Enhancing Secrecy Rates and Resourse Allocation in Single-User …shahidshah.weebly.com/uploads/1/1/2/2/11221304/colloqium... · 2018-09-07 · (Thesis Colloqium) Shahid Mehraj Shah

Introduction Single user Wiretap Channel: Alternate Secrecy Notion Single User Wiretap Channel: Time Slotted System Fading MAC: Resource Allocation Fading MAC: Game Theoretic Formulation Fading MAC: Time Slotted System Conclusions and Future Work

Enhancing Secrecy Rates and ResourseAllocation in Single-User & Multi-User Fading

Wiretap Channels(Thesis Colloqium)

Shahid Mehraj Shah ∗

∗Dept. of ECE, Indian Institute of Science, Bangalore

June 22nd 2015

Shah&Sharma — Enhancing Secrecy Rates and Resourse Allocation in Single-User & Multi-User Fading Wiretap Channels 1/84


Outline

1 Introduction

2 Single user Wiretap Channel: Alternate Secrecy Notion

3 Single User Wiretap Channel: Time Slotted System

4 Fading MAC: Resource Allocation

5 Fading MAC: Game Theoretic Formulation

6 Fading MAC: Time Slotted System

7 Conclusions and Future Work



Outline

1 Introduction









How it began

Conventional security measure taken at higher layerCryptographic techniques depend on finiteness of computationalpower while Information theoretic security (ITS) promises provablesecurityWyner in 1975 Proposed a wiretap coding scheme for degradedWiretap Channel1

Security achieved at the cost of rate lossWyner’s result extended in several ways: BCC, MAC, IC, RC 2

Cost paid in each channel model is: Rate LossIn most of the work in ITS, Eve’s complete channel knowledge isassumedBecause of passive nature of Eve, this assumption not practical

1A. Wyner, “The wire-tap channel,” Bell Syst. Tech. J., 1974.2Y. Liang, H.V. Poor, S. Shamai, “Information theoretic security,” Foundations and

Trends in Communications and Information Theory, 2009.



Motivation

We try to address these two issues in our workTo mitigate rate loss we propose alternate measure of secrecy andnovel coding scheme.We use optimization techniques to allocate power with No CSI ofEve in MACWe also use Game theoretic learning to allocate resources with lessinformation about Eve’s channel state.



Outline

1 Introduction









Channel Model: AWGN Wiretap Channel1

Message W ∈ 1, . . . , 2nR to be transmitted securely & is encodedto X n = X1, . . . ,Xn.Bob receives Yi = Xi + N1i , N1i ∼ N (0, σ2

1), iid , Eve receivesZi = Xi + N2i , N2i ∼ N (0, σ2

2), iid , and σ21 < σ2

2

Let C1 = 0.5 log(1 + P/σ21) and C2 = 0.5 log(1 + P/σ2

2)

Equivocation Based definition:(R,Re) is achievable if ∃ Wn with | Wn |= 2nR and encoder -decoder pairs (fn, gn) such that P(n)

e → 0, and the equivocation ratelimn→∞

1n H(W |Z n) ≥ Re

Code design to satisfy equivocation rate is difficult. 2

To confuse Eve random messages are sent, which decreases the rate.

2Klinc, D., Ha, J., McLaughlin, S. W., Barros, J., & Kwak, B. J. “LDPC codes forthe Gaussian wiretap channel.” IEEE Trans. on Info. Forensics and Sec., (2011)

1Part of this work has been presented in IEEE ICCS 2012, Singapore



New Notion

Natural Definition of secrecy: Probability of error for receiver → 0and that for eavesdropper → 1.Strong converse: R > C , probability of error of decoding → 1.Will use strong converse to formulate coding schemes.Eve is confused with the messages which are useful for Intendedreceiver,and Eve getting no message, i.e. no common information.Each message for Eve confused with same number of codewords asin Equivocation based approach



New Notion: Achieving Shannon Capacity

PropositionAll rates R < C1 are achievable such that Pn

e (B)→ 0, Pne (E )→ 1 as

n→∞.

where Pne (B) = Probability of decoding error at Bob, Pn

e (E ) =Probability of decoding error at Eve

Coding SchemeRegion C2 < R < C1: generate n length iid Gaussian codewordswith X ∼ N (0,P)

R < C1 ensures Pne (B)→ 0

R > C2, by strong converse, Pne (E )→ 1.

To achieve rate R < C2, select Gaussian PX with power < P suchthat I(X ; Y ) > R > I(X ; Z )



Performance (via AEP decoder at Eve)

N be the number of codewords other than that of message 1 thatare jointly typical with Z n

E[N] = (2nR − 1)2−nI(X ;Z) and Pne (E ) ≈ 2n(R−I(X ;Z))

Pne (E ) ≤ 2−n(C1−C2) possible, maximum decay rate for equivocation

based secrecy also.Higher R means more confusion for Eve.ML Decoding at Eve: If x1(1), ..., xn(1) is transmitted, decode asmessage m if

m = arg minn∑

k=1(Zk − xk(m))2

. (1)

Eve will confuse with 2n(R−C2) codewords . Max confusion as in AEPdecoder.AEP and ML decoder best for Eve.



New Notion: Relation with Equivocation

When C2 < R < C1, Pne (B)→ 0, and Pn

e (E )→ 1. Fano’s Inequalitygives: For Bob

1n H(W |Y n) ≤ H(Pn

e (B))

n + Pne (B)R (2)

Lower bound: For Eve

H(W | Z n) ≥ φ∗(π(W | Z n)) (3)

where φ∗ is a piecewise linear, continuous, non-decreasing, convexfunction.2. π(W | Z n) is the average probability of error for theMAP decoder at Eve.From Arimoto’s lower bound

π(W | Z n) ≥ 1− e−n(E0(ρ,p)−ρR), 0 ≥ ρ ≥ −1. (4)2M. Feder and N. Merhav, “Relations Between Entropy and Error Probability,”

IEEE Trans. Inform. Theory. Jan. 1994



Numerical Example

For σ21 = 0.1, σ2

2 = 1.5, and P = 20dB, Pne (B) and Pn

e (E ) areplotted for n = 50, 100 and 200.For Pn

e (B) Gallagers random coding bound and for Pne (E ) Arimoto’s

lower bound are plotted

0.3 0.4 0.5 0.6 0.70

0.2

0.4

0.6

0.8

1

Fraction of SNR(α)

Pro

babili

ty o

f E

rror Bob N=50

Eve N=50

Bob N=100

Eve N=100

Bob N=200

Eve N=200

P=20dB, σ1

2=0.1,

σ2

2=1.5, R=2.5

Figure : Probability of Error vs SNR



Fading Channel

For the fading channel

Yi = hi Xi + N1i , Zi = gi Xi + N2i , (5)

,The capacity is ([gopala2008]):

Cs =

∫ ∞0

∫ ∞r

[log(1 + qP∗(q, r))− log(1 + rP∗(q, r))]f (q)f (r)dqdr

(6)where q(i) = |h(i)|2 and r(i) = |g(i)|2 E [P∗(q, r)] = P,

PropositionAll rates R < C1 are achievable such that Pn

e (B)→ 0 and Pne (E )→ 1

where

C1 = supP(q,r)

∫ ∞0

∫ ∞r

[log(1 + qP(q, r))]f (q)f (r)dqdr . (7)

.



Outline

1 Introduction









Discrete Memoryless Wiretap Channel1

Yamamoto in 1997 used Wiretap coding and secret key to enhancesecrecy rate in degraded Wiretap channel. 1

Kang and Liu (2010) extended Yamamoto’s work to generalbroadcast channel 2.E. Ardestanizadeh et. al. (2008) used feedback from Bob to Alice toenhance secrecy rate 3.In this work we use previous secret messages as secret key toenhance the secrecy rate.

1H. Yamamoto, “Rate-distortion Theory For The Shannon Cipher System,” IEEETrans. on Info. Theory,May 1997.

2W. Kang, N. Liu,“Wiretap Channel with Shared Key,” 2010 Info. theoryWorkshop, Dublin, 2010.

3E. Ardestanizadeh et. al., “Wiretap Channel With Secure Rate-LimitedFeedback,” IEEE Trans. on Info. Theory Dec. 2009.

1Part of this work has been presented in IEEE ICC 2013 workshop on PhysicalLayer Security, Hungary



Channel Model

W ∈ W = 1, 2, . . . , 2nRs is message set, where

Rs = maxp(x)

[I(X ; Y )− I(X ; Z )] . (8)

Wm,m ≥ 1 is an independent sequence of messages to betransmitted.At time i: Xi the channel input, Yi Channel O/P at Bob & ZiChannel O/P at Eve.A mini-slot consists of n channel usesEach slot, upto λ, consists of 2 mini-slots where

λ ,

[CRs

], (9)

After λ slots each slot has only one mini-slot.



Channel Model

Wk to be transmitted in slot k consists of one or more messages Wm.Codeword for W k : X 2n

k = Xk1, . . . ,X2kn or X nk depending on the

length of the slot.Transmitter uses the secret message W k transmitted in slot k as thekey for transmitting the message in slot k + 1.



Encoder/Decoder

WiretapCoding only

n1 n2

WiretapCoding

SecretKey

0 1 2 3

k > M k + 1 Slots

Rs Rs Rs Rs 2Rs

Rs C



Encoder: To transmit W k in slot k, encoder has two parts

fs :W → X n, fd :W ×K → X n, (10)

where X ∈ X , and K set of secret keys generated and fs is theWiretap encoderDeterministic Encoder fd : Encode XOR of message and binaryversion of the key optimal usual channel encoder.Slot 1:Message encoded using the wiretap code only.Slot k: (1 < k ≤ λ), both fs and fd used simultaneously.



Encoder/Decoder

Slot 1: Decoder function at Bob is

φ1 : Y2n →W. (11)

Slot k: (k > 1), decoder is

φi : Yn ×K →W j (12)

for time slot i , with j = min(i , CRs

).Probability of error:

P(n)e = PrW 6= W (13)

where W is the decoded message.



Achievable Rate

Leakage rate is RnL = 1

n I(W ; Z 2n).

DefinitionA Leakage-rate pair (RL,R) is said to be achievable if there exists asequence of (2nR , n)-codes such that P(n)

e → 0 and lim supn→∞ RnL ≤ RL

as n→∞.

TheoremAny rate < C is achievable for all slots k ≥ λ.

Slot 1: Alice picks message W1 from W and transmits this messageusing (n, 2nRs )-Code.Slot 2: using the previous message, W 1 = W1, as a key (with keyrate Rk = Rs) Alice transmits message W 2 = (W21,W22), whereW21 = W2,W22 = W3 are taken from the iid sequence Wk , k ≥ 1.



Coding Scheme Contd.

1st message W21 is encoded to X n21 using wiretap code.

2nd message W22 is first encrypted to produce the cipher usingone-time pad with the previous message as secret key, i.e.,W22 = W22

⊕W1

We encode this encrypted message to X n22 using a point-to-point

optimal channel codeWe continue this till λ− 1 slots. In slot λ− 1, we transmit message(Wλ−1,1,Wλ−1,2, ...,Wλ−1,λ−1).Total rate:

12 (Rs + (λ− 1)Rs) =

12 (Rs + C) . (14)

Bob (Decoder): In slot k, (for 1 < k < λ) Y nk1 is decoded via usual

wiretap decoding while Y nk2 is decoded first by the channel decoder

and then XORed with Wk−1



Error Analysis

εn = message error probability for the wiretap encoder and δn =message error probability due to the channel encoder for Wk

Slot k: (1 < k < λ− 1), P(W k 6= Wk) ≤ Pr(Error in decodingWk1) + Pr(Error in decoding Wk2) + Pr(Error in decodingW k−1) ≤ kεn + (k − 1)δn.Error upper bound increases with kRestarting (as in slot 1) after some k slots ( > λ) will ensure thatP(W k 6= Wk)→ 0 as n→∞.



Leakage Rate Analysis

We show1n I(W k ; Z n

1 ,Z 2n2 , ...,Z 2n

k )→ 0 (15)

as n→∞Slot 1: Wire-tap coding is used, hence 1

n I(W 1; Z n1 )→ 0, as n→∞.

slot 2:1n I(W 1; Z n

1 ,Z 2n2 )→ 0 (16)

We use mathematical induction to show that1n I(W m; Z n

1 ,Z 2n2 , ...,Z 2n

k+1)→ 0 for all m ≤ k + 1, k ≥ 1



Strong Secrecy

Same secrecy rate can be achieved even with strong secrecy.Use Resolvability based codes4 in the first slot.In the subsequent blocks we use both the stochastic encoder and thedeterministic encoder

4Bloch, M and Laneman, N, “Strong Secrecy from Channel Resolvability,” IEEETransactions on Information Theory, vol. 51, no. 1, pp. 44–55, May 2011.



Wiretap Channel with Secret Key Buffer

We now consider the following model

Alice PY ,Z |X (.|.)

Bob

Eve

- -

-

-

-

-

W k

W k

RL

X k

Y k

Z k

-

?Bk Rk

Figure : Wiretap Channel with secret key buffer



In slot k message W k is stored in key buffer.Bk=# of bits in buffer in beginning of slot k,Rk=# of bits used as secret key in slot k, then

Bk+1 = Bk + |W k | − Rk , where |W k | = # of bits in|W k |. (17)

In slot k, (k − 1)Rsn bits from buffer removed, kRsn bits added.Message W k = (W k,1,W k , 2) transmitter s.t. W k,1 via wiretapcoding, W k,2 via secret key.Key buffer used as a FIFO queue, also Bk∞ as k →∞.



Aim is to make secrecy criterion stronger:

1n I(W k , . . . ,W k−N1 ; Z 1, . . . ,Z k

)→ 0 as n→∞. (18)

for some k > N1, N1 is arbitrarily large integer.We now have the following result

TheoremThe secrecy capacity of our coding-decoding scheme is C and it satiafies(18) for any N1 ≥ 0, for all k large enough.

We use the oldest key bits in buffer (not more than MC bits), afterk ≥ N2 we use key bits from W 1,W 2, . . . ,W k−N1−1 for messagesW k ,W k−1, . . . ,W k−N1 .can extend to strong secrecy by using resolvability based coding.



AWGN Slow Fading Wiretap Channel

We consider Fading channel

Yi =∼H Xi + N1i , Zi =

∼G Xi + N2i , (19)

Xi channel i/p, Nki ∼ N (0, σ2k), k = 1, 2 and H = |

∼H|2, G = |∼G |2.

we define C(P(H,G)) = 12 log

(1 + HP(H,G)

σ21

), Ce(P(H,G)) =

12 log

(1 + GP(H,G)

σ22

),

Now we have following result

TheoremThe secrecy rate

Cs = EH [C(P(H))] (20)

is achievable if Pr(Hk > Gk) > 0, where P(H) = P(H,G) is thewater-filling power policy for Alice → Bob channel.



Fading Wiretap Channel: No CSI of Eve

Here we assume the coherence time of (Hk ,Gk) is much smallerthan the duration n of a minislotWe use the coding-decoding scheme developed earlier in firstminislot with secrecy rate Rs , whereRs = 1

2 EH,G [C(P(H,G))− Ce(P(H,G))] .

We have the following proposition

PropositionSecrecy capacity equal to the main channel capacity without CSI of Eveat the transmitter

C =12 EH

[log(

1 +HP(H)

σ21

)](21)

is achievable subject to power constraint EH [P(H)] ≤ P, where P(H) isthe waterfilling policy.



Outline

1 Introduction









Multiple Access Channel with Eve1

Gaussian MAC with eavesdropper studied by Yener and Tekin(IEEEtrans 2008)Co-operative jamming improves secrecy sum-rateFading MAC with full CSI of eavesdropper studied by Yener andTekinIn general, Secrecy Capacity region for MAC not known.

1Part of this work has been presented in IEEE SPCOM 2012, Bangalore, India



Multiple Access Channel with Eve

Security if measured by the equivocation rate for each userR(n)

1e = 1n H(W1|Z n)

R(n)2e = 1

n H(W2|Z n)

R(n)1e + R(n)

2e = 1n H(W1W2|Z n)

Reliability: P(n)e1 = PrW1 6= W1,P(n)

e1 = PrW1 6= W1A rate-equivocation pair (R1,R2,R1e ,R2e) is achievable if ∃ asequence of message sets W1n and W2n and encoder-decoder pairs(f1n, f2n, gn) s.t. P(n)

e → 0 as n→∞ and equivocation rates satisfyR1e ≤ lim infn→∞ R(n)

1e ,R2e ≤ lim infn→∞ R(n)2e ,



MAC with eavesdropper: Secrecy rate region

The best known rate region for DM - MAC is as followsR1 ≤ [I(U1; Y | U2)− I(U1; Z )]+ (22)

R2 ≤ [I(U2; Y | U1)− I(U2; Z )]+ (23)

R1 + R2 ≤ [I(U1,U2; Y )− I(U1,U2; Z )]+ (24)

wherep(x1, x2, u1, u2, y , z) = p(u1)p(x1 | u1)p(u2)p(x2 | u2)p(y , z | x1, x2)

The rate region for Gaussian MAC and fading MAC can be obtainedfrom this rate region.



Fading MAC: Secrecy rate region

Rate region for fading MAC is as follows(Yener and Tekin)

R1 ≤ Eh,g

[log (1 + h1P1(h, g))(1 + g2P2(h, g))

1 + g1P1(h, g) + g2P2(h, g)

]+,

R2 ≤ Eh,g

[log (1 + g1P1(h, g))(1 + h2P2(h, g))

1 + g1P1(h, g) + g2P2(h, g)

]+,

R1 + R2 ≤ Eh,g

[log 1 + h1P1(h, g) + h2P2(h, g)

1 + g1P1(h, g) + g2P2(h, g)

]+,

E [Pi (h, g)] ≤ Pi , i = 1, 2. Gaussian signalling is used.



Fading MAC without CSI of eve

In passive attack of Eavesdropper, CSI cannot be estimatedIn single user wiretap channel, H.E Elgamal(IEEE trans 2008)reports secrecy capacity with optimal power control with mainchannel CSI onlyMathew Bloch et. al(IEEE trans 2008) studied outage analysis ofsingle user slow fading channel with imperfect CSI of eve.We study fading MAC which achieve secrecy sum rate withoutknowing CSI of eve.



Fading MAC without CSI of eve: Optimal powerallocation

φsx1,x2

= 1 + s1x1 + s2x2 where s is the channel state (h or g) and xkis the power used.

TheoremFor a given power control policy Pk(h), k = 1, 2, the following secrecysum-rate

Eh,g

[

log(φh

P1,P2+ φh

Q1,Q2− 1

φgP1,P2

+ φgQ1,Q2

− 1

)(φg

Q1,Q2

φhQ1,Q2

)]+ (25)

is achievable, subject to power constraint Eh,g [Pk(h)] ≤ Pk , k = 1, 2.

Optimal policy is not available in closed form. Can be computednumerically.



Fading MAC without CSI of eve: Optimal powercontrol with Cooperative Jamming

Pk(h), k = 1, 2, policy when users transmit and Qk(h),whenusers jamCooperative Jamming substantially improves secrecy sum-rate.



Without CSI of Eve: ON/OFF power control

ON/OFF power control is a simple threshold based scheme asfollows:

h1 > τ1, h2 > τ2 : Both transmit;hi > τi , hj < τj , j 6= i : User-i transmits;h1 < τ1, h2 < τ2 : No user transmits.

Optimize the secrecy sum-rate over τ1, τ2

We also employ co-operative jamming over ON/OFF scheme



Fading MAC Without CSI of Eve: Simulationresults

0 5 10 15 20 25 30 350

0.5

1

1.5

2

2.5

SNR (dB)

Sec

recy

Sum

−R

ate

(bits

/cha

nnel

use

)

Opt with full CSIOpt with only Rx CSION/OFF with only Rx CSICJ Opt with full CSICJ Opt with only Rx CSICJ ON/OFF with only Rx CSI



Outline

1 Introduction









Fading MAC without security constraint

Power allocation in Fading MAC well knownTse & Hanly assumes Global CSI available at TxThey use Polymatroid theoryH El Gamal et al use Game theory for distributed power controlThey also assume Global CSIAltman et al assume partial CSI, use Bayesian Game theory



We consider K - user fading, discrete time MAC

Y (t) =K∑

i=1Hi (t)Xi (t) + η(t), (26)

Xi (t) channel i/p, η(t) is WGN ∼ N (0, σ2)

Fading process Hi (t) ∈ Hi , h(1)i , . . . , h(M)

i for user i with pmfρ(1)

i , . . . , ρ(M)i



User i transmits at fixed rate ri

Receiver uses succesive decoding in following orderArranges users in increasing order of channel gainsLet π be permutation on K s.t. hπ(1) ≥ hπ(2) ≥ . . . , fhπ(K)

Receiver sends ACK to user π(i) if

rπ(i) ≤12 log

(1 +

hπ(i)Pπ(i)(hπ(i))

σ2 +∑K

j=i+1 hπ(j)Pπ(j)(hπ(j))

)(27)

(28)



Reward of user i in time slot t

ω(t)i

(a(t)

i , hi (t))

=

1, If user i receives ACK0, else

Average utility is

νi (Pi ,P−i ) , lim supT→∞

1T

T∑t=1

ω(t)i (Pi ,Hi (t)) = E

[ω

(t)i (Pi ,Hi )

],

(29)

By WLLN Probability of successful transmission = νi (Pi ,P−i ).



Goal of each user

maxai

νi (Pi ,P−i ) (30)

subject to ai ∈ Pi (31)

This is a stochastic game G = 〈K,Pi , νi〉.We would like to find ε-Coarse Correlated Equilibrium (CCE),defined as

DefinitionIf a distribution ψ on P satisfy the following inequality

Ea∼ψ[Cπ(i) (a)

]≤ Ea∼ψ

[Cπ(i)

(aπ(i), a−π(i)

)]+ ε (32)

then it is called ε−coarse correlated equilibrium for each userπ(i), i ∈ 1, . . . ,K, and every action aπ(i), aπ(i) ∈ Pπ(i).



We use Multiplicative Weight No-regret algorithm to obtain CCE

1: ω(t)i ← 1, i = 1, 2, . . . ,K

2: User i : Choose action w .p. Φ(t)i =

ω(t)i (ai )∑

a∈Piω

(t)i (ai )

3: Time t: User i receives average utility for choosing aiν

(t)i = Ea−i∼−i [C(ai , a−1)]

4: Update the weight5: ω

(t+1)i = ω

(t)i (ai )(1− ε)c(t)(ai )

i

6: Time t + 1: Calculate ψt =∏K

i=1 Φ(t)i



Pareto Optimal Points

We first define

Ω(a) =K∑

i=1γiνi (a) (33)

Goal is to obtain

maxa

Ω(a)

subject to a ∈ P. (34)

The equilibrium obtained is Pareto optimal, defined as

Pareto optimalityAn action profile a ∈ P is Pareto optimal if there does not exist anotherprofile a such that νi (a) ≥ νi (a), ∀ i ∈ K and νj(a) > νi (a) for j 6= i .

We use distributed algorithm to obtain Pareto points



Distributed Algorithm I

1: OUTER ITER = 1,User i: choose ai ∈ Pi uniformly.2: Use ai for T time slots.3: Update weight of each user i4: Ω(a)←

∑Ki=1 γi

(1T∑T

t=1 ω(t)i (ai ,Hi (t))

)5: After T slots: INNER ITER = 1, User i experiments w .p. Pexp6: w .p ε choose a′i 6= ai , a′i ∈ Pi7: w .p. 1− ε8: choose a′i 6= ai s.t. hi with high αi gets higher power level9: If αi same for all hi , then higher value of channel state gets

higherpower level.

10: Call new action ai11: User i : use ai for T time slots.



Distributed Algorithm II

12: User j : use aj for T time slots, j 6= i .13: User i : find Ω(ai , a−i )14: if Ω(ai , a−i ) > Ω(ai , a−i ) then15: ai ← ai16: Pbenchmark = Ω(ai , a−i )17: else18: Randomly select another action19: end if20: INNER ITER = INNER ITER + 1, goto step 5 till

INNER ITER = MAX21: OUTER ITER = OUTER ITER + 1, goto step 1 till

OUTER ITER = M



Nash Bargaining Solution

consider the strategic game G = 〈K,Pi , νi〉 where userscooperate.we need to specify disagreement outcomeWe define the set of all possible utulities as

V = (ν1(a), . . . , νK (a)) : a ∈ P (35)

disagreement action vector as δ = (δ1, . . . , δK ) whereδi = ξi (∆), i = 1, . . . ,K , and where ∆ is disagreement action.Bargaining problem is (V, δ)

Aim is to find some function, ξ, that specifies a bargaining outcomeξ(V, δ) for every bargaining problem (V, δ) that satisfies thefollowing axioms.



A-1 Pareto efficiency :A-2 Symmetry :A3 Invariance:

A-4 Independence of irrelevant alternatives:

Existence of Nash EquilibriumThere exists a unique bargaining solution (provided feasible region isnon-empty) that satisfy A1− A4 and it is given by the solution of thefollowing optimization problema

max (ν1 − δ1)(ν2 − δ2)

subject to νi ≥ δi , i = 1, 2(ν1, ν2) ∈ V (36)

aNash Jr, John F, “The bargaining problem,” Econometrica, 1950.



Generalized to K−player

maxK∏

i=1(νi − δi ) subject to νi ≥ δi , i = 1, . . . ,K (ν1, . . . ,K ) ∈ V.

(37)

Since the set V is convex, hence NBS is proportionally fair also.



Fading MAC-WT

The channel model is

Y (t) =K∑

i=1Hi (t)Xi (t) + ηb(t) (38)

Z (t) =K∑

i=1Gi (t)Xi (t) + ηe(t) (39)

ηb(t), ηe(t) ∼ N (0, 1), Hi (t) , |Hi (t)| and Gi (t) , |Gi (t)|

Hi (t) ∈ Hi , h(1)i , . . . , h(M(H)

i )i with probabilities (w .p.)

α(1)i , . . . , α

(M(H)i )

i

Gi (t) ∈ G , g (1)i , . . . , g (M(G)

i )i w .p. β(1)

i , . . . , β(M(G)

i )i



Power can be choosen from Pi , P(1)i , . . . ,P(M)

i subject to

K∑j=1

α(j)i β

(j)i P(j)

i ≤ P i . (40)

Receiver decodes via succesive decoding in the increasing order ofmain channel gain.We define

Cb(Pπ(i),Hπ(i)

),

12 log

(1 +

Hπ(i)Pπ(i)(Hπ(i),Gπ(i))

1 +∑K

j=i+1 Hπ(j)Pπ(j)(Hπ(j),Gπ(j))

)(41)

Ce(Pπ(i),Gπ(i)

),

12 log

(1 +

Gπ(i)Pπ(i)(Hπ(i),Gπ(i))

1 +∑K

j 6=i Gπ(j)Pπ(j)(Hπ(j),Gπ(j))

)(42)



Receiver sends ACK to user π(i) if

rπ(i) ≤(Cb(Pπ(i), hπ(i), gπ(i))− Ce(Pπ(i), hπ(i), gπ(i))

)+ (43)

Feasible Action Space

Pi =

Pi = (P(1)i , . . . ,P(M)

i ) : P(k)i ∈ p(1)

i , . . . , p(M)i ,

M∑j=1

αi (j)β(j)i P(j)

i ≤ P i

(44)



Fading-MAC without CSI of Eve

Outage based metric

PSO ,Pr

ri > Cb

(Pπ(i),Hπ(i)

)− Ce

(Pπ(i),Gπ(i)

)(45)

We define

Γ =1

2rπ(i)

(1 +

hπ(i)pπ(i)

1 +∑K

j=i+1 hπ(j)pπ(j)

)(46)

Hence

PSO = Pr

1 +

gπ(i)pπ(i)

1 +∑K

j 6=i gπ(j)pπ(j)> Γ

(47)

Now the receiver send ACK if PSo < ε, else NACK.

Utility is now defined as

ωπ(i)

(a(t)π(i), hπ(i)(t)

)= 1PS

O<ε(48)



Simulation Results: MAC without Eve

Figure : Comparision of Sum-Rate in Fading MAC without Eve: Fixed ratetransmission



Figure : Sum-rate comparision: Multiple rate (3 states)



Figure : Comparision of Fairness b/w NBS and PP



Figure : Fading MAC without Eve: Comparision with other schemes (Multiplerate transmission)



Figure : Fading MAC-WT: Comparision with full CSI and other scheme(Multiple rate)



Outline

1 Introduction









Discrete Memoryless Wiretap Channel1

Encoder 1

Encoder 2

PY ,Z |X1,X2 (.|.)Bob

Eve

- -

--

-

-

-

-

W (1)

W (2)

(W (1), W (2))

(R(1)L ,R(2)

L )

X(1)

X(2)

Y n

Z n

Figure : Discrete Memoryless Multiple Access Wiretap Channel

1Part of this work has been presented in IEEE WCNC 2015, New Orleans, USAShah&Sharma — Enhancing Secrecy Rates and Resourse Allocation in Single-User & Multi-User Fading Wiretap Channels 64/84


Codebook for MAC-WT

MAC-WT2-user MAC-WT (X1 ×X2, p(y , z |x1, x2),Y,Z), X1,X2,Y,Z arefinite sets and p(y , z |x1, x2) is the collection of t.p.m2 users want to send messages W (1) and W (2) to Bob reliably,keeping Eve ignorant(2nR1 , 2nR2 , n) codebook consists of

Message sets W(1) and W(2) of cardinality 2nR1 and 2nR2

Two independent Messages W (1) and W (2), uniformally distributedover W(1) and W(2)

stochastic encoders,

fi :W(i) → X ni i = 1, 2, (49)



Codebook for MAC-WT

MAC-WT4) Decoder at Bob:

g : Yn →W(1) ×W(2). (50)

average probability of error at Bob is

P(n)e , P

(W (1), W (2)

)6=(

W (1),W (2))

. (51)

Leakage Rate is

R(n)L =

1n I(W (1),W (2); Z n), (52)



Discrete Memoryless MACCapacity region

Capacity region of DM-MAC

Capacity Region for DM-MACClosure of convex hull of all rate pair (R1,R2) satisfying

R1 ≤ I(X1; Y |X2),

R2 ≤ I(X2; Y |X1),

R1 + R2 ≤ I(X1,X2; Y ), (53)

for some X1, X2 independent.



Discrete Memoryless MAC-WTInner Bound

Secrecy Capacity region not known for DM-MAC-WTInner bound was proposed by Tekin and Yener 1

Secrecy-Rate Region for MAC

Rates (R1,R2) are achievable with lim supn→∞ R(n)L = 0, if there exist

independent random variables (X1,X2) satisfying

R1 ≤ I(X1; Y |X2)− I(X1; Z ),

R2 ≤ I(X2; Y |X1)− I(X2; Z ),

R1 + R2 ≤ I(X1,X2; Y )− I(X1; Z )− I(X2; Z ). (54)

1Tekin, Ender and Yener, Aylin, “The Gaussian multiple access wire-tap channel,”IEEE Transactions on Information Theory, 2008.



Notation

W (i)k denotes message of user i in slot k.

X(i) denotes n-length codeword transmitted by user i .

W (i)k = (W (i)

k,1,W(i)k,2), W (i)

k,1 being first part and W (i)k,2 the second

part.

X(i) = (X(i)1 ,X(i)

2 ), X(i)1 is n1-length codeword for W (i)

k,1 and X(i)2 is

codeword for W (i)k,2



Encoder modified

Also the encoders will now be defined as

Encoders

f s1 :W(1) → X n1

1 , f d1 :W(1) ×K1 → X n2

1 (55)

f s2 :W(2) → X n1

2 , f d2 :W(2) ×K2 → X n2

2 , (56)

where Xi ∈ Xi , i = 1, 2,Ki , i = 1, 2 are the sets of secret keys generated for the respectiveuser, f s

i , i = 1, 2 are the wiretap encoders,f di , i = 1, 2 are the deterministic encoders , n1 + n2 = n



Coding Scheme

Wiretap Cod-ing

n1 n2

Wiretap Cod-ing Secret Key

0 1 2 3

Slots

(R1, R2) (R1, R2) (R∗1 , R∗2 )

Fix distributions pX1 , pX2 & consider time slotted system

Initialization: Take n1 = n2 = n/2 & in Slot 1 User i transmits W (i)1

via Wiretap Coding

Slot 2: Each user selects two messages (W (i)21 ,W

(i)22 ) ≡W (i)

2 , i = 1, 2

(W (1)21 ,W

(2)21 ) transmitted via Wiretap coding

For W (i)22 , first take XOR with previous message, then transmit

W (i)22 ⊕W (i)

1 via usual MAC coding



Coding Scheme

Wiretap Cod-ing

n1 n2


0 1 2 3 k > λ k + 1

Slots

(R1, R2) (R1, R2) (R1, R2) (2R1, 2R2) (R∗1 , R∗2 )

Slot 3: Transmit first part of message via Wiretap coding (WTC)2nd part by

⊕with previous message, hence rate doubled.

Slot λ: We define

Time slot of Saturation

λi ,

⌈I(Xi ; Y |Xj)

I(Xi ; Y |Xj)− I(Xi ; Z )

⌉, (i 6= j , i , j ∈ 1, 2)

λ , maxλ1, λ2+ 1 (57)



Coding Scheme

Wiretap Cod-ing

n1 n2


0 1 2 3 k > λ k + 1

Slots

(R1, R2) (R1, R2) (R1, R2) (2R1, 2R2) (R1, R2) (R∗1 , R∗2 )

After some slot λ∗ > λ sum-rate will get saturated

R1 + R2 ≤ I(X1,X2; Y ). (58)

Slot k (k > λ): For W (i)k = (W (i)

k,1,W(i)k,2), i = 1, 2, use WTC for

(W (1)

k,1,W(2)

k,1) & for the 2nd part, encode W (i)k,2 ⊕W (i)

k−1 & transmitat Shannon rate (R∗1 ,R∗2 ).Then we make n2/n1 large enough to achieve close to ShannonCapacity for the whole slot.



Secrecy Measures

We consider the following Secrecy measure

Secrecy Measure

I(W (1),W (2); Z n) ≤ nε. (59)

We note that I(W (1),W (2); Z n) ≤ I(W (1); Z n|X n2 ) + I(W (2); Z n|X n

1 )

k th slotLet the leakage rate inequality

I(W (1)

l ,W (2)

l ; Z n1 , . . . ,Z n

k ) ≤ 2n1ε, for l = 1, . . . , k (60)

hold for k, then it holds for k + 1 slot also (Mathematical Induction).



Enhancing rate in Strong Sense

The same rate region can be achieved in Strong Secrecy sense.We use resolvability based coding scheme in slot 1 .In subsequent slots use resolvability coding scheme instead ofwiretap coding.We can achieve Shannon capacity region as secrecy rate regionsatisfying

Strong Secrecy

lim supn→∞

I(W (1)

k ,W (2)

k ; Z n1 ,Z n

2 , . . . ,Z nk ) = 0, (61)

as n→∞.



MAC with Secret Key Buffer

We consider the following model

Encoder 1

Encoder 2

PY ,Z |X1,X2 (.|.)

Bob

Eve

- -

--

-

-

-

-

W (1)k

W (2)k

(W (1)k , W (2)

k )

(R(1)L ,R(2)

L )

X (1)

k

X (2)

k

Y k

Z k

-

-?

B(1)k

B(2)k

6

Figure : Discrete Memoryless Multiple Access Wiretap Channel withsecret key buffers



MAC with Secret key buffer

By using very old messages as keys we have the following result

TheoremThe secrecy-rate region of a DM-MAC-WT equals the usual Shannoncapacity region of the MAC while satisfying

I(W (1)

k ,W (1)

k−1, . . . ,W(1)

k−N1; Z 1, . . . ,Z k |X

(2)

k ) ≤ n1ε,

I(W (2)

k ,W (2)

k−1, . . . ,W(1)

k−N1; Z 1, . . . ,Z k |X

(1)

k ) ≤ n1ε,

I(W (1)

k ,W (2)

k , . . . ,W (1)

k−N1,W (2)

k−N1; Z 1, . . . ,Z k) ≤ 2n1ε. (62)



Fading MAC-WT

Channel model

Y = H1X2 + H2X2 + N1 Z = G1X1 + G2X2 + N2, (63)

Notation

C1(P1(H,G)) ,12 log

(1 +

H1P1(H,G)

σ21

)C2(P2(H,G)) ,

12 log

(1 +

H2P1(H,G)

σ21

)C e

1 (P1(H,G)) ,12 log

(1 +

G1P1(H,G)

σ22 + G2P2(H,G)

)C e

2 (P2(H,G)) ,12 log

(1 +

G2P2(H,G)

σ22 + G1P1(H,G)

)C(P1(H,G),P2(H,G)) ,

12 log

(1 +

H1P1(H,G) + H2P2(H,G)

σ21

)(64)



We have the following result If Pr(Hik > Gik) > 0, i = 1, 2, and allthe channel gains are available at all the transmitters, then thefollowing long term average rates that maintain the leakage rates(62), are achievable:

R1 ≤12 EH,G [C1 (P1(H))] ,

R2 ≤12 EH,G [C2 (P2(H))] ,

R1 + R2 ≤12 EH,G [C (P1(H),P2(H))] . (65)

where P is any policy that satisfies avg. power constraint. If onlyBob knows all the channel states but not the transmitters, then(R1,R2) satisfies (65) with Pi (H,G) ≡ P i , i = 1, 2.



Outline

1 Introduction









Conclusions

Proposed Probability of error as metric of securityAchieved Shannon capacity in wiretap channel via previous messagesPower allocation in fading MAC-WT without CSI of EveExtended CJ to enhance secrecy sum-rate in fading MAC-WTUsed Algorithmic Game theory to allocate resources in fading MACwith individual CSI onlyExtended the same algorithms to fading MAC-WTExtended coding scheme of using previous messages as key to MACto achieve Shannon Capacity region



Future Work

Develop practical codes based on probability of error at Eve assecurity metricDevelop practical codes for using previous messages as secret key toenhance secrecy ratesResource allocation with continous state space



References I



Thanks

Questions ?

Shahid M [email protected]


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Enhancing Secrecy Rates and Resourse Allocation in Single-User …shahidshah.weebly.com/uploads/1/1/2/2/11221304/colloqium... · 2018-09-07 · (Thesis Colloqium) Shahid Mehraj Shah