Optimal Power Allocation in Block Fading Gaussian Channels with Causal CSI and Secrecy Constraints

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by  Arsenia Chorti

Optimal Power Allocation in Block Fading Gaussian
Channels with Causal CSI and Secrecy Constraints
Arsenia Chorti†, Katerina Papadaki‡, H. Vincent Poor∗
†School of Computer Science and Electronic Engineering, Wivenhoe Park, Colchester, CO4 3SQ, UK
‡Department of Management, London School of Economics and Political Science, Houghton Street, London WC2A 2AE
∗Department of Electrical Engineering, EQUAD, 19 Olden Street, Princeton University, Princeton, New Jersey 08544, USA
[email protected], [email protected], {achorti, poor}@princeton.edu
4
Abstract—The optimal power allocation that maximizes the and the generalized capacity expressions in [5] need to be
1
secrecy capacity of block fading Gaussian (BF-Gaussian) net- employed.Inthiswork,similarlytotheworkin[6],webypass
0 works with causal channel state information (CSI), M-block
2 such issues by assuming that N → ∞. An alternative line
delaytoleranceandaframebasedpowerconstraintisexamined.
of work was suggested in [7] by jointly employing queues
n Inparticular,weformulatethesecrecycapacitymaximizationas
a a dynamic program. We propose suitable linear approximations of secret keys allowing for the avoidance of secrecy outage
J ofthesecrecycapacitydensityinthelowSNR,thehighSNRand events; however this option is not considered at present. The

7 theintermediateSNRregimes,accordingtotheoverall available case of M →∞ that corresponds to the ergodic channel has
2 power budget. Our findings indicate that when the available been investigated in [2] and [3].
powerresourcesareverylow(lowSNRcase)theoptimalstrategy
] is a threshold policy. On the other hand when the available The presentation of the present work is organized as fol-
T powerbudgetisinfinite(highSNRcase)aconstantpowerpolicy lows. First, we restate the secure waterfilling solution to the
I maximizes the frame secrecy capacity. Finally, when the power optimal power allocation optimization problem in M-block
s. budget is finite (medium SNR case), an approximate tractable BF-Gaussian networkswith acausal channelstate information
c power allocation policy is derived. (CSI).Thisframeworkispertinenttoapplicationswithparallel
[ Index Terms—delay constrained secrecy capacity, causal CSI,
channels (e.g. in the frequency domain) under short-term
secure waterfilling, dynamic programming
1 power constraints (e.g. OFDM networks with frame based
v power constraints). Assuming that the M-block CSI is avail-
0 I. INTRODUCTION
able at the transmitting and receiving nodes at the beginning
9
Physical layer security (PLS) investigates the potential of of the transmission frame, the secure waterfilling policy that
7
6 taking advantage of the impairments in real communication maximizes the network secrecy capacity [8] is discussed.
. media, such as fading or noise in wireless channels, in order Next, we investigate BF-Gaussian channels with long term
1
0 toachieveconfidentialityindataexchange.PLSwaspioneered power constraints. We begin with a “blind scenario” in which
4 byWyner,whointroducedthewiretapchannelandestablished theoptimalpowerallocationistobedecidedwithoutanyCSI
1 the possibility of creating perfectly secure communication information; the statistics of the channel gains are the only
v: links without relying on private (secret) keys [1]. Recently, variables in the power allocation decision process. We use
i there have been considerable efforts devoted to generalizing this setting to demonstratethat the formulationof the optimal
X
this result to the wireless fading channel and to multi-user power allocation problem, maximizing the secrecy capacity
r
a scenarios [2], [3], [4]. subject to a delay, as a dynamic program leads naturally to
Inthepresentstudyweinvestigateoptimalpowerallocation intuitive and analytic solutions. In particular, in absence of
policies in block fading Gaussian (BF-Gaussian) wireless any CSI information we show that the optimal policy is to
networks with secrecy and delay constrains. In our model, a equally distribute the power budget in the M transmission
transmitterwishestobroadcastsecretmessagestoalegitimate blocks, as long as the expected value of the difference of the
user by employingphysicallayer security approaches,subject channelgainsofthelegitimateandeavesdroppingterminalsis
toastrictM-blockdelayconstraint;accordingly,atthesource positive.
a stochastic encoder maps the confidential messages to code- Then, we examine networks with causal access to the
words of length n = MN transmitted over M independent legitimateuser’sandtheeavesdropper’sCSIsoverahorizonof
blocks, i.e., we assume that an interleaver of at most depth M transmission blocks; the pairs of the legitimate user’s and
M is employed. We assume that the fading realizations are eavesdropper’s channel gains are sequentially revealed to the
independentandidenticallydistributed(i.i.d),thattheyremain network nodes. We distinguish three subcases accounting for
constant over each block of N channel uses and that they the low, high and intermediate SNR regions. In the low SNR
change independently from one block to the next. a threshold transmission policy is shown to be approximately
In the investigated setting, in order for random coding optimal,inlinewithearlierresultsinnetworkswithoutsecrecy
arguments to hold it is required that n → ∞. For finite n, constraints [9]. On the contrary, we demonstrate that in the
the BF-Gaussian channelsare typically notinformation-stable high SNR case, the optimal strategy is to transmit with
constant power in those blocks in which a non zero secrecy and is given by the secure waterfilling algorithm
capacity can be achieved. Finally, for intermediate SNRs we
derive a tractable expression for the transmission policy that 1 1 d2 + 4d − 2 +d , m∈Q
dependson the gapbetweenthe legitimateand eavesdropping γm∗Åλã= 2 » m λ m Åαm mã
0,h i otherwise
receivers’ CSIs. 
(8)
II. SYSTEMMODEL where Q={i:λ−1 ≥δi−1}.
The functions γ∗(λ−1) are monotone increasing and con-
We assume a BF-Gaussian channel with i.i.d. realizations. m
tinuous in λ−1. As a result, there exists a unique integer
During the m-th transmission block the legitimate user’s
µ in {1,...,M} such that λ−1 ≥ δ −1 for m ≤ µ
channelgainisdenotedbyα andtheeavesdropper’schannel m
m and λ−1 < δ −1 for m > µ. The waterlevel λ−1 can
gainbyβ .WecanexploitthefactthatBF-Gaussianchannels m
m
be derived by sequentially pouring water to the functions
areweaklysymmetric[10]tosimplifytheproofsofthecoding
γ∗(λ−1) until the power constraint is met with equality, i.e.,
theorems. m
µ γ∗(λ−1)=MP.
Definition: The secrecy capacity density during one trans- m=1 m
mission blockof the BF-Gaussian channelfor an inputpower P IV. POWERCONTROL WITH LONG-TERMPOWER
γ and channel gains (α,β) can be expressed as CONSTRAINT WITHOUTCSI
+ We assume an overall long-term power constraint over M
. 1+αγ
cs(γ,α,β)= log (1) sequentialtransmissionblocksintheformof(5).Accordingly,
1+βγ
" #
the channel gains of the legitimate user and the eavesdropper
with [·]+ = max(·,0). The secrecy capacity of the M- duringthe m-thblockaredenotedasαm andβm withknown
block transmission frame for a vector of input powers distributions pA(α) and pB(β) respectively. Our objective at
γ = [γ1,γ2,...,γM] and pairs of channel gains (α,β) = block m, given that we have remaining power pm, is the
[(α ,β ),(α ,β ),...,(α ,β )], can be expressed as: identification of the power allocation γ∗ that maximizes the
1 1 2 2 M M m
instantaneoussecrecycapacityandthesecrecycapacityforthe
M
. 1 future transmission blocks from block m+1 to M.
C = c (γ ,α ,β ). (2)
s s m m m
M
mX=1 A. Blind Scenario
III. POWERCONTROL WITHSHORT-TERMPOWER
We first consider the case in which during the m-th block
CONSTRAINT AND FULLM-BLOCK CSI
we take a decision on the value of γ without having
m
The optimal power allocation policy assuming that at the information on the current channel gains (α ,β ), except
m m
beginning of the transmission frame the CSI of M parallel for their distributions and the remaining power p . In this
m
blocks is revealed to the transmitting and receiving nodeshas formulation,ourobjectiveistomaximizetheexpectedsecrecy
beenderivedin[8]andisrepeatedbelowforconvenience.This capacity over the entire horizon. Let γ = (γ ,...,γ ). The
1 M
is the baseline secure waterfilling policy and its performance stochastic optimization objective function can be written as
cannot be exceeded in the causal scenario. follows:
Without loss of generality we assume that the pairs of
M M
channelgains(αm,βm), m=1,...,M are alreadypermuted maxE c (γ ,α ,β ) =maxE c (γ ,α,β) ,
s m m m s m
so that the differences γ ( ) γ ( )
m=1 m=1
X X (9)
δ =α −β (3)
m m m wheretheexpectationtakenovertherandomvariablesα and
m
appear in non-increasing order. The optimal power allocation βm is re-written with rapport to the generic random variables
problem can be stated as: α and β.
The above problem can be written as a stochastic dynamic
maxC (4)
s program as follows: We let V (p ) be the aggregate secrecy
γ m m
M capacity density gained from block m to the end of the
s.t. γ ≤MP and γ ≥0,m=1,...,M. (5) horizon if the optimal power allocation policy is used. Then
m m
m=1 the dynamic programming equations can be written as:
X
We further define the inverse channel gaps d as:
m V (p ) = max E{c (γ ,α,β)+V (p −γ )}
m m s m m+1 m m
1 1 0≤γm≤pm
dm = − . (6) m=1,...,M
β α
m m
V (p ) = 0 (resources exhausted). (10)
The power allocation γ∗ = (γ∗,γ∗,...,γ∗ ) that maximizes M M+1
1 2 M
the secrecy capacity satisfies the M-block power constraint We perform backward dynamic programming on the opti-
with equality, i.e, mality equations (10). We define the function:
M
γ∗ =MP, (7) 1+αγ
m f(γ)≡E log . (11)
ß 1+βγ™
m=1
X
Westartthedynamicprogrammingrecursionatblockm=M, we have to solve the optimization problem
where the optimality equations are:
V (p ) = max c (α ,β ,γ )
m m s m m m
VM(pM)= max f(γM)1E{α−β}>0, (12) 0≤γm≤Mpm
0≤γM≤pM
+ E c (γ ,α,β) , (17)
s n
where 1F(·) denotes the indicator function. Since f is nonde- (n=Xm+1 )
creasing,the maximizationin (12) is achievedat γM∗ =pM if M
E{α−β}>0andforanyvalueofthepowerifE{α−β}<0. s.t. γ ≤MP. (18)
m
The ambiguity in the latter scenario is resolved by imposing m=1
X
γ∗ =0 whenever this occurs, i.e., Wedistinguishthreecases,accordingtotheavailablepower
M
budget P; the low SNR, the high SNR and the intermediate
p , if E{α−β}>0
γ∗ = M (13) SNR cases.
M ß 0, otherwise.
A. Low SNR
When the condition E{α−β} > 0 is satisfied, we have
In the low SNR, the available power is assumed small,
V (p ) = f(p ). In this case, at block m = M −1 the
M M M i.e., P ≪ 1. As a result a valid linear approximation of the
optimality equations are:
logarithmic function would be log(1+z) ≃ z, leading to an
approximateexpressionforthe secrecycapacitydensity given
V (p )= max f(γ )+f(p −γ ).
M−1 M−1 M−1 M−1 M−1
0≤γM−1≤pM−1 by:
(14)
Let h(γ) = f(γ) + f(p − γ). Note that h′(γ) = f′(γ) − cs(γ,α,β)≃[α−β]+γ =[δ]+γ, (19)
f′(p − γ), and since f′(γ) is nonincreasing and f′(p − γ)
with δ defined in (3). The function V to be optimized at
is nondecreasing in γ, we have that h′ is nonincreasing. This m
m=M could then be written as
meansthatitcanhaveatmostoneextremepointintheinterval
[0,p], and the extreme point must be a maximum. At γ = p V (p )= max [δ ]+γ . (20)
2 M M M M
we have: h′ p = f′(p)−f′(p) = 0. Therefore in (14) the 0≤γM≤pM
2 2 2
maximum is achieved at γ∗ = pM−1 and V (p )= The objective is thus approximatedas a linear functionof the
2f(pM−1). (cid:0) (cid:1) M−1 2 M−1 M−1 power, so that at m = M the optimal power allocation is
2
Continuing the recursion we get straightforwardly given by
VM−n(pM−n)=(n+1)f pnM+−1n (15) γM∗ =ßp0M,, ioftδhMerw>is0e, =[δM]+pM. (21)
and the optimal decision is γ∗ =(cid:0)pM−n.(cid:1)This implies At m=M −1 the objective function takes the form
M−n n+1
that if we have no information about the channel the optimal V (p ) = max [δ ]+γ
M−1 M−1 M−1 M−1
thing to do is to divide the power into as many equal parts as 0≤γM−1≤pM−1
there are periods remaining, i.e., for m = 1,...,M and any + E{[δ]+}(pM−1−γM−1). (22)
distributions p (α) and p (β)
A B Thus, at m = M −1, the optimal power allocation is given
P , if E{δ}>0 by
γ∗ = M (16)
m ß0, otherwise, 0, if [δ ]+ ≤E{[δ]+}
γ∗ = M−1 (23)
M−1 ßp , if [δ ]+ >E{[δ]+}
with δ defined as the gap of channel gains given in (3). M−1 M−1
The above results are intuitive; as expected, the blind Continuing on the backwards recursion, the optimal power
maximizationofafunctionoftheoutcomesofM independent policy during the m-th block is derived as:
trials can be achieved by equidistribution of the available 0, if [δ ]+ ≤E{[δ]+}
resources. What is surprising though, is that the results are γm∗ =ßp , if [δm[+>E{[δ]+} (24)
m m
independent of the statistics of the underlying processes and
form=0,...,M−1.Intheproposedthresholdpowerpolicy,
only require knowledge of the expected value of the gap
whenever a ”good enough” gap in the channel gains δ of
between the legitimate and eavesdropper’s channel gains. m
the legitimateand the eavesdroppingreceiversoccursthenwe
transmit at full power.
V. POWERCONTROL WITHLONG-TERMPOWER
Intuitively, in the low SNR there will not be many op-
CONSTRAINT AND CAUSAL CSI
portunities for achieving high values of the secrecy capacity
In the current section we investigate the case in which density, so whenever such an opportunity occurs it should be
during the m-th transmission block we causally obtain infor- seized in order to maximize the secrecy capacity over the
mation regarding the channel state, i.e., we have access to whole horizon. The threshold is fixed to the expected value
(α ,β ). In thissetting, duringthe m-th transmissionblock, of the gap between the channel gains of the legitimate user
m m
and the eavesdropper, lower bounded by zero. Even when Notably,thequantityδdefinedin(3)oftheacausalCSIsecure
the legitimate user’s channel is on average worse than the waterfilling case reappears in the optimization problem.
eavesdropper’s, it is still possible to transmit at some non- Performing backward dynamic programming on the opti-
zero rate even in the low SNR, given a long enough horizon, mality equations we get:
i.e., for M ≫. ⊲ At block m=M,
B. High SNR 2 [δ ]+γ
V (p ) = max M M ⇒
In the high SNR, i.e., for P →∞, we can transmit at very M M 0≤γM≤pM ln(2)2+(δM +2βM)γM
high power during any of the transmission blocks. A good γ∗ = [δ ]+p , (31)
M M M
approximation for the secrecy capacity density during the m-
while for p = γ∗ , we get
th block is derived as M M
2 δ2 p
α + VM(pM) = ln(2)2+(δ +M2βM)[δ ]+p . (32)
lim c (γ,α,β)= log . (25) M M M M
s
γ→∞ " β#
⊲Fortheblockm=M−1,theobjectiveequationisgiven
The optimization problem of the secrecy capacity density is at the bottom of the page. We note that the objectivefunction
as a result independentof the power allocation and any trans- is concavein the interval[0,p ]. The optimalpolicy is as
M−1
mission policy could be used. Accountingfor other important a result derived as
considerations, e.g. the minimization of the information leak-
age, it is proposed to only transmit during the blocks that γ∗ = ρ (34)
M−1 M−1
satisfy the condition δ >0, i.e.,
m
with ρ being the positive root of the quadratic equation
γ∗ = 0, if δm ≤0 =[δ ]+ pm , (26) ∂VM−1M=−01, that can be evaluated in analytic form.
m ßMp−mm, if δm >0 m M −m ∂γAMl−t1hough the analytic expression for the quantity
with p =P and m=1,...,M. V (γ∗ ) is overly complicated to be included at
1 M−1 M−1
present, we note that its expectation is easier to evaluate.
C. Intermediate SNR ⊲ Continuing backwards we get that at m = M −2 the
Aiming at producing tractable expressions for the power objective function is also concave in the interval [0,p ].
M−2
allocation, we propose using the following approximation for The optimal policy is derived as
the logarithmic function [11]:
γ∗ = ρ , (35)
∞ 1 z−1 2n+1 M−2 M−2
ln(z) = 2 ⇒
nX=02n+1(cid:16)z+1(cid:17) withρM−2 beingthepositiverootoftheequation ∂∂VγMM−−22 =0.
2 z−1 Generalizing we find that the optimal policy is expressed as:
log(z) ≃ . (27)
ln(2)z+1
γ∗ = ρ , (36)
In addition, we will use the following first order approxima- m m
tion: for the correlated random variables X and Y and linear
where ρ is the positive root of the equation ∂Vm =0.
functions F(·) and G(·), m ∂γm
F(X) E{F(X)} D. Semi-blind Scenario
E ≃ . (28)
G(Y) E{G(Y)}
( )
The derivedresults apply also in the semi-blindscenario in
Using (27), the secrecy capacity density can be expressed which only the legitimate user CSI is causally made available
as: to the transmitting and receiving nodes by substituting δ by
2 [δ]+γ α in the equations (24), (26) and (36). However, in this case
c (γ,α,β)≃ , (29)
s ln(2)2+(δ+2β)γ the minimization of the probability of secrecy outage should
inprinciplebe investigatedinsteadofthe maximizationofthe
while employing (28)
secrecycapacitydensity.Atpresentthisisleftasfuturework,
2 E{[δ]+}γ alongwith thenumericalevaluationoftheperformanceofthe
E{c (γ,α,β)}≃ . (30)
s ln(2)2+E{(δ+2β)}γ outlined policies.
2 [δ ]+γ 2 E{δ2}(p −γ )
V (p ) = max M−1 M−1 + M−1 M−1
M−1 M−1 0≤γM−1≤pM−1 ln(2)2+(δM−1+2βM−1)γM−1 ln(2)2+E{(δ+2β)E{[δ]+}(pM−1−γM−1)
(33)
VI. CONCLUSIONS [3] P. K. Gopala, L. Lai, and H. El-Gamal, “On the secrecy capacity of
fadingchannels,” IEEETrans.Information Theory,vol.54,no.10,pp.
In the present work we investigate the optimal power
4687–4698, Oct.2008.
allocation in delay constrained M-block BF-Gaussian net- [4] A.Chorti,S.M.Perlaza,Z.Han,andH.V.Poor,“Ontheresilience of
works. By studying the blind case with no CSI availability wirelessmultiusernetworkstopassiveandactiveeavesdroppers,”IEEE
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during the decision process we conclude that the optimal
[5] S.Verdu´andT.S.Han,“Ageneralformulaforchannelcapacity,”IEEE
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transmission blocks as long as a positive gap between the [6] G. Caire, G. Taricco, and E. Biglieri, “Optimum power control over
fading channels,” IEEE Trans. Information Theory, vol. 45, no. 5, pp.
channel gains of the legitimate user and the eavesdropper is
1468–1489, Jul.1999.
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[8] A.chorti,katerinaPapadaki,P.Tsakalides,andH.V.Poor,“Thesecrecy
we derived a near optimal threshold policy whereas in the capacity of block fading multiuser wireless networks,” in 2013 Inter-
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(ATC). Ho-ChiMinhCity,Vietnam:IEEE,Oct.2013,pp.247–251.
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