Inseparability of the Multiple Access Wiretap Channel

Transcription

1 Iseparability of the Multiple Access Wiretap Chael Jiawei Xie Seur Ulukus Departmet of Electrical ad Computer Egieerig Uiversity of Marylad, College Park, MD 074 Abstract We examie the separability of the parallel multiple access wiretap chael. Separability, whe exists, is useful as it eables us to code separately over parallel chaels, ad still achieve the optimum overall performace. It is well-kow that the parallel sigle-user chael, parallel multiple access chael (MAC) ad parallel broadcast chael (BC) are all separable, however, the parallel iterferece chael (IC) is ot separable i geeral. I this paper, we show that, while MAC is separable MAC wiretap chael is ot separable i geeral. We prove this via a specific liear determiistic MAC wiretap chael. We the show that eve the Gaussia MAC wiretap chael is iseparable i geeral. Fially, we show that, whe the chael gais are draw from cotiuous distributios, ad whe the secure degrees of freedom (s.d.o.f.) regio is cosidered, the the Gaussia MAC wiretap chael is almost surely separable. I. INTRODUCTION Separability, whe exists, is useful as it eables us to code separately over parallel chaels, ad still achieve the optimum overall performace. It is well-kow that the parallel sigleuser chael, parallel multiple access chael (MAC) ad parallel broadcast chael (BC) 3 are all separable, however, the parallel iterferece chael (IC) is ot separable i geeral 4 7. I particular, referece 4 studied the twouser oe-sided ergodic fadig IC ad showed that separatio ca be strictly sub-optimal i certai cases. Referece 5 studied the separability i a parallel Gaussia IC, ad showed that the parallel Gaussia IC is ot always separable by presetig a specific example where joit ecodig over the parallel chaels outperforms idividually optimal ecodig i each parallel chael. Referece 6 further cofirmed the iseparability of the parallel IC by examiig the topological iterferece chael where the parallel chaels correspod to differet etwork topologies some of which had asymmetric coectivity. Recetly, referece 7 showed that eve symmetric parallel ICs are iseparable by characterizig the capacity regio of parallel symmetric liear determiistic ICs. I this paper, we cosider the MAC wiretap chael, which is a combiatio of a MAC to the legitimate receiver ad a MAC to the eavesdropper. The MAC wiretap chael was itroduced i 8, 9 ad studied further i 0 6. Eve though, i the absece of ay secrecy costraits, MAC is the most well-uderstood multi-user chael model, its wiretap versio is sigificatly more complex. The secrecy capacity This work was supported by NSF Grats CNS , CCF , CCF ad CNS regio of the MAC wiretap chael is still ukow today, ad its secure degrees of freedom (s.d.o.f.) regio has bee fully characterized oly recetly 7, 8. I this paper, we focus o the separability of the parallel MAC wiretap chael ad show that it is ot separable i geeral. Ituitively, this ca be attributed to the observatio that, eve though MAC wiretap chael is composed of MAC legitimate ad eavesdroppig liks, as a whole, it resembles the IC more, as it has two idepedet trasmitters ad two idepedet receivers. To show the iseparability of the parallel MAC wiretap chael, we costruct a specific liear determiistic MAC wiretap chael i each compoet chael. We fid the exact secrecy capacity of each of these compoet MAC wiretap chaels, ad the determie the optimum secrecy rates achievable by separate ecodig. This step is challegig as the secrecy capacity of MAC wiretap chaels is ukow i geeral; we provide a specific achievability ad coverse for the capacity of each of the compoet chaels. We the provide a ecodig scheme that codes over the parallel chaels which outperforms the optimum separable scheme. Next, we cosider the parallel Gaussia MAC wiretap chael. Sice the secrecy capacity regio of the geeral MAC wiretap chael, icludig the Gaussia MAC wiretap chael, is ukow but exact s.d.o.f. regio is kow 7, 8, we ivestigate the sum s.d.o.f. of parallel Gaussia MAC wiretap chaels ad prove that it is iseparable. This implies the iseparability of the secrecy regio as well. Next, we observe that, if the differet chael gais which give rise to differet parallel chaels are draw idepedetly from cotiuous distributios, the the chael gai cofiguratios which give rise to iseparability fall ito a set with zero Lebesgue measure. To cofirm this observatio, ad prove the almost sure s.d.o.f. separability of parallel Gaussia MAC wiretap chaels, we cosider the flat chael, where we put the idividual chael uses of each compoet chael ito a sigle chael uses. We utilize the coverse techiques i 7, 8 to show the separability i this case. Fially, we ote that, while iseparability i s.d.o.f. implies iseparability i the secrecy capacity, separability i s.d.o.f. does ot imply separability i secrecy capacities. The almost sure separability proved for the parallel Gaussia MAC wiretap chael i this paper holds oly for the s.d.o.f., which is the pre-log factor of the secrecy capacity, ad is a weaker measure of separability.

2 II. SYSTEM MODEL AND DEFINITIONS I a two-user MAC wiretap chael p(y, y x, x ), each trasmitter i, i =,, has a message W i iteded for the legitimate receiver whose chael output is Y. For each i, message W i is uiformly ad idepedetly chose from set W i. The rate of message i is R i = log W i. Trasmitter i uses a stochastic fuctio f i : W i Xi, where the - legth vector Xi deotes the ith user s chael iput i chael uses. All messages are eeded to be kept secret from the eavesdropper whose chael output is Y. A secrecy rate pair (R, R ) is said to be achievable if for ay ɛ > 0 there exist -legth codes such that the legitimate receiver ca decode the messages reliably, i.e., the probability of decodig error is less tha ɛ Pr (W, W ) (Ŵ, Ŵ) ɛ () A X X Y U X Y (a) V B C Y X Y Y X Y X (b) A U A U B C V V B ad the messages are kept iformatio-theoretically secure agaist the eavesdropper H(W, W Y ) H(W, W ) ɛ () where Ŵ, Ŵ are the estimates of the messages based o the legitimate receiver s observatio Y. The secrecy capacity regio C is the closure of the set cotaiig all achievable secrecy rate pairs. The sum secrecy capacity is C Σ = sup(r + R ), where the supremum is over all achievable secrecy rate pairs (R, R ) C. For Gaussia MAC wiretap chael with average power costrait P for both trasmitters, the s.d.o.f. regio is defied as: { } R i D s = (d, d ) : (R, R ) C, d i = lim P log P (3) ad the sum s.d.o.f. is defied as: D s,σ = lim P C Σ log P (4) Let p(y a, y a x a, x a ) ad p(y b, y b x b, x b ) be two two-user MAC wiretap chaels. The parallel two-user MAC wiretap chael is a two-user MAC wiretap chael i which the chael iputs of trasmitter ad are (x a, x b ) ad (x a, x b ), respectively, ad the chael iputs are set simultaeously i parallel. The chael outputs of the legitimate receiver ad the eavesdropper are (y a, y b ) ad (y a, y b ), respectively, ad are distributed accordig to p (y a, y a, y b, y b x a, x a, x b, x b ) = p(y a, y a x a, x a )p(y b, y b x b, x b ) (5) We refer to each MAC wiretap chael, p(y a, y a x a, x a ) ad p(y b, y b x b, x b ), as a compoet chael of the overall parallel MAC wiretap chael. III. INSEPARABILITY OF THE MAC WIRETAP CHANNEL I this sectio, we show that the parallel MAC wiretap chael is ot separable i geeral. To this ed, we provide a specific couter example. A V B U X X (c) Y Y A V B U A U B V Fig.. A iseparable liear determiistic parallel MAC wiretap chael. There are three compoet chaels: (a), (b) ad (c). A achievable scheme that codes across the parallel chaels is show i color mageta. Cosider the liear determiistic parallel discrete memoryless MAC wiretap chael show i Fig., which has three compoet chaels: (a), (b) ad (c). I the first compoet chael, (a), trasmitter has two sub-chael iputs, i.e., (X, X ), ad trasmitter has oly oe sub-chael iput X. The legitimate receiver observes (Y, Y ) ad the eavesdropper observes Y. I the secod compoet chael, (b), the roles of the two trasmitters are swapped. I the third compoet chael, (c), the legitimate receiver ad the eavesdropper have idetical observatios. Specifically, the iput/output relatioships for sub-chael (a) are: Y = X, Y = X X, Y = X X (6) where all symbols are biary, ad additio is modulo-. While trasmitters sed idepedet data, they ca each code their data joitly across their parallel chaels. I the followig two sub-sectios, we show that the optimum separable (i.e., idepedet) codig yields bits/chael-use for the sum secrecy rate, while through codig joitly across the compoet chaels a sum secrecy rate of 3 bits/chael-use is achievable, ad hece separatio is strictly sub-optimal. A. Optimum Sum Secrecy Rate with Separable Ecodig Due to idepedet codig across the compoet chaels: C Σ,idep = C Σ,(a) + C Σ,(b) + C Σ,(c) = C Σ,(a) (7) where C Σ,(a) = C Σ,(b) is due to symmetry, ad C Σ,(c) = 0 is due to the fact that the legitimate receiver ad the eavesdropper have idetical observatios. Therefore, we oly eed to show C Σ,(a) = i order to show C Σ,idep =. The achievability of

3 this follows by the followig sigallig: The first user seds a bit (uiform) iformatio sigal i X, ad seds o sigal i the other sub-chael which leaks to the eavesdropper, i.e., X = 0, ad the secod user does ot sed ay iformatio, i.e., X = 0. This gives bit secure rate for the first user, ad hece bit sum secrecy rate for the system, i.e., C Σ,(a). Next, we eed to prove that the sum secrecy rate i the compoet chael (a) is upper bouded by, i.e., C Σ,(a). For coveiece, let us deote R Σ = (R + R ) ɛ i order ot to carry +ɛ throughout the derivatio. The, by defiitio, ad Fao s iequality, we have R Σ = H(W, W ) ɛ (8) I(W, W ; Y, Y) I(W, W ; Y ) (9) Usig the chai rule o both terms o the right had side, R Σ I(W, W ; Y) + I(W, W ; Y Y ) I(W ; Y ) I(W ; Y W ) (0) = I(W ; Y) + I(W ; Y W ) I(W ; Y ) + I(W, W ; Y Y ) I(W ; Y W ) () = I(W ; Y) + I(W ; Y, W ) I(W ; Y ) + I(W, W ; Y Y ) I(W ; Y W ) () = I(W ; Y) I(W ; Y ) + I(W, W ; Y Y ) I(W ; Y W ) (3) = I(W ; Y) I(W ; Y ) + I(W, W ; Y Y ) I(W ; Y, W ) (4) where () ad (4) come from the idepedece of W ad W, ad (3) comes from the idepedece of W ad (W, Y ). For the first part i (4), we have I(W ; Y) I(W ; Y ) I(W ; Y, Y ) I(W ; Y ) (5) = I(W ; Y Y ) (6) = I(W ; X Y ) (7) = H(X Y ) H(X Y, W ) (8) where we refer to (6). For the secod part i (4), we have I(W, W ; Y Y ) I(W ; Y, W ) = I(W ; Y Y ) + I(W ; Y Y, W ) I(W ; Y, W ) (9) = I(W ; Y Y ) + I(W ; Y, Y, W ) I(W ; Y, W ) (0) I(W ; Y Y ) + I(W ; Y, Y, Y, W ) I(W ; Y, W ) () = I(W ; Y Y ) + I(W ; Y, Y Y, W ) () I(X, X; Y Y ) + I(X ; Y, Y Y, W ) (3) = I(X; Y X ) + H(X Y, W ) (4) = I(X; Y X ) + H(X Y, W ) (5) where (0) follows from the idepedece of W ad (W, Y ), (3) follows from the Markov chais W (Y, X, X) Y W (X, Y, W ) (Y, Y), we obtai (4) by usig the chael model i (6) ad the fact that by kowig (Y, Y ) = (X, Y ), X ca be determied, ad fially, we reach (5) by usig the chael model i (6) ad through the followig derivatio H(X Y, W ) = H(X, Y, W ) H(Y, W ) (6) = H(X, X, W ) H(Y, W ) (7) = H(X, Y, W ) H(Y, W ) (8) = H(X Y, W ) (9) Substitutig (8) ad (5) ito (4), we obtai R Σ H(X Y ) + I(X ; Y X ) (30) = H(X X X ) + I(X ; X X X ) (3) where meas bitwise modulo plus. Now, ituitively, as show i (3), if trasmitter iteds to trasmit -bit message via X, the to protect it, trasmitter must sed Beroulli ( ) i.i.d radom oise; however, by performig that, the sub-chael capacity betwee X ad Y is costraied ad reduced to zero. To cofirm this, we cotiue from (3) R Σ H(X X X ) + I(X ; X X X ) (3) = H(X, X ) H(X X ) + H(X X X ) H(X X, X ) (33) = H(X ) + H(X ) H(X X ) + H(X X X ) H(X ) (34) = H(X ) H(X X ) + H(X X X ) (35) = H(X X ) H(X X ) + H(X X X ) (36) = H(X X X ) H(X X ) + H(X X X ) (37) = H(X X X) I(X ; X X ) (38) H(X X X) = H(Y X ) H(Y) (39) (40) where we repeatedly use the idepedece of X ad X, ad also the idepedece of X ad (X, X ). Fially, (40) implies C Σ,(a), cocludig, together with the achievability, that C Σ,(a) =, ad hece C Σ,idep =. B. Joit Ecodig Based Achievable Scheme Here, we provide a achievable scheme to trasmit 3 bits securely by codig across the compoet chaels, i.e., by itroducig correlatio betwee the chael iputs of compoet chaels. Let {A, B, C, U, V } be mutually idepedet Beroulli ( ) radom variables. Here, {A, B, C} represet the message carryig sigals, ad {U, V } represet the jammig sigals. The joit ecodig based achievable scheme is show

4 i color mageta i Fig., where trasmitter seds A, V ad A V i three compoet chaels, respectively (ote that we choose X = 0), ad trasmitter seds U, (B, C) ad B U i three compoet chaels, respectively. With this scheme, the legitimate receiver observes A, U, B, C V, A V B U from three compoet chaels, which meas that the legitimate receiver ca decode message A from trasmitter ad messages B, C from trasmitter with zero probability of error, i.e., the legitimate receiver ca decode 3 bits reliably. O the other had, the eavesdropper observes A U, B V ad A U B V, which implies I(A, B, C; A U, B V, A U B V ) = I(A, B, C; A U, B V ) (4) = H(A U, B V ) H(A U, B V A, B, C) (4) = H(A U, B V ) H(U, V ) (43) = = 0 (44) where we use the idepedece of {A, B, C, U, V } ad also that they are all Beroulli ( ). This derivatio implies that the eavesdropper lears othig about the messages, ad therefore, 3 bits are set to the legitimate receiver reliably ad securely. IV. GAUSSIAN MAC WIRETAP CHANNEL A. Geeral Iseparability I this sectio, we show that eve the parallel Gaussia MAC wiretap chael is ot separable i geeral. We prove this by providig a specific example. Also ote that, it suffices to show the iseparability from the s.d.o.f. poit of view, sice it implies the iseparability of the secrecy capacity. Cosider the special two-user parallel Gaussia MAC wiretap chael show i Fig., i which each compoet chael is a two-user Gaussia MAC wiretap chael defied by, Y k = h k X k + h k X k + N k (45) Y k = g k X k + g k X k + N k (46) where k = a, b, ad (h ia, h ib ) ad (g ia, g ib ) are the timeivariat chael gais of user i to the legitimate receiver ad the eavesdropper, respectively. We let h b = h b = α, ad g b = g b = β (47) The, the six radom variables {h a, h a, g a, g a, α, β} are mutually idepedetly distributed accordig to the same cotiuous distributio, ad N a, N a, N b, N b are mutually idepedet Gaussia radom variables with zero-mea ad uit-variace. The chael iputs of each user satisfy average power costraits, E Xia + ib X P, for i =,. From 7, for almost all chael gais {h a, h a, g a, g a }, the sum s.d.o.f. for compoet chael (a) is 3. From 8, compoet chael (b) is degraded, ad its sum s.d.o.f. is zero. This implies that, by idepedet ecodig across the compoet chaels, the optimum sum s.d.o.f. is 3. O the other had, by selectig X a = g a V, X a = g a U, X b = β V, X b = β U (48) Fig.. X a X a X b X b g a g b = β h a g a h b = α g b = β (a) (b) N a h a N a N b h b = α N b A example two-user parallel Gaussia MAC wiretap chael. where V ad U are idepedet radom variable draw from the followig discrete PAM costellatio: C(a, Q) = a{ Q, Q +,..., Q, Q} (49) Here, V represets the message-carryig sigal ad U represets the jammig sigal. Let us defie Ŷ as Ŷ = g a Y a β h a α Y b (50) ga h a = V + g a N a β g a h a h a α N b (5) The factor i frot of V is o-zero for almost all chael gais. Let us defie ˆV as the estimate of V obtaied by selectig the closest poit i C(a, Q) based o the observatio Ŷ. For ay small eough δ > 0, let us choose Q = P δ ad a = γp δ, where γ is a costat idepedet of P to meet the average power costrait. The, due to the Markov chai V (Y a, Y b ) Ŷ ˆV, we have I(V ; Y a, Y b ) I(V ; Ŷ ) I(V ; ˆV ) (5) = H(V ) H(V ˆV ) (53) = log(q + ) H(V ˆV ) (54) log(q + ) Pr V ˆV log(q + ) (55) { Pr V ˆV } δ log P (56) Now, due to the PAM structure, probability of error is Pr V ˆV exp ( γ a ) exp ( γ P δ) (57) where γ, γ are costats idepedet of P. The, from (56) ad (57), at high SNR (large eough P ), we have I(V ; Y a, Y b ) δ log P + o(log P ) (58) where o( ) is the little-o fuctio. Y a Y a Y b Y b

5 O the other had, for the iformatio leakage rate, I(V ; Y a, Y b ) I(V ; V + U) (59) H(V + U) H(V ) (60) log 4Q + Q + (6) By 3, Theorem, we ca achieve the sum secrecy rate of sup (R + R ) I(V ; Y a, Y b ) I(V ; Y a, Y b ) (6) δ log P + o(log P ) (63) for ay δ 0, which implies that we ca achieve sum s.d.o.f. This meas that by joit ecodig across compoet chaels, we achieve sum s.d.o.f. outperformig optimum idepedet ecodig, which ca at most achieve 3 sum s.d.o.f. B. Separability i s.d.o.f. for Almost All Chael Gais Although the Gaussia MAC wiretap chael is ot always separable, the special costructio provided i the last subsectio is ot geeral, i.e., for almost all chael gais, the costraits i (47) are ever met. Based o this observatio, we show that the s.d.o.f. regio of the parallel Gaussia MAC wiretap chael is separable for almost all chael gais. From 8, the s.d.o.f. regios of the compoet Gaussia MAC wiretap chaels are idetical, i.e., D s,(a) = D s,(b), ad D s,(a) = {(d, d ) : d + d, d + d } (64) Therefore, it suffices to show that for the overall parallel Gaussia MAC chael the s.d.o.f. regio is D s = {(d, d ) : d + d, d + d } (65) The achievability follows from 8 for almost all chael gais. I the achievability, we scale the power i each compoet chael, to meet the overall power costrait; however, this does ot affect the s.d.o.f. calculatios. For the coverse, we first flatte the parallel chael by cocateatig the chael iputs ad outputs of compoet chaels ito -legth vectors. Istead of studyig the parallel chael i chael uses, we study the flat chael i chael uses. The power costrait remais the same over chael uses. I additio, sice itroducig correlatio i time ad i compoet chaels has the same effect, the flat chael must have the same coverse as the origial oe. The, similar to the steps i 7, Eqs. (7)-(6), we have (R + R ) h( X, X, Y, Y ) h( X, X Y, Y ) h(y ) + c 0 (66) where vectors i bold-face are -legth vectors. The compoets of -vectors X j, for j =,, are X ji = X ji + Ñji, for i =,...,. Here, the sequece Ñ j is i.i.d. over time, is idepedet of all other radom variables, ad Ñji is a Gaussia radom variable with zero-mea ad variace σji, such that { } σji < mi,,, (67) h ja g ja h jb g jb The, all the remaiig steps i 7 follow, ad we have ( ) R i +R +R h(y )+c log P +c (68) for i =,. This implies d + d, ad d + d (69) which completes the proof of the coverse for this case. V. CONCLUSIONS We showed that the parallel MAC wiretap chael is ot always separable by providig a specific example i which the sum secrecy rate by joit ecodig over parallel chaels outperforms the best rate achievable by idividually optimal ecodig for each compoet chael. The, we showed that the parallel Gaussia MAC wiretap chael is iseparable i geeral as well. 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