secrecy penalty. Notice that all the interference vec-

tors are aligned within the space spanned by (N −1)(m+1)Γ

interference vectors from Transmitter 1. Therefore, the secrecy

penalty is

I(X(J−k)×I;¯ Yk|Xk×I)

(N − 1)(m + 1)Γlog(P) + o(log(P)) ∀k ∈ J

=

Hence, (5) can be written as

Rji=1

F(m + 1)Γ(1 −

1

M)log(P) + o(log(P)) i = 1

and

Rji=1

F(mΓ−(m + 1)Γ

As m → ∞, we have

M

)log(P)+o(log(P)) i = 2,...,M

Rji=

M − 1

M(M + N − 1)log(P)+o(log(P))

∀(j,i) ∈ J ×I

As a result, each message can achieve ηji =

secure degrees of freedom for a total of

degrees of freedom.

M−1

M(M+N−1)

(M−1)N

M+N−1secure

Note that in [11], it is shown that

dom can be achieved for each message Wji without secrecy

constraint. Theorem 1 shows that only a fraction

of freedom is lost under secrecy constraint 1. However, it is

interesting that if we relax the secrecy constraint a little, i.e.,

only ensure the confidentiality of the set of messages from

any M − 1 out of M transmitters at each receiver, there will

be no loss of degrees of freedom. We present the result in the

following theorem:

Theorem 2: For the M × N user X network with single

antenna nodes, each message can achieve

degrees of freedom for a total of

freedom under secrecy constraint 2.

Proof: The proof is similar to the proof of Theorem 1.

We only provide a sketch of proof here. It can be shown that

the following secrecy rate is achievable:

1

M+N+1degrees of free-

1

Mdegrees

1

M+N−1secure

MN

M+N−1secure degrees of

Rji=1

FI(Xji;¯ Yj)

1

(M − 1)(N − 1)

−1

F

max

k∈J,l∈II(X(J−k)×(I−l);¯Yk|Xk×I)

∀(j,i) ∈ J × I ∀l ∈ I

(6)

where F = N(m+1)Γ+(M−1)mΓand Γ = (M−1)(N−1).

Through interference alignment, it can be shown that

I(Xji;¯Yj) = η log(P) + o(log(P))

where η = (m + 1)Γwhen i = 1 and η = mΓwhen

i = 2,3,...,M. Then consider the secrecy penalty term

I(X(J−k)×(I−l);¯Yk|Xk×I). At each receiver, the interfer-

ence vectors from Transmitter 2,3...,M are aligned perfectly

with the interference vectors from Transmitter 1, i.e. every in-

terference signal vector from Transmitter 2,3...,M is aligned

along the same dimension with one interference signal vector

from Transmitter 1. Note that there are (m+1)Γinterference

vectors for each message from Transmitter 1, but there are only

mΓinterference vectors for each message from Transmitter

2,3...,M. If l = 1, I(X(J−k)×(I−1);¯Yk|Xk×I) denotes the

mutual information between the channel output at Receiver

k and channel inputs from Transmitter 2,...,M. Since all

vectors from Transmitter 2,3...,M are aligned perfectly with

interference vectors from Transmitter 1, it has zero degrees of

freedom, i.e., I(X(J−k)×(I−1);¯ Yk|Xk×I) = o(log(P)). For

∀l ?= 1, the interference vectors from Transmitter l occupy a

(N − 1)mΓdimensional subspace. Therefore, the remaining

transmitters can get a (N − 1)((m + 1)Γ− mΓ) dimensional

space without interference vectors from Transmitter l. There-

fore, we have

max

k∈J, l∈II(X(J−k)×(I−l);¯Yk|Xk×I)

= (N − 1)((m + 1)Γ− mΓ)log(P) + o(log(P))

∀(j,i) ∈ J × I

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Thus, (6) can be written as

Rji=(M − 1)η − ((m + 1)Γ− mΓ)

F(M − 1)

log(P) + o(log(P))

∀i = 1,2,...,M

When m → ∞, we have

ηji= lim

m→∞

(M − 1)η − ((m + 1)Γ− mΓ)

F(M − 1)

=

1

M + N − 1

Therefore, each message can achieve

of freedom for a total of

1

M+N−1secure degrees

MN

M+N−1secure degrees of freedom.

Next, we consider the achievable secure degrees of freedom

under the more conservative secrecy constraints to ensure

secrecy of any subset of transmitters even if all other transmit-

ters are compromised. We present the result in the following

theorem.

Theorem 3: For the M × N user X network with single

antenna nodes, even if any subset of transmitters, S ⊂

{1,...,M} is compromised, the remaining (M − |S|) × N

users can still achieve a total of

of freedom under secrecy constraint 3 and

degrees of freedom under secrecy constraint 4, as long as

|S| ≤ M − 2 .

Proof:

To satisfy secrecy constraint 3, we design an

achievable scheme to satisfy the following secrecy constraint:

N(M−|S|−1)

M+N−1

secure degrees

N(M−|S|)

M+N−1

secure

1

nFH(W(J−j)×Sc|¯Yn

j,Xn

(J−j)×S) ≥

?

(r,i)∈(J−j)×Sc

Rri− ǫ

∀S ⊂ I

where

Xn

(J−j)×S= {Xn

ji: ∀(j,i) ∈ (J − j) × S}

Xn

Note

1

nFH(W(J−j)×Sc|¯Yn

ji

denotes

that

the

secrecy

j,W(J−j)×S). Because

codeword

constraint

formessage

stronger

Wji.

thanthis is

H(W(J−j)×Sc|¯Yn

H(W(J−j)×Sc|¯Yn

H(W(J−j)×Sc|¯Yn

j,W(J−j)×S)

j,W(J−j)×S,Xn

j,Xn

≥

(J−j)×S)

=

(J−j)×S)

In other words, we want to ensure secrecy of any subset of

transmitters even if all other transmitters’ codewords rather

than messages are revealed to the unintended receivers. This

is possible because the achievability scheme encodes the

messages separately and each message has its codewords. The

coding scheme is similar to that used in Theorem 1. Then it

can be shown that the following secrecy rate is achievable:

Rji=1

FI(Xji;¯Yj) −1

max

F

1

(M − |S|)(N − 1)×

k∈J,S⊂II(X(J−k)×Sc;¯Yk|Xk×Sc,XJ×S)

∀(j,i) ∈ J × Sc

Consider the term I(X(J−k)×Sc;¯ Yk|Xk×Sc,XJ×S). Fol-

lowing similar analysis in Theorem 1, if |S| ≤ M − 2, it

can be shown that

max

k∈J,S⊂II(X(J−k)×Sc;¯ Yk|Xk×Sc,XJ×S)

= (N − 1)(m + 1)Γlog(P) + o(log(P))

Therefore,

Rji=1

F(η −(m + 1)Γ

M − |S|)log(P) + o(log(P))

∀(j,i) ∈ J × Sc

where η = (m + 1)Γwhen i = 1 and η = mΓwhen i =

2,3,...,M. As m → ∞,

Rji=

1

M + N − 1(1 −

1

M − |S|)log(P) + o(log(P))

∀(j,i) ∈ J × Sc

Therefore, each message can achieve

secure degrees of freedom for a total of

degrees of freedom under secrecy constraint 3.

Similarly, to satisfy secrecy constraint 4, we design an

achievable scheme to satisfy the following constraint:

1

M+N−1(1 −

N(M−|S|−1)

M+N−1

1

M−|S|)

secure

1

nFH(W(J−j)×(Sc−l)|¯Yn

?

(r,i)∈(J−j)×(Sc−l)

j,Xn

(J−j)×S)

≥

Rri− ǫ

∀S ⊂ I, ∀l ∈ Sc

I

Then it can be shown that the following secure rate is

achievable:

Rji=1

F

max

FI(Xji;¯Yj) −1

k∈J,l∈Sc,S⊂II(X(J−k)×(Sc−l);¯Yk|Xk×Sc,XJ×S)

1

(M − |S| − 1)(N − 1)×

∀(j,i) ∈ J × Sc

Following similar analysis in Theorem 2, if |S| ≤ M − 2, it

can be shown that

max

k∈J,l∈Sc,S⊂II(X(J−k)×(Sc−l);¯Yk|Xk×Sc,XJ×S)

= (N − 1)((m + 1)Γ− mΓ)log(P) + o(log(P))

Therefore,

Rji=(M − |S| − 1)η − ((m + 1)Γ− mΓ)

F(M − |S| − 1)

log(P) + o(log(P))

∀(j,i) ∈ J × Sc

where η = (m + 1)Γwhen i = 1 and η = mΓwhen i =

2,3,...,M. As m → ∞,

Rji=

1

M + N − 1log(P) + o(log(P)) ∀(j,i) ∈ J × Sc

Therefore, each message can achieve

of freedom for a total ofN(M−|S|)

1

M+N−1secure degrees

M+N−1secure degrees of freedom.