Square Root Law for Communication with Low Probability of Detection on AWGN Channels

Transcription

1 Square Root Law for Commucato wth Low Probablty of Detecto o AWGN Chaels Boulat A. Bash, Des Goeckel, Do Towsley Departmet of Computer Scece, Uversty of Massachusetts, Amherst, Massachusetts Electrcal ad Computer Egeerg Departmet, Uversty of Massachusetts, Amherst, Massachusetts Abstract We preset a square root lmt o low probablty of detecto (LPD) commucato over addtve whte Gaussa ose (AWGN) chaels. Specfcally, f a warde has a AWGN chael to the trasmtter wth o-zero ose power, we prove that o( ) bts ca be set from the trasmtter to the recever AWGN chael uses wth probablty of detecto by the warde less tha ɛ for ay ɛ > 0, ad, f a lower boud o the ose power o the warde s chael s kow, the O( ) bts ca be covertly set chael uses. Coversely, tryg to trasmt more tha O( ) bts ether results detecto by the warde wth probablty oe or a o-zero probablty of decodg error as. Further, we show that LPD commucato o the AWGN chael allows oe to sed a ozero symbol o every chael use, cotrast to what mght be expected from the square root law foud recetly mage-based stegaography. I. INTRODUCTION Securg formato trasmtted over wreless lks s of paramout cocer for cosumer, dustral, ad mltary applcatos. Tradtoal ecrypto ad key exchage protocols secure data from tercepto by a utrusted thrd party. However, there are may real-lfe stuatos where t s mperatve to prevet the trasmsso from beg detected the frst place, as ecrypted data arouses suspco, ad eve the most theoretcally robust ecrypto ca ofte be defeated by a determed adversary usg o-computatoal methods such as sde-chael aalyss. I spte of ts mportace, low probablty of detecto (LPD) commucato has bee relatvely uderexplored. I ths work we exame the fudametal lmtatos of LPD commucato over wreless lks. I our scearo, Alce commucates wth Bob over a chael subject to addtve whte Gaussa ose (AWGN), whle Wlle attempts to detect her trasmsso (wthout actvely jammg Alce s chael). The chael betwee Wlle ad Alce s also subject to AWGN. Alce seds low-power covert sgals to Bob that Wlle attempts to classfy as ether ose Ths research was sposored by the Natoal Scece Foudato uder grats CNS ad CNS , ad by the U.S. Army Research Laboratory ad the U.K. Mstry of Defece uder Agreemet Number W911NF The vews ad coclusos cotaed ths documet are those of the author(s) ad should ot be terpreted as represetg the offcal polces, ether expressed or mpled, of the U.S. Army Research Laboratory, the U.S. Govermet, the U.K. Mstry of Defece or the U.K. Govermet. The U.S. ad U.K. Govermets are authorzed to reproduce ad dstrbute reprts for Govermet purposes otwthstadg ay copyrght otato hereo. o hs chael from Alce or Alce s sgals to Bob. If the ose o the chael betwee Wlle ad Alce has o-zero power, Alce ca commucate wth Bob whle toleratg a certa probablty of detecto, whch she ca drve dow by trasmttg wth low eough power. Our problem s related to the problem of mperfect stegaography, but the two problems are ot the same. Stegaography cosders hdg formato by alterg the propertes of fxed-sze, fte-alphabet covertext objects (lke mages or software bary code) wth mperfect stegaography systems allowg a fxed probablty of detecto of hdde formato. Covertext ca be cosdered a type of lossless fte-alphabet chael. However, the square root law recetly foud ths evromet [1], whch states that O( ) symbols the orgal covertext of sze may safely be modfed to hde a message, s lmted ts scope. The cotuousvalued chael allows us to spread hdde formato across every symbol used the trasmsso, thus showg that a drect applcato of the stegaographc result quckly leads to cotradcto ad demostratg the dstcto betwee the two problems. I fact, our square root law ca be vewed as geeralzg the square root law for mperfect stegaography. We state our ma result that lmts mutual formato o the covert chael betwee Alce ad Bob usg asymptotc otato where f() = O(g()) deotes a asymptotcally tght upper boud o f(), ad f() = o(g()) ad f() = ω(g()) deote upper ad lower bouds, respectvely, that are ot asymptotcally tght [, Ch. 3.1]: Theorem (Square root law). Suppose the chael betwee Alce ad each of Bob ad Wlle expereces depedet addtve whte Gaussa ose (AWGN) wth power σ b > 0 ad σ w > 0, respectvely, where σ b ad σ w are costats. The, for ay ɛ > 0 ad ukow σ w, Alce ca sed o( ) formato bts to Bob chael uses whle matag a probablty of detecto of Alce s trasmsso by Wlle of less tha ɛ. Moreover, f Alce ca lower-boud σ w ˆσ w, she ca sed O( ) bts chael uses whle matag a probablty of detecto of less tha ɛ. Coversely, f Alce attempts to trasmt ω( ) bts chael uses, the, as, ether Wlle detects her wth arbtrary low probablty of error or Bob caot decode her message relably (.e. wth arbtrary low probablty of decodg error).

2 After troducg our dscrete-tme chael model ad hypothess testg backgroud Secto II, we sketch the proofs of the achevablty ad the coverse of the square root law Sectos III ad IV, respectvely. Detaled proofs ad remarks are avalable [3]. We dscuss the mappg to the cotuoustme chael ad the mplcatos of chael fadg o our results, as well as the relatoshp to prevous work Secto V, ad coclude Secto VI. Alce f 1, f,..., f z (w) II. PREREQUISITES z (b) Wlle Bob decode f 1, f,..., f decde z (w) 1, z (w),..., z (w) or f 1 + z (w) 1, f + z (w),..., f + z (w)? Fg. 1. System framework: Alce ecodes formato to a vector of real symbols f = {f } =1 ad trasmts t o a AWGN chael to Bob, whle Wlle attempts to classfy hs vector of observatos of the chael from Alce y w as ether a AWGN vector z w = {z (w) } =1 or a vector {f +z (w) } =1 of trasmssos corrupted by AWGN. A. Chael Model We use the dscrete-tme AWGN chael model wth realvalued symbols depcted Fgure 1 (ad defer dscusso of the mappg to a cotuous-tme chael as well as a fadg chael to Secto V). Alce trasmts a vector of real-valued symbols f y b = {y (b) } =1 where y(b) ad detcally dstrbuted (..d.) z (b) observes vector y w = { } =1 wth..d. z (w) N (0, σ w). B. Hypothess Testg = {f } =1. Bob receves vector = f + z (b) wth a depedet N (0, σb ). Wlle where y(w) = f + z (w), Wlle expects vector y w to be cosstet wth hs chael ose model. He performs a statstcal hypothess test o ths vector, wth the ull hypothess H 0 beg that Alce s ot covertly commucatg. Ths correspods to each sample N (0, σw)..d. The alterate hypothess H 1 s that Alce s trasmttg, whch correspods to samples from a dfferet dstrbuto. Wlle tolerates some cases whe hs test correctly accuses Alce. Followg the stadard omeclature, we deote the probablty of such rejecto of H 0 whe t s true by α [4]. Wlle s test may also accept H 0 whe t s false ad mss Alce s covert trasmsso. We deote the probablty of a mss by β. The sum α + β determes the performace of a hypothess test [4]. III. ACHIEVABILITY OF SQUARE ROOT LAW I our scearo, Alce ad Bob costruct a covert commucatos system, wth all the detals kow to Wlle except for a secret key shared before commucato. Ths follows best practces securty system desg, as ts securty depeds oly o the key [5]. Sce ths work cocers the lmts of covert commucato, key sze s ot a costrat ad we defer the study of key effcecy to future work. Wlle tres to determe whether Alce trasmtted covert data gve the vector of observatos y w of hs chael from Alce. Deote the probablty dstrbuto of Wlle s chael observatos whe H 0 s true as P 0, ad whe H 1 s true as P 1. To stregthe the achevablty result, we assume that Alce s chael put dstrbuto, as well as the dstrbuto of AWGN o the chael betwee Alce ad Wlle are kow to Wlle. The P 0 ad P 1 are kow to Wlle, ad he ca costruct a optmal statstcal hypothess test that mmzes the sum of error probabltes α + β [4, Ch. 13]. The: Fact 1 (Theorem [4]). For the optmal test: α + β = 1 T V (P 0, P 1 ) where T V (P 0, P 1 ) = 1 p 0(x) p 1 (x) dx s the total varato dstace betwee P 0 ad P 1 ad p 0 (x) ad p 1 (x) are destes of P 0 ad P 1, respectvely. Ufortuately, the total varato metrc s uweldy for the products of probablty measures, whch are used the aalyss of the vectors of observatos. We thus use [6, Lemma ]: ( 1 p 0 (x) p 1 (x) dx) D(P 0 P 1 ) (1) where relatve etropy D(P 0 P 1 ) = X p 0(x) l p0(x) p 1(x) dx wth X beg the support of p 1 (x). If P s the dstrbuto of a sequece {X } =1 where each X P s..d., the D(P 0 P 1 ) = D(P 0 P 1 ). Now we are ready to prove the achevablty theorem uder a average power costrat. Theorem 1.1 (Achevablty). Let Wlle s chael be subject to AWGN wth power σ w > 0. The Alce ca mata Wlle s sum of the probabltes of detecto errors α + β 1 ɛ for ay ɛ > 0 whle covertly trasmttg o( ) bts to Bob uses of a AWGN chael f σ w s ukow ad O( ) bts chael uses f she ca lower-boud σ w ˆσ w. Proof: Costructo: Alce s chael ecoder takes put blocks of sze M bts ad ecodes them to codewords of legth at the rate of R = M/ bts/symbol. We employ radom codg argumets ad depedetly geerate R codewords {c(w k ), k = 1,,..., R } from R for messages W k, each accordg to p X (x) = =1 p X(x ), where X N (0, P f ) ad P f s defed later. The codebook s the secret key shared betwee Alce ad Bob, ad s ot revealed to Wlle. 1 However, Wlle kows how t s costructed, cludg the value of P f, ad he uses statstcal hypothess testg o chael readgs y w to decde whether Alce trasmtted. Aalyss: Cosder the case whe Alce trasmts codeword c(w k ). Suppose that Wlle employs a detector that mplemets a optmal hypothess test o hs chael readgs. 1 Wlle s lack of kowledge of the codebook s crtcal to our result, as the sparsty of the codewords mples that, f the codeword s correctly decoded by Wlle, the the trasmsso s detected.

3 Hs ull hypothess H 0 s that he observed ose o hs chael. Hs alterate hypothess H 1 s that Alce trasmtted ad he observed Alce s codeword corrupted by ose. By Fact 1, the sum of the probabltes of Wlle s detector s errors s expressed by α + β = 1 T V (P 0, P 1 ), where the total varato dstace s betwee the dstrbuto P 0 of ose readgs that Wlle expects to observe uder hs ull hypothess ad the dstrbuto P 1 of the covert codeword trasmtted by Alce corrupted by ose. Alce ca lowerboud α + β by upper-boudg T V (P 0, P 1 ) ɛ. The realzatos of ose z (w) vector z w are zero-mea..d. Gaussa radom varables wth varace σw, ad, thus, P 0 = P w where P w = N (0, σw). Sce he does ot kow the codebook, Wlle s probablty dstrbuto of the trasmtted symbols s of zero-mea..d. Gaussa radom varables wth varace P f. Sce ose s depedet of the trasmtted symbols, whe Alce trasmts, Wlle observes y w, where N (0, P f + σw) = P s s..d., ad P 1 = P s. The, usg (1), the propertes of relatve etropy, ad the Taylor seres expaso of D(P w P s ) wth respect to P f aroud P f = 0, we obta the upper boud: T V (P w, P s ) P f σw () Suppose Alce sets her average covert symbol power P f cf(), where c = ɛ. I most practcal scearos Alce ca lower-boud σw ˆσ w ad set f() = ˆσ w. If σw s ukow, select f() such that f() = o(1) ad f() = ω(1/ ) (the latter codto s used to boud Bob s decodg error probablty). I ether case, for large, P f < σw satsfes the Taylor seres covergece crtero for () to be vald, ad Alce upper-bouds T V (P w, P s ) ɛ, lmtg the performace of Wlle s detector. As stadard results for costat symbol power are ot drectly applcable to our system where P f s a fucto of codeword sze, we exame the probablty P e of Bob s decodg error averaged over all possble codebooks. Let Bob employ a maxmum-lkelhood (ML) decoder (.e. mmum dstace) to process the receved vector y b whe c(w k ) was set. The decoder suffers a error evet E (c(w k )) whe y b s closer to aother codeword c(w ), k: [ )] P e = E c(wk ) P ( R =0, ke (c(w k )) R =0, k E c(wk ) [P (E (c(w k )))] (3) where (3) follows from the uo boud ad the learty of expectato. The dstace betwee two codewords s d = c(w k ) c(w ), where s the L orm. Sce codewords are depedet ad Gaussa, c(w k ) c(w ) N (0, P f ) ad d = P f U, where U χ, wth χ deotg the ch-squared dstrbuto wth degrees of freedom. Therefore, by [7, (3.44)]: ( )] P f U E c(wk ) [P (E (c(w k )))] = E U [Q σb where Q(x) = x expectato yelds: e t / π dt 1 / e x [8, (5)]. Takg the E c(wk ) [P (E (c(w k )))] log ( ) 1+ cf() σ b The summad (3) does ot deped o, ad (3) becomes: ( ) 1+ cf() σ b P e R log Sce f() = ω(1/ ( ) ), f rate R = ρ log 1 + cf() for σb a costat ρ < 1, as creases, the probablty of Bob s decodg error decays expoetally to zero ad Bob obtas R = ( ) ρ log 1 + cf() σ covert bts chael b uses. Sce R ρcf() 4σb l, approachg equalty as gets very large, Bob receves R = o( ) bts chael uses, ad R = O( ) bts chael uses f f() = ˆσ w. Implcatos of a peak power costrat Sce most practcal systems are peak-power costraed, we show that the square root law holds for the bary put Gaussa output chael usg a proof smlar to that of Theorem 1.1. Theorem 1. (Achevablty uder a peak power costrat). Suppose Alce s trasmtter s subject to the peak power costrat b ad Wlle s chael s subject to AWGN wth power σ w > 0. The Alce ca mata Wlle s sum of the probabltes of detecto errors α + β 1 ɛ for ay ɛ > 0 whle covertly trasmttg o( ) bts to Bob over uses of a AWGN chael f σ w s ukow ad O( ) bts chael uses f she ca lower-boud σ w ˆσ w. Proof: Costructo: Alce ecodes the put blocks of sze M bts to codewords of legth at the rate R = M/ bts/symbol wth the symbols draw from alphabet { a, a}, where a satsfes the peak power costrat a < b ad s defed later. We depedetly geerate R codewords {c(w k ), k = 1,,..., R } for messages W k from { a, a} accordg to p X (x) = =1 p X(x ), where p X ( a) = p X (a) = 1. As the proof of Theorem 1.1, the codebook s a secret key shared betwee Alce ad Bob, but Wlle kows how t s costructed, cludg the value of a. Aalyss: Whe Alce trasmts a covert symbol durg the th symbol perod, she trasmts a or a equprobably by costructo ad Wlle observes the covert symbol corrupted ( by AWGN. Therefore, P s = 1 N ( a, σ w ) + N (a, σw) ). Aga, usg (1), the propertes of relatve etropy, ad the Taylor seres expaso of D(P w P s ) wth wth respect to a aroud a = 0, we obta the upper boud: T V (P w, P s ) a σ w Sce the power of Alce s covert symbol s a = P f, (4) s detcal to () ad Alce sets a cf(), where c ad f() are defed as Theorem 1.1. The, for large eough, a < σ w satsfes the Taylor seres covergece crtero, ad (4)

4 Alce obtas the upper boud T V (P w, P s ) ɛ, lmtg the performace of Wlle s detector. Lke Theorem 1.1, we caot apply stadard costatpower chael codg results. Thus, we upper-boud Bob s decodg error probablty by aalyzg a suboptmal decodg scheme. Suppose Bob uses a hard-decso devce o each receved { covert symbol y (b) } = f + z (b) va the rule ˆf = a f y (b) 0; a otherwse, ad apples a ML decoder o ts output. The effectve chael for the ecoder/decoder par s a bary symmetrc chael wth cross-over probablty p e = Q(a/σ b ) ad the probablty of the decodg error averaged over all possble codebooks s P e R (1 H(pe)) [9], where H(p) = p log p (1 p) log (1 p) s the bary etropy fucto. Extedg the aalyss [10, Secto I..1], we frst use the Taylor seres expaso of p e wth respect to a aroud a = 0 to upper-boud p e p (UB) e. Sce H(p e ) H(p (UB) e ) o the terval [0, 1 ], we perform Taylor seres expaso of H(p (UB) e ) wth respect to a aroud a = 0 to obta P e R cf() σ b +O(1) π l. As f() = ω(1/ ), f rate R = ρcf() σ bts/symbol for a costat ρ < 1, the b π l probablty of Bob s decodg error decays expoetally to zero as creases ad Bob obtas R = o( ) bts chael uses, ad R = O( ) bts chael uses f f() = ˆσ w. Relatoshp wth Square Root Law Stegaography It has recetly bee show that fte-alphabet mperfect stegaographc systems at most O( ) symbols the orgal covertext of sze may safely be modfed to hde a stegaographc message [1]. From the stegaographc perspectve, our covertext s the ose o Wlle s chael to Alce. But our result does ot obey ther coverse, as we ca modfy all symbols our covertext, hghlghtg the dfferet ature of the problem scearos. To demostrate the rchess of our scearo ad the geeralty of our square root law, we costruct a codebook where roughly τ out of of symbols are used to carry the message ad whe Alce s trasmttg a codeword, the dstrbuto of each of Wlle s observatos s P s = (1 τ)n (0, σw) + τn (0, P f + σw). Aga, usg (1), the propertes of relatve etropy, ad the Taylor seres expaso of D(P w P s ) wth respect to P f aroud P f = 0 yelds the followg boud: T V (P w, P s ) τp f σw (5) The oly dfferece (5) from () s τ the umerator. Thus, f Alce sets the product τp f = cf(), wth c ad f() as prevously defed, she lmts the performace of Wlle s detector. Ths product s the average symbol power used by Alce. It s easy to verfy that the peak power costraed scearo Alce should set product τa = cf() ad that the umber of bts that Alce ca covertly trasmt to Bob obeys the square root bouds. IV. CONVERSE Here, as the achevablty, the chael betwee Alce ad Bob s subject to AWGN of power σb. Alce s objectve s to covertly trasmt a message W k that s M = ω( ) bts log to Bob chael uses wth arbtrarly small probablty of decodg error as gets large. For a upper boud o the reducto etropy, the messages are chose equprobably. Alce ecodes each message W k arbtrarly to symbols at the rate R = M/ symbols/bt. I the coverse that follows Wlle observes all of Alce s chael uses, but, to stregthe the result, he s oblvous to her sgal propertes. Theorem. If over chael uses, Alce attempts to trasmt a covert message to Bob that s ω( ) bts log, the, as, ether Wlle ca detect her wth arbtrarly low sum of error probabltes α + β, or Bob caot decode wth arbtrarly low probablty of error. Proof: To detect Alce s covert trasmssos, Wlle performs the followg hypothess test: H 0 : H 1 : = z (w), = 1,..., = f + z (w), = 1,..., Rejecto of H 0 meas that Alce s covertly commucatg wth Bob. To perform the test, Wlle collects a vector of depedet readgs y w from hs chael to Alce ad geerates the test statstc S = ywyt w where x T deotes traspose of vector x. Uder the ull hypothess H 0 Alce does ot trasmt ad Wlle reads AWGN. Thus, N (0, σw), ad the mea ad the varace of S are: E [S] = σ w (6) var [S] = σ4 w Suppose Alce trasmts codeword c(w k ) wth the average power per symbol P k = c(w k)c T (W k ). The the mea ad varace of S whe Alce trasmts message W k are: (7) E [S] = σ w + P k (8) var [S] = 4P kσw + σw 4 (9) The varace of Wlle s test statstc (9) s computed by addg the varaces codtoed o c(w k ) of the squared dvdual observatos var [ ] y (ad dvdg by ) sce the ose o the dvdual observatos s depedet. Deote P 0 as the dstrbuto whe H 0 holds, ad P (k) 1 whe H 1 holds wth Alce trasmttg message W k. If H 0 s true, the S should be close to (6). Wlle pcks some threshold t ad compares the value of S to σw + t. He accepts H 0 f S < σw + t ad rejects t otherwse. Suppose that he desres false postve probablty α, whch s the probablty that S σw + t whe H 0 s true. We boud t usg (6) ad (7) wth Chebyshev s Iequalty [6, (3.3)]: α P 0 ( S σ w t ) σ4 w t

5 d, where d = σ w α Thus, to obta α, Wlle sets t = s a costat. As creases, t decreases, whch s cosstet wth Wlle gag greater cofdece wth more observatos. Now let s boud the probablty of a mss β gve t, whch s the probablty that S < σw + t whe Alce trasmts W k. We use Chebyshev s Iequalty wth (8) ad (9): β P (k) 1 ( S σ w P k P k t ) 4P kσ w + σ 4 w ( P k d) (10) If P k = ω(1/ ), lm β = 0. Thus, wth eough observatos, Wlle ca detect wth arbtrarly low error probablty Alce s codewords wth average symbol power P k = ω(1/ ). Note that Wlle s detector s oblvous to ay detals of Alce s codebook costructo. By (10), f Alce wats to lower-boud the sum of the probabltes of error of Wlle s statstcal test by α + β ζ > 0, she must use low-power codewords; partcular, a fracto γ > 0 of the codewords must have P U = O(1/ ). Deotg ths set of codewords by U, we ca lower-boud the probablty of Bob s decodg error P e γp (U) e, where P (U) e s the probablty of decodg error whe a message from U s set. Focusg o U, we adapt the proof of the coverse to the codg theorem for Gaussa chaels [6, Ch. 9.] to obta: P (U) e 1 P U/σb + 1/ log γ + R (11) Sce Alce trasmts ω( ) bts chael uses, her rate s R = ω(1/ ) bts/symbol. However, P U = O(1/ ), ad, as, P (U) e s bouded away from zero. Thus, P e s bouded away from zero f Alce tres to beat Wlle s smple hypothess test. V. DISCUSSION A. Mappg to Cotuous-tme Chael Cosder the mappg of the dscrete-tme model employed throughout ths paper to the physcal (cotuous-tme) chael. For a system that has a (basebad) badwdth costrat of W Hz, f Alce employs the optmal badlmted pulse shape sc(w t), all of the formato s extracted from the chael by Wlle (ad Bob) by samplg at rate W samples/secod. Ths results the dscrete-tme model of Secto II, ad the results preseted here apply drectly. Whe the pulse shape s stead chose to have some excess badwdth, the samplg at a rate hgher tha W has utlty for Wlle whe attemptg to detect Alce s trasmsso. Although we fd t ulkely that the asymptotcs cosdered here wll be altered, techques volvg cyclostatoary detecto are applcable ad could potetally mpact practcal system mplemetatos. B. Fadg ad Shadowg Fadg ad shadowg wll mpact both the capacty of the chael from Alce to Bob ad the ablty for Wlle to detect Alce s trasmsso. There are a umber of dfferet models that could be employed to corporate these effects. However, whle these models wll have a mpact as we move toward practcal systems, they have lttle mpact o the asymptotc results preseted here. C. Relatoshp to Prevous Work Aalytcal evaluato of LPD commucato has bee sparse. Hero studes LPD chaels [11] a multple-put multple-output (MIMO) settg. Whle he recogzes that a LPD commucato system s costraed by average power, he does ot aalyze the costrat asymptotcally ad, thus, does ot obta the square root law. Ulke LPD commucato, much aalytcal work has bee doe o stegaography. As oted the remark Secto III, the square root law was foud fte-alphabet mperfect stegaography [1]. However, although ther goal s the same as ours (hdg formato wth low probablty of detecto by Wlle), ther model based o hdg formato fte-alphabet mages s very dfferet from ours. Our scearo s arguably rcher, ad ts addtoal degree of freedom the choce of trasmsso power allows Alce to alter all symbols used trasmsso whle matag a fxed detecto probablty, whch stads cotrast to the fte-alphabet stegaography result. VI. CONCLUSION We proved that the LPD commucato s subject to a square root law that the umber of bts that ca be covertly trasmtted chael uses s O( ). A terestg result our work s the fact that oe ca use all of the symbols wth postve power to trasmt the covert messages. A promsg drecto of future research s the exteso of ths work to a practcal etworked settg. 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