Enhancing Secrecy Rates and Resourse Allocation in Single-User & Multi-User Fading Wiretap Channels

Transcription

1 Enhancing Secrecy Rates and Resourse Allocation 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

2 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 Shah&Sharma Enhancing Secrecy Rates and Resourse Allocation in Single-User & Multi-User Fading Wiretap Channels 2/84

3 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 Shah&Sharma Enhancing Secrecy Rates and Resourse Allocation in Single-User & Multi-User Fading Wiretap Channels 3/84

4 How it began Conventional security measure taken at higher layer Cryptographic techniques depend on finiteness of computational power while Information theoretic security (ITS) promises provable security Wyner in 1975 Proposed a wiretap coding scheme for degraded Wiretap Channel 1 Security achieved at the cost of rate loss Wyner s result extended in several ways: BCC, MAC, IC, RC 2 Cost paid in each channel model is: Rate Loss In most of the work in ITS, Eve s complete channel knowledge is assumed Because of passive nature of Eve, this assumption not practical 1 A. Wyner, The wire-tap channel, Bell Syst. Tech. J., Y. Liang, H.V. Poor, S. Shamai, Information theoretic security, Foundations and Trends in Communications and Information Theory, Shah&Sharma Enhancing Secrecy Rates and Resourse Allocation in Single-User & Multi-User Fading Wiretap Channels 4/84

5 Motivation We try to address these two issues in our work To mitigate rate loss we propose alternate measure of secrecy and novel coding scheme. We use optimization techniques to allocate power with No CSI of Eve in MAC We also use Game theoretic learning to allocate resources with less information about Eve s channel state. Shah&Sharma Enhancing Secrecy Rates and Resourse Allocation in Single-User & Multi-User Fading Wiretap Channels 5/84

6 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 Shah&Sharma Enhancing Secrecy Rates and Resourse Allocation in Single-User & Multi-User Fading Wiretap Channels 6/84

7 Channel Model: AWGN Wiretap Channel 1 Message W {1,..., 2 nr } to be transmitted securely & is encoded to X n = {X 1,..., X n }. Bob receives Y i = X i + N 1i, N 1i N (0, σ1 2 ), iid, Eve receives Z i = X i + N 2i, N 2i N (0, σ2 2), iid, and σ2 1 < σ2 2 Let C 1 = 0.5 log(1 + P/σ1 2) and C 2 = 0.5 log(1 + P/σ2 2) Equivocation Based definition: (R, R e) is achievable if W n with W n = 2 nr and encoder - decoder pairs (f n, g n) such that P e (n) 0, and the equivocation rate 1 lim n H(W Z n ) R n e Code design to satisfy equivocation rate is difficult. 2 To confuse Eve random messages are sent, which decreases the rate. 2 Klinc, D., Ha, J., McLaughlin, S. W., Barros, J., & Kwak, B. J. LDPC codes for the Gaussian wiretap channel. IEEE Trans. on Info. Forensics and Sec., (2011) 1 Part of this work has been presented in IEEE ICCS 2012, Singapore Shah&Sharma Enhancing Secrecy Rates and Resourse Allocation in Single-User & Multi-User Fading Wiretap Channels 7/84

8 New Notion Natural Definition of secrecy: Probability of error for receiver 0 and 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 Intended receiver,and Eve getting no message, i.e. no common information. Each message for Eve confused with same number of codewords as in Equivocation based approach Shah&Sharma Enhancing Secrecy Rates and Resourse Allocation in Single-User & Multi-User Fading Wiretap Channels 8/84

9 New Notion: Achieving Shannon Capacity Proposition All rates R < C 1 are achievable such that P n e (B) 0, P n e (E) 1 as n. where P n e (B) = Probability of decoding error at Bob, P n e (E) = Probability of decoding error at Eve Coding Scheme Region C 2 < R < C 1 : generate n length iid Gaussian codewords with X N (0, P) R < C 1 ensures P n e (B) 0 R > C 2, by strong converse, P n e (E) 1. To achieve rate R < C 2, select Gaussian P X with power < P such that I(X; Y ) > R > I(X; Z) Shah&Sharma Enhancing Secrecy Rates and Resourse Allocation in Single-User & Multi-User Fading Wiretap Channels 9/84

10 Performance (via AEP decoder at Eve) N be the number of codewords other than that of message 1 that are jointly typical with Z n E[N] = (2 nr 1)2 ni(x;z) and P n e (E) 2 n(r I(X;Z)) P n e (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 x 1 (1),..., x n (1) is transmitted, decode as message ˆm if n ˆm = arg min (Z k x k ( ˆm)) 2. (1) k=1 Eve will confuse with 2 n(r C2) codewords. Max confusion as in AEP decoder. AEP and ML decoder best for Eve. Shah&Sharma Enhancing Secrecy Rates and Resourse Allocation in Single-User & Multi-User Fading Wiretap Channels 10/84

11 New Notion: Relation with Equivocation When C 2 < R < C 1, P n e (B) 0, and P n e (E) 1. Fano s Inequality gives: For Bob 1 n H(W Y n ) H(Pn e (B)) + Pe n (B)R (2) n Lower bound: For Eve H(W Z n ) φ (π(w Z n )) (3) where φ is a piecewise linear, continuous, non-decreasing, convex function. 2. π(w Z n ) is the average probability of error for the MAP decoder at Eve. From Arimoto s lower bound π(w Z n ) 1 e n(e0(ρ,p) ρr), 0 ρ 1. (4) 2 M. Feder and N. Merhav, Relations Between Entropy and Error Probability, IEEE Trans. Inform. Theory. Jan Shah&Sharma Enhancing Secrecy Rates and Resourse Allocation in Single-User & Multi-User Fading Wiretap Channels 11/84

12 Numerical Example For σ 2 1 = 0.1, σ2 2 = 1.5, and P = 20dB, Pn e (B) and P n e (E) are plotted for n = 50, 100 and 200. For P n e (B) Gallagers random coding bound and for P n e (E) Arimoto s lower bound are plotted Probability of Error Bob N=50 Eve N=50 Bob N=100 Eve N=100 Bob N=200 Eve N=200 P=20dB, σ 1 2 =0.1, σ 2 2 =1.5, R= Fraction of SNR(α) Shah&Sharma Enhancing Secrecy Rates and Resourse Allocation in Single-User & Multi-User Fading Wiretap Channels 12/84

13 Fading Channel For the fading channel, The capacity is ([gopala2008]): C s = 0 r Y i = h i X i + N 1i, Z i = g i X i + N 2i, (5) [log(1 + qp (q, r)) log(1 + rp (q, r))]f (q)f (r)dqdr where q(i) = h(i) 2 and r(i) = g(i) 2 E[P (q, r)] = P, Proposition All rates R < C 1 are achievable such that Pe n (B) 0 and Pe n (E) 1 where. C 1 = sup P(q,r) 0 r (6) [log(1 + qp(q, r))]f (q)f (r)dqdr. (7) Shah&Sharma Enhancing Secrecy Rates and Resourse Allocation in Single-User & Multi-User Fading Wiretap Channels 13/84

14 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 Shah&Sharma Enhancing Secrecy Rates and Resourse Allocation in Single-User & Multi-User Fading Wiretap Channels 14/84

15 Discrete Memoryless Wiretap Channel 1 Yamamoto in 1997 used Wiretap coding and secret key to enhance secrecy rate in degraded Wiretap channel. 1 Kang and Liu (2010) extended Yamamoto s work to general broadcast channel 2. E. Ardestanizadeh et. al. (2008) used feedback from Bob to Alice to enhance secrecy rate 3. In this work we use previous secret messages as secret key to enhance the secrecy rate. 1 H. Yamamoto, Rate-distortion Theory For The Shannon Cipher System, IEEE Trans. on Info. Theory,May W. Kang, N. Liu, Wiretap Channel with Shared Key, 2010 Info. theory Workshop, Dublin, E. Ardestanizadeh et. al., Wiretap Channel With Secure Rate-Limited Feedback, IEEE Trans. on Info. Theory Dec Part of this work has been presented in IEEE ICC 2013 workshop on Physical Layer Security, Hungary Shah&Sharma Enhancing Secrecy Rates and Resourse Allocation in Single-User & Multi-User Fading Wiretap Channels 15/84

16 Channel Model W W = {1, 2,..., 2 nrs } is message set, where R s = max [I(X; Y ) I(X; Z)]. (8) p(x) {W m, m 1} is an independent sequence of messages to be transmitted. At time i: X i the channel input, Y i Channel O/P at Bob & Z i Channel O/P at Eve. A mini-slot consists of n channel uses Each slot, upto λ, consists of 2 mini-slots where [ ] C λ, (9) After λ slots each slot has only one mini-slot. R s Shah&Sharma Enhancing Secrecy Rates and Resourse Allocation in Single-User & Multi-User Fading Wiretap Channels 16/84

17 Channel Model W k to be transmitted in slot k consists of one or more messages W m. Codeword for W k : Xk 2n = {X k1,..., X 2kn } or Xk n depending on the length of the slot. Transmitter uses the secret message W k transmitted in slot k as the key for transmitting the message in slot k + 1. Shah&Sharma Enhancing Secrecy Rates and Resourse Allocation in Single-User & Multi-User Fading Wiretap Channels 17/84

18 Encoder/Decoder R s R s R s R s 2R s n 1 n Wiretap Coding only Wiretap Coding Secret Key k > M k + 1 Slots R s C Shah&Sharma Enhancing Secrecy Rates and Resourse Allocation in Single-User & Multi-User Fading Wiretap Channels 18/84

19 Encoder: To transmit W k in slot k, encoder has two parts f s : W X n, f d : W K X n, (10) where X X, and K set of secret keys generated and f s is the Wiretap encoder Deterministic Encoder f d : Encode XOR of message and binary version of the key optimal usual channel encoder. Slot 1:Message encoded using the wiretap code only. Slot k: (1 < k λ), both f s and f d used simultaneously. Shah&Sharma Enhancing Secrecy Rates and Resourse Allocation in Single-User & Multi-User Fading Wiretap Channels 19/84

20 Encoder/Decoder Slot 1: Decoder function at Bob is Slot k: (k > 1), decoder is for time slot i, with j = min(i, C R s ). Probability of error: where Ŵ is the decoded message. φ 1 : Y 2n W. (11) φ i : Y n K W j (12) P (n) e = Pr{Ŵ W } (13) Shah&Sharma Enhancing Secrecy Rates and Resourse Allocation in Single-User & Multi-User Fading Wiretap Channels 20/84

21 Achievable Rate Leakage rate is R n L = 1 n I(W ; Z 2n ). Definition A Leakage-rate pair (R L, R) is said to be achievable if there exists a sequence of (2 nr, n)-codes such that P e (n) 0 and lim sup n RL n R L as n. Theorem Any rate < C is achievable for all slots k λ. Slot 1: Alice picks message W 1 from W and transmits this message using (n, 2 nrs )-Code. Slot 2: using the previous message, W 1 = W 1, as a key (with key rate R k = R s ) Alice transmits message W 2 = (W 21, W 22 ), where W 21 = W 2, W 22 = W 3 are taken from the iid sequence {W k, k 1}. Shah&Sharma Enhancing Secrecy Rates and Resourse Allocation in Single-User & Multi-User Fading Wiretap Channels 21/84

22 Coding Scheme Contd. 1 st message W 21 is encoded to X21 n using wiretap code. 2 nd message W 22 is first encrypted to produce the cipher using one-time pad with the previous message as secret key, i.e., W 22 = W 22 W1 We encode this encrypted message to X22 n using a point-to-point optimal channel code We continue this till λ 1 slots. In slot λ 1, we transmit message (W λ 1,1, W λ 1,2,..., W λ 1,λ 1 ). Total rate: 1 2 (R s + (λ 1)R s ) = 1 2 (R s + C). (14) Bob (Decoder): In slot k, (for 1 < k < λ) Yk1 n is decoded via usual wiretap decoding while Yk2 n is decoded first by the channel decoder and then XORed with Ŵk 1 Shah&Sharma Enhancing Secrecy Rates and Resourse Allocation in Single-User & Multi-User Fading Wiretap Channels 22/84

23 Error Analysis ɛ n = message error probability for the wiretap encoder and δ n = message error probability due to the channel encoder for W k Slot k: (1 < k < λ 1), P(W k Ŵk) Pr(Error in decoding W k1 ) + Pr(Error in decoding W k2 ) + Pr(Error in decoding W k 1 ) kɛ n + (k 1)δ n. Error upper bound increases with k Restarting (as in slot 1) after some k slots ( > λ) will ensure that P(W k Ŵk) 0 as n. Shah&Sharma Enhancing Secrecy Rates and Resourse Allocation in Single-User & Multi-User Fading Wiretap Channels 23/84

24 Leakage Rate Analysis We show as n 1 n I(W k; Z n 1, Z 2n 2,..., Z 2n k ) 0 (15) Slot 1: Wire-tap coding is used, hence 1 n I(W 1; Z1 n ) 0, as n. slot 2: 1 n I(W 1; Z1 n, Z2 2n ) 0 (16) We use mathematical induction to show that 1 n I(W m; Z1 n, Z 2 2n 2n,..., Zk+1 ) 0 for all m k + 1, k 1 Shah&Sharma Enhancing Secrecy Rates and Resourse Allocation in Single-User & Multi-User Fading Wiretap Channels 24/84

25 Strong Secrecy Same secrecy rate can be achieved even with strong secrecy. Use Resolvability based codes 4 in the first slot. In the subsequent blocks we use both the stochastic encoder and the deterministic encoder 4 Bloch, M and Laneman, N, Strong Secrecy from Channel Resolvability, IEEE Transactions on Information Theory, vol. 51, no. 1, pp , May Shah&Sharma Enhancing Secrecy Rates and Resourse Allocation in Single-User & Multi-User Fading Wiretap Channels 25/84

26 Wiretap Channel with Secret Key Buffer We now consider the following model B k R k Y k Bob W k W k X k Alice P Y,Z X (..) Eve Z k R L Figure : Wiretap Channel with secret key buffer Shah&Sharma Enhancing Secrecy Rates and Resourse Allocation in Single-User & Multi-User Fading Wiretap Channels 26/84

27 In slot k message W k is stored in key buffer. B k =# of bits in buffer in beginning of slot k, R k =# of bits used as secret key in slot k, then B k+1 = B k + W k R k, where W k = # of bits in W k. (17) In slot k, (k 1)R s n bits from buffer removed, kr s n bits added. Message W k = (W k,1, W k, 2) transmitter s.t. W k,1 via wiretap coding, W k,2 via secret key. Key buffer used as a FIFO queue, also B k as k. Shah&Sharma Enhancing Secrecy Rates and Resourse Allocation in Single-User & Multi-User Fading Wiretap Channels 27/84

28 Theorem Aim is to make secrecy criterion stronger: 1 n I ( W k,..., W k N1 ; Z 1,..., Z k ) 0 as n. (18) for some k > N 1, N 1 is arbitrarily large integer. We now have the following result The secrecy capacity of our coding-decoding scheme is C and it satiafies (18) for any N 1 0, for all k large enough. We use the oldest key bits in buffer (not more than MC bits), after k N 2 we use key bits from W 1, W 2,..., W k N1 1 for messages W k, W k 1,..., W k N1. can extend to strong secrecy by using resolvability based coding. Shah&Sharma Enhancing Secrecy Rates and Resourse Allocation in Single-User & Multi-User Fading Wiretap Channels 28/84

29 AWGN Slow Fading Wiretap Channel We consider Fading channel Y i = H X i + N 1i, Z i = G X i + N 2i, (19) X i channel i/p, N ki N (0, σk 2), k = 1, 2 and H = H 2, G = G 2. ( ) we define C(P(H, G)) = 1 2 log 1 + HP(H,G), C σ1 2 e (P(H, G)) = ( ) 1 2 log 1 + GP(H,G), σ2 2 Now we have following result Theorem The secrecy rate C s = E H [C(P(H))] (20) is achievable if Pr(H k > G k ) > 0, where P(H) = P(H, G) is the water-filling power policy for Alice Bob channel. Shah&Sharma Enhancing Secrecy Rates and Resourse Allocation in Single-User & Multi-User Fading Wiretap Channels 29/84

30 Fading Wiretap Channel: No CSI of Eve Here we assume the coherence time of (H k, G k ) is much smaller than the duration n of a minislot We use the coding-decoding scheme developed earlier in first minislot with secrecy rate R s, where R s = 1 2 E H,G [C(P(H, G)) C e (P(H, G))]. We have the following proposition Proposition Secrecy capacity equal to the main channel capacity without CSI of Eve at the transmitter C = 1 [ ( 2 E H log 1 + HP(H) )] σ1 2 (21) is achievable subject to power constraint E H [P(H)] P, where P(H) is the waterfilling policy. Shah&Sharma Enhancing Secrecy Rates and Resourse Allocation in Single-User & Multi-User Fading Wiretap Channels 30/84

31 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 Shah&Sharma Enhancing Secrecy Rates and Resourse Allocation in Single-User & Multi-User Fading Wiretap Channels 31/84

32 Multiple Access Channel with Eve 1 Gaussian MAC with eavesdropper studied by Yener and Tekin(IEEE trans 2008) Co-operative jamming improves secrecy sum-rate Fading MAC with full CSI of eavesdropper studied by Yener and Tekin In general, Secrecy Capacity region for MAC not known. 1 Part of this work has been presented in IEEE SPCOM 2012, Bangalore, India Shah&Sharma Enhancing Secrecy Rates and Resourse Allocation in Single-User & Multi-User Fading Wiretap Channels 32/84

33 Multiple Access Channel with Eve Security if measured by the equivocation rate for each user R (n) 1e = 1 n H(W 1 Z n ) R (n) 2e = 1 n H(W 2 Z n ) R (n) 1e + R(n) 2e = 1 n H(W 1W 2 Z n ) Reliability: P (n) e1 = Pr{ W 1 W 1 }, P (n) e1 = Pr{ W 1 W 1 } A rate-equivocation pair (R 1, R 2, R 1e, R 2e ) is achievable if a sequence of message sets W 1n and W 2n and encoder-decoder pairs (f 1n, f 2n, g n ) s.t. P e (n) 0 as n and equivocation rates satisfy R 1e lim inf n R (n) 1e, R 2e lim inf n R (n) 2e, Shah&Sharma Enhancing Secrecy Rates and Resourse Allocation in Single-User & Multi-User Fading Wiretap Channels 33/84

34 MAC with eavesdropper: Secrecy rate region The best known rate region for DM - MAC is as follows R 1 [I(U 1 ; Y U 2 ) I(U 1 ; Z)] + (22) R 2 [I(U 2 ; Y U 1 ) I(U 2 ; Z)] + (23) R 1 + R 2 [I(U 1, U 2 ; Y ) I(U 1, U 2 ; Z)] + (24) where p(x 1, x 2, u 1, u 2, y, z) = p(u 1 )p(x 1 u 1 )p(u 2 )p(x 2 u 2 )p(y, z x 1, x 2 ) The rate region for Gaussian MAC and fading MAC can be obtained from this rate region. Shah&Sharma Enhancing Secrecy Rates and Resourse Allocation in Single-User & Multi-User Fading Wiretap Channels 34/84

35 Fading MAC: Secrecy rate region Rate region for fading MAC is as follows(yener and Tekin) { [ R 1 E h,g log (1 + h ] } + 1P 1 (h, g))(1 + g 2 P 2 (h, g)), 1 + g 1 P 1 (h, g) + g 2 P 2 (h, g) { [ R 2 E h,g log (1 + g ] } + 1P 1 (h, g))(1 + h 2 P 2 (h, g)), 1 + g 1 P 1 (h, g) + g 2 P 2 (h, g) { [ R 1 + R 2 E h,g log 1 + h ] } + 1P 1 (h, g) + h 2 P 2 (h, g), 1 + g 1 P 1 (h, g) + g 2 P 2 (h, g) E [P i (h, g)] P i, i = 1, 2. Gaussian signalling is used. Shah&Sharma Enhancing Secrecy Rates and Resourse Allocation in Single-User & Multi-User Fading Wiretap Channels 35/84

36 Fading MAC without CSI of eve In passive attack of Eavesdropper, CSI cannot be estimated In single user wiretap channel, H.E Elgamal(IEEE trans 2008) reports secrecy capacity with optimal power control with main channel CSI only Mathew Bloch et. al(ieee trans 2008) studied outage analysis of single user slow fading channel with imperfect CSI of eve. We study fading MAC which achieve secrecy sum rate without knowing CSI of eve. Shah&Sharma Enhancing Secrecy Rates and Resourse Allocation in Single-User & Multi-User Fading Wiretap Channels 36/84

37 Fading MAC without CSI of eve: Optimal power allocation φ s x 1,x 2 = 1 + s 1 x 1 + s 2 x 2 where s is the channel state (h or g) and x k is the power used. Theorem For a given power control policy {P k (h)}, k = 1, 2, the following secrecy sum-rate [ ( ) ( )] φ h P1,P E h,g log 2 + φ h Q 1,Q 2 1 φ g + Q 1,Q 2 φ g P 1,P 2 + φ g Q 1,Q 2 1 φ h (25) Q 1,Q 2 is achievable, subject to power constraint E h,g [P k (h)] P k, k = 1, 2. Optimal policy is not available in closed form. Can be computed numerically. Shah&Sharma Enhancing Secrecy Rates and Resourse Allocation in Single-User & Multi-User Fading Wiretap Channels 37/84

38 Fading MAC without CSI of eve: Optimal power control with Cooperative Jamming {P k (h)}, k = 1, 2, policy when users transmit and {Q k (h)},when users jam Cooperative Jamming substantially improves secrecy sum-rate. Shah&Sharma Enhancing Secrecy Rates and Resourse Allocation in Single-User & Multi-User Fading Wiretap Channels 38/84

39 Without CSI of Eve: ON/OFF power control ON/OFF power control is a simple threshold based scheme as follows: h 1 > τ 1, h 2 > τ 2 : Both transmit; h i > τ i, h j < τ j, j i : User-i transmits; h 1 < τ 1, h 2 < τ 2 : No user transmits. Optimize the secrecy sum-rate over τ 1, τ 2 We also employ co-operative jamming over ON/OFF scheme Shah&Sharma Enhancing Secrecy Rates and Resourse Allocation in Single-User & Multi-User Fading Wiretap Channels 39/84

40 Fading MAC Without CSI of Eve: Simulation results 2.5 Secrecy Sum Rate (bits/channel use) Opt with full CSI Opt with only Rx CSI ON/OFF with only Rx CSI 0.5 CJ Opt with full CSI CJ Opt with only Rx CSI CJ ON/OFF with only Rx CSI SNR (db) Shah&Sharma Enhancing Secrecy Rates and Resourse Allocation in Single-User & Multi-User Fading Wiretap Channels 40/84

41 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 Shah&Sharma Enhancing Secrecy Rates and Resourse Allocation in Single-User & Multi-User Fading Wiretap Channels 41/84

42 Fading MAC without security constraint Power allocation in Fading MAC well known Tse & Hanly assumes Global CSI available at Tx They use Polymatroid theory H El Gamal et al use Game theory for distributed power control They also assume Global CSI Altman et al assume partial CSI, use Bayesian Game theory Shah&Sharma Enhancing Secrecy Rates and Resourse Allocation in Single-User & Multi-User Fading Wiretap Channels 42/84

43 We consider K- user fading, discrete time MAC Y (t) = K H i (t)x i (t) + η(t), (26) i=1 X i (t) channel i/p, η(t) is WGN N (0, σ 2 ) Fading process H i (t) H i {h (1) i,..., h (M) i } for user i with pmf {ρ (1) i,..., ρ (M) i } Shah&Sharma Enhancing Secrecy Rates and Resourse Allocation in Single-User & Multi-User Fading Wiretap Channels 43/84

44 User i transmits at fixed rate r i Receiver uses succesive decoding in following order Arranges users in increasing order of channel gains Let π be permutation on K s.t. h π(1) h π(2)..., fh π(k) Receiver sends ACK to user π(i) if ) r π(i) (1 1 2 log h π(i) P π(i) (h π (i)) + σ 2 + K j=i+1 h π(j)p π(j) (h π (j)) (27) (28) Shah&Sharma Enhancing Secrecy Rates and Resourse Allocation in Single-User & Multi-User Fading Wiretap Channels 44/84

45 Reward of user i in time slot t ( ω (t) i Average utility is a (t) i ν i (P i, P i ) lim sup T ), h i (t) = 1 T T t=1 { 1, If user i receives ACK 0, else [ ] ω (t) i (P i, H i (t)) = E ω (t) i (P i, H i ), By WLLN Probability of successful transmission = ν i (P i, P i ). (29) Shah&Sharma Enhancing Secrecy Rates and Resourse Allocation in Single-User & Multi-User Fading Wiretap Channels 45/84

46 Goal of each user This is a stochastic game G = K, P i, {ν i }. max a i ν i (P i, P i ) (30) subject to a i P i (31) We would like to find ɛ-coarse Correlated Equilibrium (CCE), defined as Definition If a distribution ψ on P satisfy the following inequality E a ψ [ Cπ(i) (a) ] E a ψ [ Cπ(i) (âπ(i), a π(i) )] + ɛ (32) then it is called ɛ coarse correlated equilibrium for each user π(i), i {1,..., K}, and every action a π(i), â π(i) P π(i). Shah&Sharma Enhancing Secrecy Rates and Resourse Allocation in Single-User & Multi-User Fading Wiretap Channels 46/84

47 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 (a i ) â Pi ω (t) i (â i ) 3: Time t: User i receives average utility for choosing a i ν (t) i = E a i Φ i [C(a i, a 1 )] 4: Update the weight 5: ω (t+1) i = ω (t) i (a i )(1 ɛ) c(t)(a i ) i 6: Time t + 1: Calculate ψ t = K i=1 Φ(t) i Shah&Sharma Enhancing Secrecy Rates and Resourse Allocation in Single-User & Multi-User Fading Wiretap Channels 47/84

48 Pareto Optimal Points We first define Goal is to obtain Ω(a) = K γ i ν i (a) (33) i=1 max Ω(a) a subject to a P. (34) The equilibrium obtained is Pareto optimal, defined as Pareto optimality An action profile a P is Pareto optimal if there does not exist another profile ã such that ν i (ã) ν i (a), i K and ν j (ã) > ν i (a) for j i. We use distributed algorithm to obtain Pareto points Shah&Sharma Enhancing Secrecy Rates and Resourse Allocation in Single-User & Multi-User Fading Wiretap Channels 48/84

49 Distributed Algorithm I 1: OUTER ITER = 1,User i: choose a i P i uniformly. 2: Use a i for T time slots. 3: Update weight of each user i 4: ˆΩ(a) K i=1 γ i ( 1 T T t=1 ω(t) i (a i, H i (t)) 5: After T slots: INNER ITER = 1, User i experiments w.p. P exp 6: w.p ɛ choose a i a i, a i P i 7: w.p. 1 ɛ 8: choose a i a i s.t. h i with high α i gets higher power level 9: If α i same for all h i, then higher value of channel state gets higher power level. 10: Call new action â i 11: User i: use â i for T time slots. ) Shah&Sharma Enhancing Secrecy Rates and Resourse Allocation in Single-User & Multi-User Fading Wiretap Channels 49/84

50 Distributed Algorithm II 12: User j: use a j for T time slots, j i. 13: User i: find ˆΩ(â i, a i ) 14: if ˆΩ(â i, a i ) > ˆΩ(a i, a i ) then 15: a i â i 16: P benchmark = ˆΩ(â i, a i ) 17: else 18: Randomly select another action 19: end if 20: INNER ITER = INNER ITER + 1, goto step 5 till INNER ITER = MAX 21: OUTER ITER = OUTER ITER + 1, goto step 1 till OUTER ITER = M Shah&Sharma Enhancing Secrecy Rates and Resourse Allocation in Single-User & Multi-User Fading Wiretap Channels 50/84

51 Nash Bargaining Solution consider the strategic game G = K, P i, {ν i } where users cooperate. we need to specify disagreement outcome We define the set of all possible utulities as V = {(ν 1 (a),..., ν K (a)) : a P} (35) disagreement action vector as δ = (δ 1,..., δ K ) where δ i = ξ i ( ), i = 1,..., K, and where is disagreement action. Bargaining problem is (V, δ) Aim is to find some function, ξ, that specifies a bargaining outcome ξ(v, δ) for every bargaining problem (V, δ) that satisfies the following axioms. Shah&Sharma Enhancing Secrecy Rates and Resourse Allocation in Single-User & Multi-User Fading Wiretap Channels 51/84

52 A-1 Pareto efficiency: A-2 Symmetry: A 3 Invariance: A-4 Independence of irrelevant alternatives: Existence of Nash Equilibrium There exists a unique bargaining solution (provided feasible region is non-empty) that satisfy A1 A4 and it is given by the solution of the following optimization problem a max (ν 1 δ 1 )(ν 2 δ 2 ) subject to ν i δ i, i = 1, 2 (ν 1, ν 2 ) V (36) a Nash Jr, John F, The bargaining problem, Econometrica, Shah&Sharma Enhancing Secrecy Rates and Resourse Allocation in Single-User & Multi-User Fading Wiretap Channels 52/84

53 Generalized to K player max K (ν i δ i ) subject to ν i δ i, i = 1,..., K (ν 1,..., K) V. i=1 (37) Since the set V is convex, hence NBS is proportionally fair also. Shah&Sharma Enhancing Secrecy Rates and Resourse Allocation in Single-User & Multi-User Fading Wiretap Channels 53/84

54 Fading MAC-WT The channel model is Y (t) = Z(t) = K H i (t)x i (t) + η b (t) (38) i=1 K G i (t)x i (t) + η e (t) (39) i=1 η b (t), η e (t) N (0, 1), H i (t) H i (t) and G i (t) G i (t) H i (t) H i {h (1) i {α (1) i,..., α (M(H) i ) i } G i (t) G {g (1) i,..., h (M(H) } with probabilities (w.p.) i i ),..., g (M(G) i } w.p. {β (1),..., β (M(G) i ) i i ) i } Shah&Sharma Enhancing Secrecy Rates and Resourse Allocation in Single-User & Multi-User Fading Wiretap Channels 54/84

55 Power can be choosen from P i {P (1) i K j=1 α (j) i β (j) i P (j),..., P (M) } subject to i i P i. (40) Receiver decodes via succesive decoding in the increasing order of main channel gain. We define C b ( Pπ(i), H π(i) ) 1 2 log ( ( ) 1 C e Pπ(i), G π(i) 2 log 1 + ( ) H π(i) P π(i) (H π(i), G π(i) ) K j=i+1 H π(j)p π(j) (H π(j), G π(j) ) (41) ) G π(i) P π(i) (H π(i), G π(i) ) 1 + K j i G π(j)p π(j) (H π(j), G π(j) ) (42) Shah&Sharma Enhancing Secrecy Rates and Resourse Allocation in Single-User & Multi-User Fading Wiretap Channels 55/84

56 Receiver sends ACK to user π(i) if r π(i) ( C b (P π(i), h π(i), g π(i) ) C e (P π(i), h π(i), g π(i) ) ) + Feasible Action Space { P i = P i = (P (1) i,..., P (M) i ) : P (k) i {p (1) i,..., p (M) i }, } M α i (j)β (j) i P (j) i P i j=1 (43) (44) Shah&Sharma Enhancing Secrecy Rates and Resourse Allocation in Single-User & Multi-User Fading Wiretap Channels 56/84

57 Fading-MAC without CSI of Eve Outage based metric P S O Pr { r i > C b ( Pπ(i), H π(i) ) Ce ( Pπ(i), G π(i) )} (45) We define Hence ( ) Γ = 1 h π(i) p π(i) 2 r 1 + π(i) 1 + K j=i+1 h π(j)p π(j) P S O = Pr { 1 + g π(i) p π(i) 1 + K j i g π(j)p π(j) > Γ } (46) (47) Now the receiver send ACK if Po S < ɛ, else NACK. Utility is now defined as ( ) ω π(i) a (t) π(i), h π(i)(t) = 1 {P S O <ɛ} (48) Shah&Sharma Enhancing Secrecy Rates and Resourse Allocation in Single-User & Multi-User Fading Wiretap Channels 57/84

58 Simulation Results: MAC without Eve Figure : Comparision of Sum-Rate in Fading MAC without Eve: Fixed rate transmission Shah&Sharma Enhancing Secrecy Rates and Resourse Allocation in Single-User & Multi-User Fading Wiretap Channels 58/84

59 Figure : Sum-rate comparision: Multiple rate (3 states) Shah&Sharma Enhancing Secrecy Rates and Resourse Allocation in Single-User & Multi-User Fading Wiretap Channels 59/84

60 Figure : Comparision of Fairness b/w NBS and PP Shah&Sharma Enhancing Secrecy Rates and Resourse Allocation in Single-User & Multi-User Fading Wiretap Channels 60/84

61 Figure : Fading MAC without Eve: Comparision with other schemes (Multiple rate transmission) Shah&Sharma Enhancing Secrecy Rates and Resourse Allocation in Single-User & Multi-User Fading Wiretap Channels 61/84

62 Figure : Fading MAC-WT: Comparision with full CSI and other scheme (Multiple rate) Shah&Sharma Enhancing Secrecy Rates and Resourse Allocation in Single-User & Multi-User Fading Wiretap Channels 62/84

63 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 Shah&Sharma Enhancing Secrecy Rates and Resourse Allocation in Single-User & Multi-User Fading Wiretap Channels 63/84

64 Discrete Memoryless Wiretap Channel 1 W (1) Encoder 1 X (1) P Y,Z X1,X 2 (..) Y n Bob (Ŵ (1), Ŵ (2) ) W (2) Encoder 2 X (2) Z n Eve (R (1) L, R(2) L ) Figure : Discrete Memoryless Multiple Access Wiretap Channel 1 Part of this work has been presented in IEEE WCNC 2015, New Orleans, USA Shah&Sharma Enhancing Secrecy Rates and Resourse Allocation in Single-User & Multi-User Fading Wiretap Channels 64/84

65 Codebook for MAC-WT MAC-WT 2-user MAC-WT (X 1 X 2, p(y, z x 1, x 2 ), Y, Z), X 1, X 2, Y, Z are finite sets and p(y, z x 1, x 2 ) is the collection of t.p.m 2 users want to send messages W (1) and W (2) to Bob reliably, keeping Eve ignorant (2 nr1, 2 nr2, n) codebook consists of Message sets W (1) and W (2) of cardinality 2 nr 1 and 2 nr 2 Two independent Messages W (1) and W (2), uniformally distributed over W (1) and W (2) stochastic encoders, f i : W (i) X n i i = 1, 2, (49) Shah&Sharma Enhancing Secrecy Rates and Resourse Allocation in Single-User & Multi-User Fading Wiretap Channels 65/84

66 Codebook for MAC-WT MAC-WT 4) Decoder at Bob: average probability of error at Bob is Leakage Rate is g : Y n W (1) W (2). (50) P (n) e P {(Ŵ (1), Ŵ (2)) ( W (1), W (2))}. (51) R (n) L = 1 n I(W (1), W (2) ; Z n ), (52) Shah&Sharma Enhancing Secrecy Rates and Resourse Allocation in Single-User & Multi-User Fading Wiretap Channels 66/84

67 Discrete Memoryless MAC Capacity region Capacity region of DM-MAC Capacity Region for DM-MAC Closure of convex hull of all rate pair (R 1, R 2 ) satisfying R 1 I(X 1 ; Y X 2 ), R 2 I(X 2 ; Y X 1 ), R 1 + R 2 I(X 1, X 2 ; Y ), (53) for some X 1, X 2 independent. Shah&Sharma Enhancing Secrecy Rates and Resourse Allocation in Single-User & Multi-User Fading Wiretap Channels 67/84

68 Discrete Memoryless MAC-WT Inner Bound Secrecy Capacity region not known for DM-MAC-WT Inner bound was proposed by Tekin and Yener 1 Secrecy-Rate Region for MAC Rates (R 1, R 2 ) are achievable with lim sup n R (n) L = 0, if there exist independent random variables (X 1, X 2 ) satisfying R 1 I(X 1 ; Y X 2 ) I(X 1 ; Z), R 2 I(X 2 ; Y X 1 ) I(X 2 ; Z), R 1 + R 2 I(X 1, X 2 ; Y ) I(X 1 ; Z) I(X 2 ; Z). (54) 1 Tekin, Ender and Yener, Aylin, The Gaussian multiple access wire-tap channel, IEEE Transactions on Information Theory, Shah&Sharma Enhancing Secrecy Rates and Resourse Allocation in Single-User & Multi-User Fading Wiretap Channels 68/84

69 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 part. = (W (i) k,1, W (i) k,2), W (i) k,1 being first part and W (i) k,2 the second X (i) = (X (i) 1, X(i) 2 ), X(i) codeword for W (i) k,2 1 is n 1 -length codeword for W (i) k,1 and X (i) 2 is Shah&Sharma Enhancing Secrecy Rates and Resourse Allocation in Single-User & Multi-User Fading Wiretap Channels 69/84

70 Encoder modified Encoders Also the encoders will now be defined as f s 1 : W (1) X n1 1, f d 1 : W (1) K 1 X n2 1 (55) f s 2 : W (2) X n1 2, f d 2 : W (2) K 2 X n2 2, (56) where X i X i, i = 1, 2, K i, i = 1, 2 are the sets of secret keys generated for the respective user, fi s, i = 1, 2 are the wiretap encoders, f d i, i = 1, 2 are the deterministic encoders, n 1 + n 2 = n Shah&Sharma Enhancing Secrecy Rates and Resourse Allocation in Single-User & Multi-User Fading Wiretap Channels 70/84

71 Coding Scheme (R 1, R 2 ) (R 1, R 2 ) (R 1, R 2 ) n 1 n 2 Slots Wiretap Coding Wiretap Coding Secret Key Fix distributions p X1, p X2 & consider time slotted system Initialization: Take n 1 = n 2 = 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 Shah&Sharma Enhancing Secrecy Rates and Resourse Allocation in Single-User & Multi-User Fading Wiretap Channels 71/84

72 Coding Scheme (R 1, R 2 ) (R 1, R 2 ) (R 1, R 2 ) (2R 1, 2R 2 ) (R 1, R 2 ) n 1 n 2 Slots k > λ k + 1 Wiretap Coding Wiretap Coding Secret Key Slot 3: Transmit first part of message via Wiretap coding (WTC) 2 nd part by with previous message, hence rate doubled. Slot λ: We define Time slot of Saturation λ i I(X i ; Y X j ) I(X i ; Y X j ) I(X i ; Z), (i j, i, j {1, 2}) λ max{λ 1, λ 2 } + 1 (57) Shah&Sharma Enhancing Secrecy Rates and Resourse Allocation in Single-User & Multi-User Fading Wiretap Channels 72/84

73 Coding Scheme (R 1, R 2 ) (R 1, R 2 ) (R 1, R 2 ) (2R 1, 2R 2 ) (R 1, R 2 ) (R 1, R 2 ) n 1 n 2 Slots k > λ k + 1 Wiretap Coding Wiretap Coding Secret Key After some slot λ > λ sum-rate will get saturated Slot k (k > λ): For W (i) k R 1 + R 2 I(X 1, X 2 ; Y ). (58) (W k,1, (1) W (2) k,1) & for the 2 nd part, encode W (i) at Shannon rate (R1, R 2 ). = (W (i) k,1, W (i) k,2), i = 1, 2, use WTC for k,2 W (i) k 1 & transmit Then we make n 2 /n 1 large enough to achieve close to Shannon Capacity for the whole slot. Shah&Sharma Enhancing Secrecy Rates and Resourse Allocation in Single-User & Multi-User Fading Wiretap Channels 73/84

74 Secrecy Measures We consider the following Secrecy measure Secrecy Measure I(W (1), W (2) ; Z n ) nɛ. (59) We note that I(W (1), W (2) ; Z n ) I(W (1) ; Z n X n 2 ) + I(W (2) ; Z n X n 1 ) k th slot Let the leakage rate inequality I(W (1) l, W (2) l ; Z n 1,..., Z n k ) 2n 1 ɛ, for l = 1,..., k (60) hold for k, then it holds for k + 1 slot also (Mathematical Induction). Shah&Sharma Enhancing Secrecy Rates and Resourse Allocation in Single-User & Multi-User Fading Wiretap Channels 74/84

75 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 of wiretap coding. We can achieve Shannon capacity region as secrecy rate region satisfying Strong Secrecy as n. lim sup I(W (1) k, W (2) k ; Z1 n, Z2 n,..., Zk n ) = 0, (61) n Shah&Sharma Enhancing Secrecy Rates and Resourse Allocation in Single-User & Multi-User Fading Wiretap Channels 75/84

76 MAC with Secret Key Buffer We consider the following model W (1) k B (1) k Encoder 1 X (1) k Y k Bob (Ŵ (1) k, Ŵ (2) k ) P Y,Z X1,X 2 (..) W (2) k Encoder 2 X (2) k Z k Eve (R (1) L, R(2) L ) B (2) k Figure : Discrete Memoryless Multiple Access Wiretap Channel with secret key buffers Shah&Sharma Enhancing Secrecy Rates and Resourse Allocation in Single-User & Multi-User Fading Wiretap Channels 76/84

77 MAC with Secret key buffer Theorem By using very old messages as keys we have the following result The secrecy-rate region of a DM-MAC-WT equals the usual Shannon capacity region of the MAC while satisfying I(W (1) k, W (1) k 1,..., W (1) k N 1 ; Z 1,..., Z k X (2) k ) n 1 ɛ, I(W (2) k, W (2) k 1,..., W (1) k N 1 ; Z 1,..., Z k X (1) k ) n 1 ɛ, I(W (1) k, W (2) k,..., W (1) k N 1, W (2) k N 1 ; Z 1,..., Z k ) 2n 1 ɛ. (62) Shah&Sharma Enhancing Secrecy Rates and Resourse Allocation in Single-User & Multi-User Fading Wiretap Channels 77/84

78 Fading MAC-WT Channel model Notation Y = H 1 X 2 + H 2 X 2 + N 1 Z = G 1 X 1 + G 2 X 2 + N 2, (63) C 1 (P 1 (H, G)) 1 ( 2 log 1 + H ) 1P 1 (H, G) σ1 2 C 2 (P 2 (H, G)) 1 ( 2 log 1 + H ) 2P 1 (H, G) σ1 2 C1 e (P 1 (H, G)) 1 ( 2 log 1 + G 1P 1 (H, G) σ2 2 + G 2P 2 (H, G) C2 e (P 2 (H, G)) 1 ( 2 log 1 + G ) 2P 2 (H, G) σ2 2 + G 1P 1 (H, G) C(P 1 (H, G), P 2 (H, G)) 1 2 log ( 1 + H 1P 1 (H, G) + H 2 P 2 (H, G) σ 2 1 ) ) (64) Shah&Sharma Enhancing Secrecy Rates and Resourse Allocation in Single-User & Multi-User Fading Wiretap Channels 78/84

79 We have the following result If Pr(H ik > G ik ) > 0, i = 1, 2, and all the channel gains are available at all the transmitters, then the following long term average rates that maintain the leakage rates (62), are achievable: R E H,G [C 1 (P 1 (H))], R E H,G [C 2 (P 2 (H))], R 1 + R E H,G [C (P 1 (H), P 2 (H))]. (65) where P is any policy that satisfies avg. power constraint. If only Bob knows all the channel states but not the transmitters, then (R 1, R 2 ) satisfies (65) with P i (H, G) P i, i = 1, 2. Shah&Sharma Enhancing Secrecy Rates and Resourse Allocation in Single-User & Multi-User Fading Wiretap Channels 79/84

80 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 Shah&Sharma Enhancing Secrecy Rates and Resourse Allocation in Single-User & Multi-User Fading Wiretap Channels 80/84

81 Conclusions Proposed Probability of error as metric of security Achieved Shannon capacity in wiretap channel via previous messages Power allocation in fading MAC-WT without CSI of Eve Extended CJ to enhance secrecy sum-rate in fading MAC-WT Used Algorithmic Game theory to allocate resources in fading MAC with individual CSI only Extended the same algorithms to fading MAC-WT Extended coding scheme of using previous messages as key to MAC to achieve Shannon Capacity region Shah&Sharma Enhancing Secrecy Rates and Resourse Allocation in Single-User & Multi-User Fading Wiretap Channels 81/84

82 Future Work Develop practical codes based on probability of error at Eve as security metric Develop practical codes for using previous messages as secret key to enhance secrecy rates Resource allocation with continous state space Shah&Sharma Enhancing Secrecy Rates and Resourse Allocation in Single-User & Multi-User Fading Wiretap Channels 82/84

83 References I Shah&Sharma Enhancing Secrecy Rates and Resourse Allocation in Single-User & Multi-User Fading Wiretap Channels 83/84

84 Thanks Questions? Shahid M Shah shahid.nit@gmail.com Shah&Sharma Enhancing Secrecy Rates and Resourse Allocation in Single-User & Multi-User Fading Wiretap Channels 84/84