The Discrete Memoryless Multiple Access Channel with Confidential Messages

Abstract

A multiple-access channel is considered in which messages from one encoder are confidential. Confidential messages are to be transmitted with perfect secrecy, as measured by equivocation at the other encoder. The upper bounds and the achievable rates for this communication situation are determined.

Full text

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arXiv:cs/0605005v2 [cs.IT] 17 Feb 2007
The Discrete Memoryless Multiple Access Channel
with Confidential Messages
Ruoheng Liu, Ivana Mari´ c, Roy D. Yates and Predrag Spasojevi´ c
WINLAB, Rutgers University
Email: {liurh,ivanam,ryates,spasojev}@winlab.rutgers.edu
Abstract—A multiple-access channel is considered in which
messages from one encoder are confidential. Confidential mes-
sages are to be transmitted with perfect secrecy, as measured
by equivocation at the other encoder. The upper bounds and the
achievable rates for this communication situation are determined.
I. INTRODUCTION
We consider a two-user discrete multiple-access channel in
which one user wishes to communicate confidential messages
to a common receiver while the other user is permitted to
eavesdrop. We refer to this channel as the multiple access
channel with confidential messages (MACC) and denote it
(X1×X2,p(y,y1|x1,x2),Y ×Y1). The communications sys-
tem is shown in Figure 1. The ignorance of the other user is
measured by equivocation. This approach was introduced by
Wyner [1] for the wiretap channel, a scenario in which a single
source-destination communication is eavesdropped. Under the
assumption that the channel to the wire-tapper is a degraded
version of that to the receiver, Wyner determined the capacity-
secrecy tradeoff. This result was generalized by Csisz´ ar and
K¨ orner who determined the capacity region of the broadcast
channel with confidential messages [2]. The Gaussian wire-tap
channel was considered in [3].
In this paper, we determine the bounds on the capacity re-
gion of the MACC, under the requirement that the eavesdrop-
ping user is kept in total ignorance. The results characterize
the rate penalty when compared to the conventional MAC [4],
[5] due to the requirement that one message is kept secret.
It is apparent from the results that eavesdropping by user
1 will hurt the achievable rate of user 2. As illustrated in the
last section by an example in which the half-duplex constraint
is imposed, the eavesdropper should give up on listening all
together, thus maximizing rates of both users. The moral of
the example is that either user 1 will make both himself and
the other user miserable by eavesdropping more and thus
reducing both its own and other user’s ability to transmit; or,
it will make both of them happy if it decides not to listen.
We note that, although user 2 cannot know exact times when
user 1 is eavesdropping, it is enough for user 2 to know the
eavesdropping probability (or equivalently, the fraction of time
user 1 is listening), to adjust its code rate accordingly. This
information can be considered public, since it is known to the
common receiver.
1This work was supported by NSF Grant NSF ANI 0338805.
Y1
Y
Enc 2W2
X2
Enc 1
X1
W1
H(W2|Y1X1)
(W1,
DecW2)
User 1
User 2
Receiver
?
i
?
iiii
xxyyp
p
),|,(
),|,(
211
211
xxyy
Channel
Fig. 1.
System Model
II. CHANNEL MODEL AND STATEMENT OF RESULT
A discrete memoryless MAC with confidential messages
consists of finite sets X1,X2,Y,Y1 and a conditional proba-
bility distribution p(y,y1|x1,x2). Symbols (x1,x2) ∈ X1×X2
are channel inputs and (y,y1) ∈ Y × Y1are channel outputs
at the receiver and encoder 1, respectively. The channel
p(y|x1,x2) is a MAC channel, and the channel p(y,y1|x1,x2)
is a wire-tap channel. Each encoder t, t = 1,2, wishes to send
an independent message Wt ∈ {1,...,Mt} to a common
receiver in n channel uses. The channel is memoryless and
time-invariant in the sense that
p(y1,i,y2,i|xi
1,xi
2,yi−1
1
,yi−1
2
) = p(y1,i,y2,i|x1,i,x2,i)
(1)
where xi
the superscript when i = n. A deterministic encoder g for user
1 is a mapping g : W1→ Xn
t=?xt,1,...,xt,i
?. To simplify notation, we drop
1generating codewords
x1= g(w1).
(2)
A stochastic encoder f for user 2 is specified by a matrix
of conditional probabilities f(x2|w2), where x2∈ Xn
W2, is the private message set, and
2, w2∈
?
x2
f(x2|w2) = 1.
Note that f(x2|w2) is the probability that the message w2is
encoded as channel input x2.
The decoding function is given by a mapping φ : Yn→
W1× W2.
The implicit assumption in our model is that user 1 observes
the sequence Y1in block fashion. This prevents user 1 from
using symbols Y1for encoding its own messages, as reflected
in the encoding function (2). This restriction of our model is

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made for the sole purpose of making the problem easier to
solve and understand.
An (M1,M2,n,Pe) code for the channel consists of two
encoding functions f,g, decoding function φ such that the
average probability of error of the code is
Pe=
?
(w1,w2)
1
M1M2P{φ(y) ?= (w1,w2)|(w1,w2) sent}(3)
The level of ignorance of user 1 with respect to the confi-
dential message is measured by the normalized equivocation
(1/n)H(W2|X1,Y1).
A rate pair (R1,R2) is achievable for the MACC if, for any
ǫ > 0, there exists a (M1,M2,n,Pe) code such that
Mt≥ 2nRtt = 1,2, Pe≤ ǫ
R2−1
nH(W2|X1,Y1) ≤ ǫ.
The capacity region of the MACC is the closure of the set
of all achievable rate pairs (R1,R2).
The next two theorems show the outer bound and the
achievable rates and are the main results of this paper.
Let CU be a closure of the union of all (R1,R2) satisfying
(4)
(5)
R1≤ I(X1;Y |X2)
R2≤ I(V ;Y |U,X1) − I(V ;Y1|U,X1)
R1+ R2≤ I(X1,V ;Y ) − I(V ;Y1|U,X1)
(6)
for some joint distribution
p(u,v,x1,x2,y,y1)
= p(u)p(v|u)p(x1|u)p(x2|v)p(y,y1|x1,x2)
(7)
where U and V are auxiliary random variables satisfying U →
V → (X1,X2) → (Y,Y1).
Theorem 1: (Outer Bound) For any achievable rate pair
(R1,R2) in MACC it holds that (R1,R2) ∈ CU.
Theorem 2: (Achievability) The rates in the closure of the
union of all (R1,R2) satisfying
R1≤ I(X1;Y |U,V )
R2≤ I(V ;Y |U,X1) − I(V ;Y1|U,X1)
R1+ R2≤ I(X1,V ;Y |U) − I(V ;Y1|U,X1)
(8)
for a joint distribution p(u,v,x1,x2,y,y1) that factors as (7).
III. OUTER BOUND
Proof: (Theorem 1)
We next show that any achievable rate pair satisfies
R1≤ I(X1;Y |X2,Q)
R2≤ I(V ;Y |U,X1,Q) − I(V ;Y1|U,X1,Q)
R1+ R2≤ I(U,X1,V ;Y |Q) − I(V ;Y1|U,X1,Q)
(9)
(10)
(11)
for some product distribution U → V → (X1,X2) → (Y,Y1)
that factor as (7) and an independent timesharing random
variable Q. Then, the approach of [6, Thm. 14.3.3] and the
observation that Markovity U → V → (X1,X2) → Y implies
U → (V,X1) → Y , will prove the claim.
Consider a code (M1,M2,n,Pe) for the MACC. Applying
Fano’s inequality results in
H(W1,W2|Y) ≤ Pelog(M1M2− 1) + h(Pe) ? nδn (12)
where δn→ 0 as Pe→ 0. It follows that
H(W1,W2|Y) = H(W1|Y) + H(W2|Y,W1) ≤ nδn (13)
We first consider the bound on R1.
nR1= H(W1)
= I(W1;Y) + H(W1|Y)
≤(a)I(W1;Y) + nδn
≤(b)I(X1(W1);Y) + nδn
≤(c)I(X1;Y|X2) + nδn
n
?
i=1
+ nδn
n
?
i=1
n
?
i=1
=(d)
H(Yi|X2,Yi−1) −
n
?
i=1
H(Yi|Yi−1,X1,X2)
≤(e)
H(Yi|X2i) −
n
?
i=1
H(Yi|X1i,X2i) + nδn
=
I(X1,i;Yi|X2,i) + nδn
(14)
where (a) follows from from Fano’s inequality (13); (b) from
(2); (c) from the independence of X1,X2; (d) from the chain
rule; (e) from the fact that the conditioning decreases entropy
and from the memoryless property of the channel (1).
Following the approach in [6, Sec. 14.3.4], we introduce
a uniformly distributed random variable Q,Q ∈ {1,...,n}.
Equation (14) becomes
nR1≤
n
?
i=1
n
?
i=1
I(X1,i;Yi|X2,i) + nδn
=
I(X1,i;Yi|X2,i,Q = i) + nδn
= nI(X1,Q;YQ|X2,Q,Q) + nδn
= nI(X1;Y |X2,Q) + nδn
(15)
where X1= X1,Q,X2= X2,Q,Y = YQ. Distributions of new
variables depend on Q in the same way as the distributions of
X1,i,X2,i,Yidepend on i.
Next, we derive the bound on R2. Note that the perfect
security (5) implies
nR2− nǫ ≤ H(W2|X1,Y1).
(16)
Hence, we consider the bound on H(W2|X1,Y1).
H(W2|X1,Y1)
= H(W2|X1) − I(W2;Y1|X1)
= I(W2;Y|X1) + H(W2|Y,X1) − I(W2;Y1|X1)
≤ I(W2;Y|X1) − I(W2;Y1|X1) + nδn
(17)

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where the inequality follows from Fano’s inequality (13).
We next use a similar approach as in [2, Sect.V] to bound
equivocation H(W2|X1,Y1) in (17).
We denote˜Yi+1
1
= [Y1,i+1,...,Y1,n] and use the chain rule
to obtain
I(W2;Y|X1)
=
n
?
i=1
n
?
i=1
I(W2;Yi|Yi−1,X1)
=
I(W2,;Yi|˜Yi+1
1
,Yi−1,X1) + Σ1− Σ2
(18)
I(W2;Y1|X1)
=
n
?
i=1
n
?
i=1
I(W2;Y1i|˜Yi+1
1
,X1)
=
I(W2,;Y1,i|˜Yi+1
1
,Yi−1,X1) +ˆΣ1−ˆΣ2 (19)
where
Σ1=
n
?
i=1
n
?
i=1
n
?
i=1
n
?
i=1
I(˜Yi+1
1
;Yi|Yi−1,X1)
Σ2=
I(˜Yi+1
1
;Yi|Yi−1,X1,W2)
ˆΣ1=
I(Yi−1;Y1,i|˜Yi+1
1
,X1)
ˆΣ2=
I(Yi−1;Y1,i|˜Yi+1
1
,X1,W2).
Lemma 1: Σ1=ˆΣ1and Σ2=ˆΣ2.
Proof: Proof follows the approach in [2, Lemma 7].
We let
Ui= (Yi−1˜Yi+1
1
Xi−1
1
˜Xi+1
1
)
(20)
Vi= (W2,Ui)
(21)
in (18) and (19) and obtain respectively
I(W2;Y|X1) =
n
?
i=1
I(Vi;Yi|Ui,X1,i) + Σ1− Σ2
(22)
I(W2;Y1|X1) =
n
?
i=1
I(Vi;Y1,i|Ui,X1,i) +ˆΣ1−ˆΣ2
(23)
We follow the same approach as in (15) to obtain
1
n
n
?
i=1
I(Vi;Yi|Ui,X1,i) =1
n
n
?
i=1
I(Vi;Yi|Ui,X1,i,Q = i)
= I(VQ;YQ|UQ,X1,Q,Q)
= I(V ;Y |U,X1,Q)
(24)
where V = VQ,Y = YQ,X1= X1,Q,U = UQ. Similarly,
1
n
n
?
i=1
I(Vi;Y1,i|Ui,X1,i) = I(V ;Y1|U,X1,Q)
(25)
where Y1 = Y1,Q. From the memoryless property of the
channel (1), it follows that V → (X1,X2) → (Y,Y1).
Using (24) in (22), we obtain
I(W2;Y|X1) = nI(V ;Y |U,X1,Q) + Σ1− Σ2.
(26)
Similarly, using (25) in (23)
I(W2;Y1|X1) = nI(V ;Y1|U,X1,Q) +ˆΣ1−ˆΣ2.
(27)
Substituting (26) and (27) in (17) results in
1
nH(W2|X1,Y1)
≤ I(V ;Y |U,X1,Q) − I(V ;Y1|U,X1,Q) + δn.
(28)
Using (16) in (28), we obtain the desired the bound (10) on
rate R2.
We next prove the bound on the sum rate (11).
n(R1+ R2) = I(W1,W2;Y) + H(W1,W2|Y)
≤ I(X1,W2;Y) + nδn
≤ I(X1,W2;Y) − [H(W2)
− H(W2|X1,Y1) − nǫ] + nδn
= I(X1;Y) + I(W2;Y|X1)
− I(W2;Y1|X1) + n(δn+ ǫ)
(29)
where the second inequality follows from the perfect secrecy
(5). Using (22), (23) and Lemma 1, we have
I(W2;Y|X1) − I(W2;Y1|X1)
n
?
i=1
=
?I(W2;Yi|Ui,X1,i) − I(W2;Y1,i|Ui,X1,i)?
(30)
Hence, (29) can be rewritten as
n(R1+ R2)
n
?
i=1
− I(W2;Y1,i|Ui,X1,i)?+ n(δn+ ǫ)
≤
?
i=1
− I(W2;Y1,i|Ui,X1,i)?+ n(δn+ ǫ)
=
?
i=1
n
?
i=1
≤
?I(X1;Yi|Yi−1) + I(W2;Yi|Ui,X1,i)
n
?I(X1,Yi−1,˜Yi+1
1
;Yi) + I(W2;Yi|Ui,X1,i)
n
?I(Vi,X1,i;Yi) − I(Vi;Y1,i|Ui,X1,i)?+ n(δn+ ǫ)
≤
?I(Vi,Ui,X1,i;Yi) − I(Vi;Y1,i|Ui,X1,i)?+ n(δn+ ǫ)
where Uiand Viare defined in (20) and (21). Using the same
time-sharing variable approach as before we obtain the sum
rate bound (11). Moreover, the Markovity X1,i−Ui−Vican
easily be verified.

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IV. ACHIEVABILITY
Proof: (Theorem 2)
Fix p(u), p(x1|u), p(v|u) and p(x2|v). Let
R3= R2+ I(V ;Y1|X1,U).
(31)
Codebook generation: Generate a random typical sequence
u, with probability p(u) =?n
transmitters and the common receiver know the sequence u.
Generate M1 = 2nR1sequences x1, each with probabil-
ity p(x1|u) =
?n
{1,...,M1}.
Generate M3
=2nR3 sequences v with probability
p(v|u)=
?n
{1,...,2nR2}, l ∈ {1,...,2nI(V ;Y1|X1,U)}.
Encoding: To send message w1 ∈ W1, user 1 sends
codeword x1(w1). To send message w2 ∈ W2, user 2 uses
stochastic encoder f, and encoder 2 uniformly randomly
chooses an codeword v(w2,l). That is, the encoder chooses
randomly a codeword v(w2,l) from a bin w2. Finally, user 2
generates the channel input sequences x2accordingto p(x2|v).
Decoding: Let A(n)
ǫ
denote the set of typical (u,x1,v,y)
sequences. Decoder chooses the pair (w1,w2) such that
(u,x1(w1),v(w2,l),y) ∈ A(n)
ǫ
and is unique; otherwise, an error is declared.
Probability of error: Define the events
i=1p(ui). We assume that both
i=1p(x1,i|ui). Label them x1(w1), w1 ∈
i=1p(vi|ui). Label them v(w2,l), w2
∈
if such a pair (w1,w2) exists
Ew1,w2= {(u,x1(w1),v(w2,l),y) ∈ A(n)
ǫ }.
(32)
Without loss of generality, we can assume that (w1,w2) =
(1,1) was sent. From the union bound, the error probability
is given by
Pe≤P{Ec
1,1|(1,1)} +
?
w1?=1
P{Ew1,1|(1,1)}
+
?
w2?=1
?
w1?=1
?
l
?
w2?=1
P{E1,w2|(1,1)}
+
?
l
P{Ew1,w2|(1,1)}
(33)
From the AEP and [6, Thm. 14.2.1, 14.2.3], it follows that
P{Ec
P{Ew1,1|(1,1)} ≤ 2−n[I(X1;Y |V,U)−δ]
P{E1,w2|(1,1)} ≤ 2−n[I(V ;Y |X1,U)−δ]
P{Ew1,w2|(1,1)} ≤ 2−n[I(X1,V ;Y |U)−δ]
1,1|(1,1)} ≤ δ
(34)
(35)
(36)
(37)
where δ → 0 as n → ∞. Hence, (33) is bounded by
Pe≤ δ + 2nR12−n(I(X1;Y |V,U)−δ)+ 2nR32−n(I(V ;Y |X1,U)−δ)
+ 2n(R1+R3)2−n(I(X1,V ;Y |U)−δ)
(38)
implying that we must choose
R1≤ I(X1;Y |V,U)
R3≤ I(V ;Y |X1,U)
R1+ R3≤ I(X1,V ;Y |U)
(39)
(40)
(41)
to guarantee Pe→ 0 as n gets large.
Equivocation: We consider the normalized equivocation.
H(W2|Y1,X1)
≥ H(W2|Y1,X1,U)
= H(W2,Y1|X1,U) − H(Y1|X1,U)
= H(W2,Y1,V|X1,U) − H(V|W2,Y1,X1,U)
− H(Y1|X1,U)
= H(W2,V|X1,U) + H(Y1|W2,V,X1,U)
− H(V|W2,Y1,X1,U) − H(Y1|X1,U)
≥ H(V|X1,U) + H(Y1|V,X1,U)
− H(V|W2,Y1,X1,U) − H(Y1|X1,U)
= H(V|X1,U) − H(V|W2,Y1,X1,U)
− I(V;Y1|X1,U)
(42)
The first term in (42) is given by
H(V|X1,U) = H(V|U) = nR3
(43)
where the first equality follows from the Markov chain V −
U − X1, and the second equality because given U = u, V
has 2nR3possible values with equal probability.
We next show that H(V|W2,Y1,X1,U) ≤ nδ1, where
δ1 → 0 as n → ∞. Let W2 = w2. User 2 then sends a
codeword v(w2,l). Let λw2denote the average probability of
error that user 1 does not decode v(w2,l) correctly given the
information W2 = w2. Following the joint typical decoding
approach, we have λw2→ 0 as n → ∞. Therefore, Fano’s
inequality implies that
H(V|W2= w2,Y1,X1,U) ≤ 1 + λw2(nR3− nR2) ? nδ1.
Hence
H(V|W2,Y1,X1,U) =
?
w2∈W2
p(W2= w2)H(V|W2= w2,Y1,X1,U) ≤ nδ1.
(44)
Finally, the third term in (42) can be bounded by
I(V;Y1|X1,U) ≤ nI(V ;Y1|X1,U) + nδ2
(45)
where δ2→ 0 as n → ∞. The proof follows the proof in [1,
Lemma 8].
Therefore, by using (31), (43), (44), and (45), we can rewrite
(42) as
H(W2|X1,Y1) ≥ nR3− nI(V ;Y1|X1,U) − n(δ1+ δ2)
= nR2− nǫ
(46)
where ǫ ? δ1+ δ2.

Page 5

V. DISCUSSION AND IMPLICATIONS
To show the impact of secret communication on the achiev-
able rates in MACC, we present two examples: the half-duplex
MACC and the Gaussian MACC. To simplify calculations, we
consider the following corollary which gives a weaker inner
bound used in the rest of the paper.
Corollary 1: The rates in the closure of the convex hull of
all (R1,R2) satisfying
R1≤ I(X1;Y |X2)
R2≤ I(X2;Y |X1) − I(X2;Y1|X1)
R1+ R2≤ I(X1,X2;Y ) − I(X2;Y1|X1)
(47)
(48)
(49)
for fixed product distribution p(x1)p(x2) on X1 × X2 is
achievable in MACC.
Proof:
Corollary follows by choosing V
independent from X1and X2in Theorem 2.
Binary inputs are to be communicated from the both users
under a half-duplex model in which user 1 cannot listen and
transmit at the same time. Therefore X2∈ {0,1} and X1∈
{∅,0,1}. Null symbol ∅ models the listening period of user
1. When X1= ∅, user 1 observes the output Y1= Y ; when
user 1 transmits, X1∈ {0,1}, the output Y1is the null symbol,
no matter what user 2 sends. When both users transmit, the
MAC channel to the destination is given by the mod 2 sum
Y = X1⊕ X2. Otherwise, Y = X2. In summary,
= X2 and U
Y = X1⊕ X2,
Y = X2,
Y1= ∅,
Y1= Y,
if X1?= ∅
if X1= ∅
(50)
(51)
Denote P = P[X1= 1] and D = P[X1= ∅]. Rates (47)-(49)
for this channel can be shown to be
R1≤ h(P)
R2≤ H(X2)(1 − D)
R1+ R2≤ H(Y ) − H(X2)D.
(52)
(53)
(54)
If we assume the inputs at user 2 are equally likely, then
H(Y ) = 1. The rates (52)-(54) become
R1≤ h(P)
R2≤ 1 − D
R1+ R2≤ 1 − D
(55)
(56)
(57)
and the secrecy constraint (56) becomes irrelevant. The achiev-
able rates are determined by the amount of time user 1 listens:
the more user 1 listens, the more user 2 must equivocate rather
than communicate. The best strategy is then for user 1 to
transmit all the time (D = 0), thus achieving the full capacity
region of the conventional MAC.
In the other limiting case in which user 1 only listens (D =
1), user 2 cannot send information because user 1 hears it
(Y1= X2). In fact, the channel reduces to the special case of
the channel considered in [2] and the conclusion is agreeable
with that of [2]. In the example, the fact that R2= 0 is due to
the very special channel Y1= Y . In the more general case in
which Y1is a noiser observation of X2than Y , user 2 can still
“squeeze” some information through even if user 1 listens all
the time. Nonetheless, this example illustrates the fundamental
behavior in the MACC, that can be observed from Corollary 1,
Eq. (48): the more user 1 decides to listen, the more user 2
has to equivocate and his achievable rate is lower.
We next consider the Gaussian channel
Y =X1+ X2+ Z
Y1=X2+ Z1
(58)
(59)
where Z and Z1are independent zero-mean Gaussian random
variables with variance N and N1, respectively. The code
definition is the same as given in Section II with the addition
of the power constraints
1
n
n
?
i=1
E[X2
ti] ≤ Pt,t = 1,2.
(60)
Corollary 2: The rates in the closure of the convex hull of
all (R1,R2) satisfying
R1≤ C
?P1
N
?P2
N
?P1+ P2
N
?
?
(61)
R2≤ C
− C
?P2
N1
?
?
?P2
N1
(62)
R1+ R2≤ C
− C
?
.
(63)
Corollary follows from Theorem 2 by independently choosing
Xt∼ N[0,Pt] for t = 1,2.
Future Work
It is conceivable that the outer bounds given in Theorem 1
can be strengthened to coincide with the lower bounds of
Theorem 2. Investigating this possibility and determining the
MACC capacity are the subjects of our future work. Moreover,
the formulation of this problem in which the objective is
to maximize rates under the secrecy constraint follows the
definition of Wyner [1]. However, different objectives can be
envisioned, in which user 1 is more interested in eavesdrop-
ing than in maximizing its rate. It would be interesting to
compare the conclusions that follow from the two problem
formulations.
REFERENCES
[1] A. Wyner, “The wire-tap channel,” Bell. Syst. Tech. J., vol. 54, no. 8, pp.
1355–1387, Jan. 1975.
[2] I. Csisz´ ar and J. K¨ orner, “Broadcast channels with confidential messages,”
IEEE Trans. on Inf. Theory, vol. 24, no. 3, pp. 339–348, May 1978.
[3] S. K. Leung-Yan-Cheong and M. E. Hellman, “The Gaussian wire-tap
channel,” IEEE Trans. on Inf. Theory, vol. 24, no. 4, pp. 451–456, July
1978.
[4] H. Liao, “Multiple access channels,” Ph.D. Thesis, University of Hawaii,
1972.
[5] R. Ahlswede, “Multi-way communication channels,” in Int. Symp. Inf,
Th., 1971, pp. 23–52.
[6] T. Cover and J. Thomas, Elements of Information Theory.
Sons, Inc., 1991.
John Wiley