State Amplification Subject To Masking Constraints

1 State Ampliﬁcation Subject T o Masking Constraints O. Ozan Ko yluoglu, Raji v Sou ndararajan, and Sriram V ishwanath Abstract This paper considers a stat e dependent broadcast channel w ith one transmitter, Alice, and two recei v ers, Bob and Eve. The problem is to effecti ve ly con ve y (“amplify”) the channel st ate sequence to Bob while “masking” it from Eve. The extent to which the state sequence cannot be masked from Eve i s referred to as leakage. T his can be vie wed as a secrecy problem, where we desire that the channel state itself be minimally leaked to Eve while being communicated to B ob . The paper is aimed at characterizing the t rade-of f region between ampliﬁcation and leakage rates for such a system. An achiev able coding scheme is presented, wherein the transmitter transmits a partial state information ov er the chann el to facilitate the ampliﬁcation proces s. For the case when Bob ob serves a stronger signal than Eve, the achiev able coding scheme is enhanced wit h secure reﬁnement. Outer bounds on t he trade-off region are also deriv ed, and used in characterizing some special case results. In particular, the optimal ampliﬁcation-leakage rate differe nce, called as differential ampliﬁcation capacity , is characterized for the reverse ly degraded discrete memoryless channel, the degrade d binary , and the degraded Gaussian channels. In addition, for t he degraded Gaussian model, the extremal corner points of the t rade-of f region are characterized, and the gap between the outer bound and achiev able rate-regions i s sho wn to be less t han half a bit for a wide set of channel parameters. I . I N T R O D U C T I O N A. Pr oblem S tatement In this paper, we con sider a state d epende nt b roadcast channe l m odel with two u sers, and in vestigate the q uestion of to what extent the state of the chann el c an be ampliﬁed at th e receiver ( Bob) an d ma sked from th e oth er rece i ver (referre d to as Eve). Th e en tire ch annel state seq uence is presumed to be known non-cau sally to the transmitter (Alice). T he only manner to affect th e state info rmation at Bob and Eve is by the encodin g scheme used at the transmitter . For such a system, we aim to c haracterize the trade-o ff be tween the “amp liﬁcation”-rate R a (at which the legitimate pair c an opera te) an d the “leakag e”-rate R l (to the e av esdropp er). Formally , co nsider a discrete memoryless chan nel g i ven by p ( y , z | x, s ) , wh ere x ∈ X is th e chan nel inp ut, s ∈ S is the channel state, an d ( y , z ) ∈ ( Y × Z ) is the chan nel outp ut. Here , y cor respond s to th e r eceived chann el ou tput O. Ozan Koylu oglu is with the Department of Electrica l and Computer Engineeri ng, The Unive rsity of Arizona, Tucson, AZ. Email: ozan@ema il.arizon a.edu. Rajiv Soundararaja n is with Qualcomm R esearch India, Bangalore, In dia. Email: raji vs@ute xas.edu. Sriram V ishwa nath is with the Department of Electric al and Computer Engineering, The Uni versi ty of T ex as, Austin, TX. Email: sriram@austin.ute xas.edu. The m ateria l in this paper was presented in part at the Forty-Nint h Annual Allerton Conference on Communication, Control, and Computing, Montice llo, IL in S eptember 2011. Novem ber 19, 2018 DRAFT 2 Z n , R l ≥ 1 n I ( S n ; Z n ) X n ( S n ) Enco der Channel p ( y , z | x , s ) S n ∼ Q n i =1 p ( s i ) Y n , R a ≤ 1 n I ( S n ; Y n ) Fig. 1. The s ystem model for ampliﬁcation subject to masking problem. at the legitimate receiver (Bob) and z is th e output at the eavesdropper (Eve). Th e channe l is memoryless in the sense that p ( Y n = y n , Z n = z n | X n = x n , S n = s n ) = n Y i =1 p ( y i , z i | x i , s i ) , and th e state seq uence S n is independe nt an d ide ntically d istributed (i.i.d.) according to a prob ability distribution indicated b y p ( s ) . I t is assumed that the chan nel state sequence is non- causally known at the tr ansmitter . The system model is g iv en in Fig 1. The task of the e ncoder is to “amplify ” th e state sequence at Bob (chan nel o utput Y n ) and to “ma sk” the state sequence f rom Eve ( Z n ). Formally , the encoder Enc ( S n , n ) is a mapp ing of channe l state S n to the chan nel inpu t X n , i.e., En c : S n → X n , which can b e characterize d by a co nditional probab ility distribution p ( x n | s n ) . ( R a , R l ) is said to be achiev able, if for any given ǫ > 0 , ther e exist a seque nce o f encod ers { E nc ( S n , n ) } such that 1 n I ( S n ; Y n ) ≥ R a − ǫ (1) 1 n I ( S n ; Z n ) ≤ R l + ǫ (2) for suf ﬁciently large n , where the mutual information terms are with respect to the jo int distrib ution p ( s n , x n , y n , z n ) = p ( x n | s n ) n Q i =1 p ( s i ) p ( y i , z i | s i , x i ) . The p roblem is to character ize all ach iev able ( R a , R l ) pair s, which we de note by the trad e-off region C . The perform ance of the enco der is also quantiﬁed by m easuring th e d ifference between the ac hiev able ampliﬁ- cation and leaka ge r ates. Th e differ ential amp liﬁcation rate R d is said to be achievable if R d = R a − R l for some ( R a , R l ) ∈ C . The m aximum value of the differential ampliﬁcation rate is called as th e differ ential ampliﬁca tion capacity , deno ted by C d , wher e C d = sup ( R a ,R l ) ∈C R a − R l . Note that this quantity is a p roper ty of a giv en trad e-off region C and can play a role for so me applications, as this difference measures the knowledge difference between the two receivers regarding the state of the chann el. (This quantity , C d , is d iscussed further in the later p arts of the seq uel.) In general, we ar e not only in terested in the knowledge difference, but also in the en tire rate trad e-off region, which con stitutes the main foc us of th is paper . Novem ber 19, 2018 DRAFT 3 A cost c onstraint may also be impo sed on the chann el inp ut with 1 n n X i =1 E { c ( X i ) } ≤ C, (3) where c : X → R + deﬁnes the cost per in put letter and the expectation is over th e distribution of the channel inp ut. In this scenario, we say ( R a , R l ) is achie vable und er th e cost function c ( . ) a nd at expec ted co st C , if th ere exists a sequence of encod ers satisfying (1 ), ( 2), and (3) in the limit of large n . (W e use this con straint f or the Gaussian channel, where th e cost is the av erage transmitted p ower .) Finally , we n ote that the equivocation r ate (denoted by ∆ l ) can be used in this fo rmulation as well. In particular, ∆ l is said to be achiev able, if there exists a seq uence of e ncoder s satisfying ∆ l ≤ 1 n H ( S n | Z n ) + ǫ, for sufﬁciently large n . Acco rding ly , the ach iev able ( R a , ∆ l ) pairs ca n be deﬁned . Hen ce, the pro blem can be re-form ulated in term s of equivocation rate, where we seek to characterize all achievable ( R a , ∆ l ) pairs in the limit of large n . Since both the eq uiv ocation and leakage rate n otions char acterize the same trade-o ff, both notions can be used interchangeab ly . B. Related works a nd app lications The problem of communicatio n over state dep enden t channels is studied by Gel’fand a nd Pinsker [1 ], where a message has to be reliab ly transmitted over the chann el with n on-cau sal state knowledge at the transmitter . The Gaussian version of th e prob lem is solved in [2] th rough the famou s d irty paper codin g scheme. While th e wiretap channel is intr oduced and solved in [3], these results are extended to a broadcast setting in [4]. T he p roblem of sending secure message s over state d epend ent wir etap chann els is studied in [5], [6]. On the other hand , th e problem s of state amp liﬁcation and state masking are individually so lved in [7], [ 8], [9] for point-to-po int chan nels. Both [7], [8] and [9] con sider the pr oblem of reliable tr ansmission of messages in addition to state amp liﬁcation and state masking, r espectively . I n this p aper, we consider th e p roblem o f amplifying the state to a desired recei ver while trying to minimize the leakage to ( or mask the state from) th e ea vesdropper . W e note that, if we set R a = 0 in our pr oblem deﬁnition, it reduces to the state masking problem as studied in [9 ]. In other word s, R a = 0 R l = min p ( x | s ) s.t. E { c ( X ) }≤ C I ( S ; Z ) can b e shown to be achievable [9]. Also, when R l ≥ H ( S ) , the pro blem red uces to a state amp liﬁcation pro blem [8], and one can achieve the fo llowing r ate pair . R a = min { H ( S ) , max p ( x | s ) I ( X, S ; Y ) } R l ≥ H ( S ) Novem ber 19, 2018 DRAFT 4 These repr esent two extreme s of the trade-o ff r egion between the ampliﬁcatio n and maskin g rates. The main focus of this pa per is to character ize the entire trade-off r egion of ampliﬁcation a nd leakage r ates. In the following, we list some a pplications of the proposed model. 1) Br oadcast channels: Communication ov er state depende nt channels ha ve a wide set of applications in broadcast channels, especially for the MIMO systems [ 10], [11], [12]. In such multi-user channels, a n inform ation-carr ying signal can be modeled as a state sequence for an other . As the transmitter knows the infor mation-car rying signal for the ﬁrst user, it can b e treated as a known state for the transmission to the seco nd user . I n a ddition, th e secon d sig nal (intended for the second user) can be con sidered as an overlay to th e ﬁrst one, espec ially for multicastin g scenarios. (Here, th e codeword of the ﬁrst signal m ay fo llow an i.i.d . pr ocess, or this can be an un coded inf ormation [9].) In such multi-user settings, the secur ity of the commu nication arise as an impo rtant pro blem due to the br oadcast nature of the m edium. Utilizing the f ramework g i ven in this paper, the amp liﬁcation an d leakag e o f the signals can be analyzed . 2) Cognitive radio and r elay channels: Another relev ant setting wher e ou r r esults will b e of signiﬁcant interest is cogn itiv e r adio systems [13], [1 4], [15], [ 16], [ 17], [18], [19], [20]. For example, in an overlay c onginitive radio scenario [16], the cog nitive en coder (Alice) can facilitate the secure co mmunica tion of the prim ary signal (th e state sequ ence of the chann el) by amplifying the signal at the prim ary receiver (Bob) while masking it fro m the eav esdropp er (Eve). In such an application, the codeboo k inform ation of the primary signal may be ab sent at th e cognitive radio, an d the for mulation we h av e in this paper would be r elev ant. Mor e generally , this setting belong s to user cooper ation o r relay ing arch itectures that ca n incr ease the security of the com munication systems [21], [2 2], [23], [ 24]. In an anoth er cognitive radio setup, two c ognitive users (Alice and Bob ) can utilize the primary sign al (interferin g sequence S n ) to sha re a secre t key betwee n eac h o ther in the presence of an e av esdropp er . (Ap plications to key sharing from chan nel states are detailed in the following.) 3) Secr et ke y generation fr om channel states: In recent years, a b ridge between crypto graphy and informatio n theoretic security has emerged in the f orm o f channe l state-depen dent key g eneration [25], [26], [27], [28]. In this line of work , a state depen dent channel (such as the wireless chann el) is conside red, and a fun ction of this state is intended to be a “shared secret” between the legitimate transmitter (Alice) and legitimate receiv er ( Bob) while aiming to keep th e eavesdropper ( Eve) as much in the d ark as p ossible. I n the b est case, th e state(s) seen by Bob and Eve will b e com pletely d ifferent (indepen dent). Howev er , th e states of the chann els are usually depen dent. Moreover , for some mode ls (e.g ., the dirty pap er codin g setup [2]), there is a single channel state deﬁn ing the channel for b oth Bob and Eve. (For example, Y = X + S + N Bob at Bob an d Z = X + S + N Eve at Eve as considered in [ 29]. See also the discussion ab ove for c ognitive radio chann els.) For su ch scenarios, a s long as there is a n on-trivial d ifference between the am pliﬁcation an d leakag e rates, the state knowledge can b e u sed to develop shared keys an d enable cryptog raphic algorithms. For in stance, u tilizing the coding sch emes pr oposed here, Bob can have nR a bits of inf ormation regarding the ch annel state, where a t most nR l number of these bits a re leaked to Eve. Then, priv acy amp liﬁcation [3 0] can b e utilized to distill secret keys. T hat is, the m ethods p rovided in this paper can be utilized for the “advantage distillation” phase of a key agreem ent pro tocol [30]. In this for m, the Novem ber 19, 2018 DRAFT 5 problem at hand is very much related to the gen eration of secrecy using sources and channels. Here, no t only the source in the m odel is the chan nel state (different from the models studied in [3 1], [32]), but also it is non -trivially combined with the e ncoded signals of Alice (via the state dependen t ch annel) to prod uce the ob servations at Bob and Eve. Henc e, our formu lation can be closely associated with th e p roblem of secret key agre ement over state depend ent chan nels [33], [29]. In such pro blems, one is interested in th e design of coding strategies that allow an agreemen t of secu re bits between the legitimate user s utilizing the state depen dent channel. F or exam ple, sending secure b its over the chann el will increase th e secret key rate [29], [ 5], [ 6]. On the o ther h and, the pro blem studied in this work, when specialized to the differential am pliﬁcation measure, is an an alysis of the channel state knowledge difference. T his analy sis is very much related to th e question of how m any secr et bits ca n be extracted from a giv en channel state ([2 5], [26], [27], [28]). W e inten d to provide information -theor etic gu arantees (achiev able r ates and upper bou nds) to th e extent to wh ich such a shared secret can be realized for our system mo del 1 . Once obtain ed, this shared secret can now be e mployed to seed n umero us symmetric key cryp tosystems [ 34], [35]. 4) W atermarking and chann el estimation : Consider that a host im age S is utilized in a watermarking scenario, where th e encode r modiﬁes the image in or der to amp lify it at the receiv er Y and mask it from rec eiv er Z . T his model is similar to c hannel estimation scenario s in wirele ss com munication s. F or exam ple, a pilo t signal can be constructed at th e transmitter such th at it n ot only facilitates the fading g ain estimation at the receiver but also hides the chann el state f rom the eav esdropp ers as much as possible. For in stance, while co mmun icating to a ba se station for c hannel state estimation, a mo bile user may want to hid e this informa tion f rom the external no des in order to establish pr iv acy of her location . C. Summary of results a nd organization In this p aper, we aim at developing an u nderstand ing o f the trade-off between ampliﬁcation and ma sking r ates in a state depende nt broa dcast chann el throu gh ach iev able regions and outer bou nds, an d ch aracterizing spec ial cases when they match. The main results of this paper can be summarized as fo llows. 1) Achievable Regions: The main achie vability argum ent o f the paper utilizes the transmission of a state dependent message over the state d epend ent chan nel. T o facilitate this, we construct a (Gel’fand-Pinsker) co deboo k, where in the corr espondin g code word (denoted by U n ) carry this re ﬁnement infor mation in such a way that reliab le commu ni- cation c an b e ach iev ed over the state-d epende nt channel. Su bsequently , we utilize this r eﬁnement in formatio n within a W yner-Ziv co ding scheme to derive expressions f or the achiev able amp liﬁcation rates. In particular, we show that, ev en though the side inf ormation is no t gener ated in an i.i.d . fashion, W yner-Ziv appr oach can b e used to facilitate ampliﬁcation proc ess. The leak age rates are th en determin ed by d eriving single- letter b ound s on 1 n I ( S n ; Z n ) , and the ach iev able r egions are established over the input pr obability distributions p ( u , x | s ) . The b ounds show that the 1 W e note that differen tial ampliﬁcation capacity is relate d to the secret ke y rate that can be achie v ed utilizing the corresponding channe l state. For example , consider Y = S 1 , Z = S 2 , where S 1 and S 2 are uniform bits. Here, C d = 0 w hen the state is deﬁned as S = [ S 1 , S 2 ] , and C d = 1 when the state is taken as S 1 , implying that 1 bit of secret key rate can be supported. Novem ber 19, 2018 DRAFT 6 rate o f the reﬁnemen t m essage n ot only increases th e amp liﬁcation rate R a , but also increases th e leak age rate R l , thereby establishing a trade-off for im plementatio n. 2) Secure Reﬁnement: W e show that it is possible to extend th e prop osed region by transmitting secur e reﬁnement informa tion when Bob obser ves a “stronger ” chan nel than Eve. In prec ise term s, this correspon ds to instances of p ( u, x | s ) satisfy ing I ( U ; Y ) ≥ I ( U ; Z ) . Note that a channe l is said to have a less noisy struc ture if I ( U ; Y ) ≥ I ( U ; Z ) fo r all input probability distrib utions [36], [37 ]. W e ﬁnd t hat the utilization of the notion of secure reﬁnement approa ch is critical to such chann els, and we show tha t the le akage due to transmission o f the me ssage can be minimized by secur ing the message. In the proc ess of establishing these results, w e also develop an alter nate proof of secure me ssage transmission over state dep endent wiretap chann els. 3) Special Classes o f Chann els: Our outer b ound argumen ts are based on up per bou nding 1 n I ( S n ; Y n ) an d lower bo unding 1 n I ( S n ; Z n ) . The ach iev able schem es and outer bou nds presented are used to establish op timality results for a class o f chan nels. In par ticular, we show that the p ropo sed schem e achieves the optimal differential ampliﬁcation cap acity (i.e., the maxim um value of R a − R l over th e set of achievable ( R a , R l ) pa irs) for the reversely degrade d d iscrete mem oryless chan nel (DMC), the degrad ed binary ch annel, and the d egraded Gaussian channel. 4) Gaussian Cha nnels: W e character ize the cor ner points o f the region for the degrade d Gaussian chann el. In th is scenario, we fu rther boun d the gap b etween achiev able an d converse regions, and show th e following: Let us deno te the message capacity of Bob’ s channel as C b = 1 2 log(1 + SNR b ) an d that of E ve’ s channel as C e = 1 2 log(1 + SNR e ) , where SNR b (SNR e ) is the signal-to-n oise ratio o f Bob (respectively , E ve). Then , for any given leak age rate R l , w e show that the gap between th e up per and lower bou nds o n the amp liﬁcation rate R a is bo unded by C b . Similarly , for any given ampliﬁcatio n r ate R a , we show that the achiev able leaka ge r ate is within C e of the lower bound on R l . In p articular, the correspon ding gap s a re within half a bit when SNR b ≤ 1 and SNR e ≤ 1 , respectively . The rest of the pa per is organized as follows. Section II details the prop osed c oding schemes and presen ts the correspo nding ac hiev able regions. Section I II pr ovides the outer bound s to the trade-off region. Section IV includes optimality discussions and nu merical r esults for special classes of DMCs in cluding th e reversely degraded ch annel, modulo additiv e binary c hannel model, an d the memo ry with defective cells mode l. The Gau ssian c hannel mo del is considered in Section V alon g with corre sponding optimality results. Finally , we conclude the paper in Section VI. Some of the proofs are c ollected in app endices to improve the ﬂow o f the p aper . I I . A C H I E V A B L E R E G I O N S W e consider transmitting state depen dent information over the state dep endent ch annel. This reﬁnemen t infor- mation can be utilized at Bob in resolving some ambigu ity regard ing the chan nel state. W e divide the discussion into two parts, eac h providing a different scheme fo r the transmission o f this reﬁn ement informa tion. A. State enha nced messaging : Reﬁneme nt In order to facilitate th e tr ansmission of information regarding the state sequen ce, we co nsider a n encode r that transmits a state depende nt message over the state depen dent chan nel. Here, by emp loying the Gel’fand-Pin sker Novem ber 19, 2018 DRAFT 7 coding [1], a rate o f I ( U ; Y ) − I ( U ; S ) can be reliably commun icated over the state depe ndent ch annel. Utilizing this co mmun ication rate, Alice can provide a reﬁnement in formatio n to Bob. In particular, we co nsider a W yner-Zi v coding [38] appro ach to pr ovide such a r eﬁnement informa tion (as discussed in [8]) . Here, the side inform ation at the re ceiv er is ( U n , Y n ) - con sisting of Gel’fand-Pinsker codeword U n and the o bserved sequence from the channel Y n . For this side informatio n scenario, we utilize the Gel’fand-Pin sker info rmation r ate a s the bin index o f a covering co dew ord V n . W e note tha t, in the original setup of source cod ing with side informatio n (aka W yn er-Zi v model), the side inf ormation is gener ated i.i.d . with the sour ce sequence. For the scenario con sidered here, th e side informa tion is not in th is fo rm. In th e f ollowing, we show th at this issue can b e resolved, and th e W yner-Ziv coding scheme can be utilized to incre ase the ampliﬁcation rate. On the other hand, this transmission scheme m ay leak some state in formation to Eve. Acco rdingly , we obtain appro priate boun ds on th e leak age r ate th at dep ends on the Gel’fand-Pisker c oding rate u sed fo r the re ﬁnement. The correspon ding achievable region is giv en b y the f ollowing. Theor em 1: [Reﬁnemen t] Let R 1 be the clo sure of the union of all ( R a , R l ) p airs satisfying R a ≤ I ( S ; Y , U ) + R r R l ≥ min  I ( U, S ; Z ) , I ( S ; Z, U ) + R r  R r ≤ min { I ( U ; Y ) − I ( U ; S ) , H ( S | Y , U ) } , over all distributions p ( u, x | s ) satisfying I ( U ; Y ) ≥ I ( U ; S ) . T hen, R 1 ⊆ C . 1) Pr oof of Theorem 1 : W e pr ovide main step s o f th e coding argum ent here and relegate some of the d etails to append ices. Codeboo k Generation: Fix a p ( u | s ) and a p ( v | s ) . (Th e requirem ent on these distributions will be speciﬁed later .) Rand omly and indepen dently gen erate 2 nR v sequences V n ( l ) , l ∈ [1 , 2 nR v ] , each accord ing to Q n i =1 p V ( v i ) . Partition indices l in to equal-size subsets r eferred to a s b ins B ( b ) = [( b − 1)2 n ( R v − R r ) + 1 : b 2 n ( R v − R r ) ] , b ∈ [1 : 2 nR r ] . (Th is is the cod ebook used for W yner-Ziv codin g [38], [39].) For each m ∈ [1 : 2 nR m ] , gen erate 2 n ( R u − R m ) number o f U n sequences ra ndomly and in depend ently according to Q n i =1 p U ( u i ) . In dex these sequences as U n ( m, k ) with k ∈ [1 : 2 n ( R u − R m ) ] . (Th is is the codeb ook used for Gel’fand -Pinsker codin g [ 1], [39]. ) Encodin g: Here, the en coder sets R m = R r and m = b and transmits message b of rate R r with Gel’ fand- Pinsker coding in o rder to perf orm reﬁnemen t via W yn er-Zi v co ding. Given s n , the en coder ﬁn ds an ind ex l such that ( s n , v n ( l )) ∈ T ( n ) ǫ ′ . If there exists more than o ne index, it selects one u niformly ra ndom amo ng these. If th ere exists no such ind ex, it selects on e un iformly rand om f rom [1 , 2 nR v ] . Fr om l , encod er determin es th e bin index b in the V n codebo ok such th at l ∈ B ( b ) . Then, it sets m = b an d ﬁnd s a n ind ex k such that ( s n , u ( n ) ( m, k )) ∈ T ( n ) ǫ ′ . If no such (covering ) index k exists or if ther e are more than on e, the encod er pic ks o ne unifor mly at r andom . The encoder then tr ansmits x n that is g enerated i.i.d. ac cording to Q n i =1 p X | U,S ( x i | u i ( m, k ) , s i ) . Ampliﬁcatio n rate an alysis: In the following, we show that the decoder ca n obtain U n codeword u sing Gel’ fand- Pinsker decodin g and then V n codeword by utilizing the side in formatio n ( U n , Y n ) . (This is th e discussion provide d in [8] f or a state am pliﬁcation mec hanism, which we d etail h ere.) W e then pr ovide a deriv ation of the state ampliﬁcation rate utilizing this deco ding m echanism. Novem ber 19, 2018 DRAFT 8 Let ǫ > ǫ ′′ > ǫ ′ . Upon receiving y n , decoder declares th at ˆ m ∈ [1 : 2 nR r ] and ˆ k ∈ [1 : 2 n ( R u − R r ) ] are chosen at the encoder, if these are the uniq ue indice s su ch that ( u n ( ˆ m, ˆ k ) , y n ) ∈ T ( n ) ǫ , otherwise it declares an erro r . (W e rema rk that compare d to the Gel’fand-Pinsker setup [1], [3 9], wher e on ly the message m has to be u niquely decoded , we require decode r to obtain the co deword u n chosen at th e encod er . This w ill be u tilized in de coding error analysis.) Then, the decoder ﬁnds the unique index ˆ l ∈ B ( ˆ m ) such that ( v n ( ˆ l ) , u n ( ˆ m, ˆ k ) , y n ) ∈ T ( n ) ǫ . Otherwise, it declares an error . Consider th e erro r event E = { ˆ M 6 = M , ˆ K 6 = K, ˆ L 6 = L } , wh ere ( K, L, M ) are th e indices chosen at the transmitter, and ( ˆ K , ˆ L, ˆ M ) are the indices decoded at the receiver . Decod er makes an er ror on ly if on e or mor e of the fo llowing e vents occur . E 1 = { ( U n ( M , k ) , S n ) / ∈ T ( n ) ǫ ′ for all k } E 2 = { ( U n ( M , K ) , Y n ) / ∈ T ( n ) ǫ } E 3 = { ( U n ( m, k ) , Y n ) ∈ T ( n ) ǫ for some m 6 = M } E 4 = { ( U n ( M , k ) , Y n ) ∈ T ( n ) ǫ for some k 6 = K } E 5 = { ( V n ( l ) , S n ) / ∈ T ( n ) ǫ ′ for all l } E 6 = { ( V n ( L ) , S n , U n ( M , K ) , Y n ) / ∈ T ( n ) ǫ } E 7 = { ( V n ( l ) , U n ( M , K ) , Y n ) ∈ T ( n ) ǫ for some l 6 = L, l ∈ B ( M ) } Here, E = ∪ 7 j =1 E j . By the union of events b ound , we h av e Pr {E } ≤ Pr {E 1 } + Pr {E c 1 ∩ E 2 } + Pr {E 3 } + Pr {E c 3 ∩ E 4 } + Pr {E 5 } + Pr {E c 1 ∩ E c 5 ∩ E 6 } + Pr {E 7 } . W e bou nd each term above in th e fo llowing. a) By covering lemma [39], P r {E 1 } → 0 as n → ∞ , if R u − R m > I ( U ; S ) + δ ( ǫ ′ ) . (4) b) E c 1 implies that ( U n ( M , K ) , S n ) ∈ T ( n ) ǫ ′ , which implies that ( U n ( M , K ) , S n , X n ) ∈ T ( n ) ǫ ′′ due to i.i.d. generation o f x i from ( s i , u i ) a nd the cond itional typicality lemma [3 9] fo r som e ǫ ′′ > ǫ ′ . Similarly , as Y n is generated i.i.d. f rom ( s i , u i , x i ) thro ugh ( s i , x i ) , we have Pr {E c 1 ∩ E 2 } → 0 as n → ∞ fo r some ǫ > ǫ ′′ , ag ain due to the co nditional typicality lemma [39]. c) E ach U n ( m, k ) for some m 6 = M is distributed i.i.d. n Q i =1 p U ( u i ) and indepen dent of Y n . By packing lemma [39], we hav e Pr {E 3 } → 0 as n → ∞ , if R u < I ( U ; Y ) − δ ( ǫ ) . (5) The three error a nalyses above are the same a rguments u sed for dec oding the m essage index in Gel’fand- Pinsker coding [1], [39]. d) The analysis for showing that P r {E c 3 ∩ E 4 } → 0 as n → ∞ is d etailed in Append ix A. W e use the arguments giv en in [40] in or der to show this. Th e original pro blem stud ied in [40] do es n ot in volve a state-depend ent chan nel, Novem ber 19, 2018 DRAFT 9 but the cod ing scheme con structs chann el in puts via x ( u, s ) , which can be viewed as a state dep endent chann el. (The argumen t we have here - the Gel’fand-Pinsker cod ew ord decoded at the d ecoder is the same as the o ne chosen at the encoder - also app ears in [41], which states that this obser vation follows f rom the argu ments in [40], [42].) e) By covering lemma [39], P r {E 5 } → 0 as n → ∞ , if R v > I ( S ; V ) + δ ( ǫ ′ ) . (6) f) For analyzin g Pr {E c 1 ∩ E c 5 ∩ E 6 } , we n ote that one m ay no t u tilize c ondition al typicality le mma as d one in the pr oof of W yner-Ziv co ding (see, e. g., [39, Section 11.3 .1]). Because, the side informatio n her e, ( U n , Y n ) , is not generated i.i.d. throug h p U,Y | S ( u i , y i | s i ) . Therefo re, o ne can not g o from joint typicality of ( V n ( L ) , S n ) to the joint typicality o f ( V n ( L ) , S n , U n ( M , K ) , Y n ) . Instead , we consider th e following ap proach: E c 5 implies that ( V n ( L ) , S n ) ∈ T ( n ) ǫ ′ , (7) and E c 1 implies th at ( U n ( M , K ) , S n ) ∈ T ( n ) ǫ ′ . Now , consider a pair ( v n , s n ) ∈ T ( n ) ǫ ′ . In addition , we have Pr { U n ( M , K ) = u n | S n = s n , V n ( L ) = v n } = Pr { U n ( M , K ) = u n | S n = s n } such th at the pm f p ( u n | s n ) satisﬁes the fo llowing two cond itions: T he ﬁrst condition is that lim n →∞ Pr { ( s n , U n ( M , K )) ∈ T ( n ) ǫ ′ ( S, U ) } = 1 , (8) which is d ue to the fact that probab ility of th e E c 1 ev ent vanishes. An d, the secon d cond ition is that, for every u n ∈ T ( n ) ǫ ′ ( U | s n ) and fo r sufﬁciently large n , 2 − n ( H ( U | S )+ δ ( ǫ ′ )) ≤ p ( u n | s n ) ≤ 2 − n ( H ( U | S ) − δ ( ǫ ′ )) (9) for some δ ( ǫ ′ ) → 0 as ǫ ′ → 0 . This secon d assertion, i.e., (9), follows f rom [39, Lem ma 12. 3]. Now , from (7 ), (8), (9), and the fact that V → S → U forms a Markov chain, we obtain from Markov lemma [39] that, for some ǫ ′′ > ǫ ′ , lim n →∞ Pr { V n ( L ) , S n , U n ( M , K ) ∈ T ( n ) ǫ ′′ | V n ( L ) = v n , S n = s n } = 1 . Denoting ˜ E 6 = { V n ( L ) , S n , U n ( M , K ) / ∈ T ( n ) ǫ ′′ } , th e analysis a bove implies that Pr {E c 1 ∩ E c 5 ∩ ˜ E 6 } → 0 as n → ∞ . (W e n ote that, an analy sis similar to the on e above is gi ven for the B erger-T un g inner bound in [39]. The dif ference here is that the bin index of the co vering code- word V n determines the M index of U n . Still, given a realization of v n , and hence m , we have covering sequ ences, i.e., the set { U n ( m, k ) } | 2 n ( R u − R r ) k =1 , for s n , fro m which the argumen t follows.) Then, by condition al typic ality lemma [ 39], for som e ǫ > ǫ ′′ , we have ( V n ( L ) , S n , U n ( M , K ) , Y n ) ∈ T ( n ) ǫ , as ( V n ( L ) , S n , U n ( M , K )) ∈ T ( n ) ǫ ′′ (due to having E c 1 ∩ E c 5 ∩ ˜ E 6 w .h. p.), and Y n | ( V n ( L ) = v n , S n = s n , U n ( M , K ) = u n ) ∼ n Q i =1 p Y | U,S ( y i | u i , s i ) . This imp lies th at Pr {E c 1 ∩ E c 5 ∩ E 6 } → 0 as n → ∞ . g) Finally , Pr {E 7 } analysis fo llows as that f or the pro of o f W yner-Ziv c oding (see, e.g. , [39, Section 11 .3.1] ). In particular, con sidering the ev ent ˜ E 7 = { ( V n ( l ) , U n ( M , K ) , Y n ) ∈ T ( n ) ǫ for some l ∈ B (1) , M 6 = 1 } , [39, Lemma 11.1] shows that Pr {E 7 } ≤ P r { ˜ E 7 } . Th en, as each V n ( l ) with l ∈ B (1) is genera ted b y n Q i =1 p V ( v i ) an d indepen dent of ( U n ( M , K ) , Y n ) , fro m packing lemm a [39], Pr { ˜ E 7 } → 0 as n → ∞ , if R v − R r < I ( V ; U, Y ) − δ ( ǫ ) . (10) Novem ber 19, 2018 DRAFT 10 Therefo re, Pr {E 7 } → 0 as n → ∞ , if (10 ) h olds. The analysis above imp lies that Pr {E } → 0 as n → ∞ if (4), (5), (6 ), and (10) hold. Here, we set R m = R r and R u − R r = I ( U ; S ) + δ 1 and R v = I ( V ; S ) + δ 2 , for some arbitrarily small δ 1 and δ 2 . This will satisfy (4) and (6). Furthermore , we set R r = I ( V ; S ) − I ( V ; U, Y ) + δ 2 = I ( V ; S | U, Y ) + δ 2 , which is the r ate req uired to describe V n to the decoder . Th is will satisfy (1 0). Finally , f or a g iv en p ( u | s ) , we c hoose some p ( v | s ) to suppor t transmission o f the reﬁnement message b = m with rate R r = R m < I ( U ; Y ) − I ( U ; S ) − δ 1 . This will satisfy (5), as R u = R r + ( I ( U ; S ) + δ 1 ) < I ( U ; Y ) . W e note that, R r ≤ H ( S | U , Y ) + δ 2 as I ( V ; S | U, Y ) ≤ H ( S | U, Y ) in above. Therefo re, for a g iv en p ( u | s ) , th e propo sed coding scheme su pports any state reﬁn ement rate R r satisfying R r ≤ min { I ( U ; Y ) − I ( U ; S ) , H ( S | U, Y ) } . Utilizing the analysis above, we now detail the der iv ation o f the ampliﬁcation rate. W e start by d eﬁning an ev ent indicator random variable E . Consider setting E = 1 , if a decodin g error ( E ) occur s o r the state seq uence observed at the enco der ( S n ) turn s ou t to be no n-typica l. Set E = 0 , otherwise. W e con tinue with the following set of relations. (W e dr op the ind icies of codewords.) 1 n I ( S n ; Y n ) ( a ) ≥ 1 n I ( S n ; Y n | E ) − 1 n = 1 n I ( S n ; Y n | E = 0) Pr { E = 0 } + 1 n I ( S n ; Y n | E = 1) Pr { E = 1 } − 1 n ( b ) ≥ 1 n I ( S n ; Y n | E = 0) Pr { E = 0 } − 1 n ( c ) = 1 n I ( S n ; Y n , U n , V n | E = 0 )(1 − Pr { E = 1 } ) − 1 n ( d ) ≥ 1 n I ( S n ; Y n , U n , V n | E = 0 ) − ( H ( S ) + ǫ 1 ) Pr { E = 1 } ) − 1 n ( e ) ≥ ( H ( S ) − ǫ 1 ) − 1 n H ( S n | Y n , U n , V n , E = 0) − ǫ 2 ( f ) ≥ ( H ( S ) − ǫ 1 ) − H ( S | Y , U , V ) − ǫ 2 ( g ) = I ( S ; Y , U ) + R r − ( ǫ 1 + ǫ 2 + δ 2 ) , where ( a) follows by the ind icator e vent cond itioning lemm a given in Appendix B, ( b) f ollows as I ( S n ; Y n | E = 1) ≥ 0 , (c) is d ue to the deﬁn ition of E , wherein E = 0 imp lies that E c , i.e., the d ecodab ility of the c odewords ( U n ( M , K ) , V n ( L )) form Y n , (d) is by I ( S n ; Y n , U n , V n | E = 0) ≤ H ( S n | Y n , U n , V n , E = 0) ≤ H ( S n | E = 0) ≤ n ( H ( S ) + ǫ 1 ) fo r some a rbitrarily small ǫ 1 as E = 0 imp lies that S n is ty pical (and S n is gene rated i.i.d. he re), (e) is by taking ǫ 2 = ( H ( S ) + ǫ 1 ) Pr { E = 1 } ) − 1 n , which can be made arbitrarily small as n → ∞ (as Pr { E = 1 } → 0 as n → ∞ ), and lower boundin g H ( S n | E = 0) ≥ n ( H ( S ) − ǫ 1 ) , which follows as E = 0 imp lies that S n is typical (and S n is gene rated i.i. d. here), (f) fo llows as H ( S n | Y n , U n , V n , E = 0) = n P i =1 H ( S i | S i − 1 1 , Y n , U n , V n , E = 0) ≤ n P i =1 H ( S i | Y i , U i , V i ) = n ( H ( S | Y , U, V )) , and (g) is b y H ( S ) − H ( S | Y , U, V ) = I ( S ; Y , U, V ) = I ( S ; Y , U ) + I ( S ; V | Y , U ) = I ( S ; Y , U ) + R r − δ 2 . ( This fo llows, as we set R r = I ( S ; V | Y , U ) + δ 2 in the coding sch eme.) From the last expression, we identify that any R a ≤ I ( S ; Y , U ) + R r is achiev able. Novem ber 19, 2018 DRAFT 11 Leakage rate an alysis: W e ﬁrst show the ach iev ability of R l ≥ I ( U, S ; Z ) with the f ollowing. 1 n I ( S n ; Z n ) ≤ 1 n I ( U n , S n ; Z n ) = 1 n H ( Z n ) − 1 n H ( Z n | U n , S n ) = 1 n n X i =1 H ( Z i | Z i − 1 1 ) − 1 n n X i =1 H ( Z i | Z i − 1 1 , U n , S n ) ( a ) ≤ 1 n n X i =1 H ( Z i ) − 1 n n X i =1 H ( Z i | U i , S i ) ( b ) = I ( U, S ; Z ) , where (a) fo llows as H ( Z i | Z i − 1 1 ) ≤ H ( Z i ) due to the fact that conditionin g does no t increase the en tropy an d H ( Z i | Z i − 1 1 , U n , S n ) = H ( Z i | U i , S i ) as ( Z i − 1 1 , U i − 1 1 , S i − 1 1 , U n i +1 , S n i +1 ) → ( U i , S i ) → Z i forms a Markov c hain (this is due to i.i.d. g eneration of x i from ( u i , s i ) , and i.i.d. ge neration of z i from ( x i , s i ) ), an d (b) follows as 1 n n P i =1 I ( U i , S i ; Z i ) = I ( U, S ; Z ) . (W e note that sin gle letterizatio n argument for H ( Z n | U n , S n ) above is also given in (2 6) of [9].) Next, we focu s on the achievability of R l ≥ I ( S ; Z , U ) + R r . W e have the f ollowing. 1 n I ( S n ; Z n ) ≤ 1 n I ( U n ( M , K ) , S n ; Z n ) ( a ) = 1 n I ( U n ( M , K ) , M ; Z n ) + 1 n I ( S n ; Z n | U n ( M , K )) ( b ) ≤ 1 n H ( M ) + 1 n H ( U n ( M , K ) | M ) + 1 n I ( S n ; Z n | U n ( M , K )) = 1 n H ( M ) + 1 n H ( U n ( M , K ) | M ) + 1 n H ( Z n | U n ( M , K )) − 1 n H ( Z n | U n ( M , K ) , S n ) ( c ) ≤ R r + I ( U ; S ) + δ 1 + H ( Z | U ) − H ( Z | U, S ) = I ( S ; Z , U ) + R r + δ 1 , where (a) is by adding I ( M ; Z n | U n ( M , K )) = 0 as M can be determ ined f rom U n ( M , K ) f or a g i ven codeboo k, (b) fo llows as I ( U n ( M , K ) , M ; Z n ) ≤ H ( U n ( M , K ) , M ) = H ( M ) + H ( U n ( M , K ) | M ) , (c) is du e to having H ( M ) ≤ nR r (a ran dom variable with 2 nR r values has entropy at most nR r ), H ( U n ( M , K ) | M ) ≤ n ( R u − R r ) = n ( I ( U ; S ) + δ 1 ) (her e, given M , th ere are 2 n ( R u − R r ) number o f U n codewords, and the entro py of th is set is maximized if the index K h as the uniform d istribution), H ( Z n | U n ) = n P i =1 H ( Z i | Z i − 1 1 , U n ) ≤ n P i =1 H ( Z i | U i ) = nH ( Z | U ) , an d having H ( Z n | U n , S n ) = n P i =1 H ( Z i | Z i − 1 1 , U n , S n ) = n P i =1 H ( Z i | U i , S i ) = nH ( Z | U, S ) , where th e second ineq uality holds as ( Z i − 1 1 , U i − 1 1 , S i − 1 1 , U n i +1 , S n i +1 ) → ( U i , S i ) → Z i forms a Markov chain (as d etailed in the previous paragr aph). (W e n ote that sing le letterization arguments f or H ( Z n | U n ) and H ( Z n | U n , S n ) above are also given in (24) a nd (26 ) o f [9].) This con cludes the proof of Theorem 1 , whe re we take the union of the achievable pair s over all p ( u | s ) × p ( x | u, s ) = p ( u, x | s ) d istributions. Novem ber 19, 2018 DRAFT 12 2) Message transmission: W e no te that it is possible to a llocate a p art of the Gel’fand -Pinsker co ding rate in th e scheme propo sed above in ord er to transmit messages. In particular, if ( R a , R l , R r ) satisﬁes the inequalities given in Theo rem 1, the n ( R a , R l , R m ) is achiev able with a message rate of R m = I ( U ; Y ) − I ( U ; S ) − R r . That is, the Gel’fand-Pinsker codin g rate given by I ( U ; Y ) − I ( U ; S ) can be divided into r eﬁnement r ate R r and message rate R m . 3) State seq uence covering : In the region g iv en by Theorem 1 , in creasing R r will not only increase the ampliﬁcation rate but will also incr ease the leak age rate. Th erefor e, for some scenarios, im plementing only a covering scheme migh t be advantageous. By ch oosing an arbitrarily sm all reﬁneme nt rate R r in Theorem 1, the following region can b e achieved. Cor o llary 2: [Covering] Le t R 2 be the clo sure of the u nion of all ( R a , R l ) pairs satisfying R a ≤ I ( S ; Y , U ) R l ≥ min  I ( U, S ; Z ) , I ( S ; Z , U )  over all distributions p ( u, x | s ) satisfying I ( U ; Y ) > I ( U ; S ) . Th en, R 2 ⊆ C . W e no te that the rate of reﬁnem ent, R r = I ( V ; S | U , Y ) + δ 2 , is set to an arbitrarily small value here, and the codeword V n only serves as a covering of th e state sequence. Further im plications of this covering scheme on the ampliﬁcation -leakage region is discussed in Section IV -A. W e note that, by transmission of a covering of the state, the leakag e rate is shown to satisfy R l ≥ min  I ( U, S ; Z ) , I ( S ; Z , U )  . Remarkab ly , o ne can gua rantee su ch a bo und, even if some state d epend ent info rmation is transmitted over the channel. I n particu lar , if the chann el seen b y Bob is stronger th an the one seen by Eve, on e can send the reﬁnem ent info rmation securely over the state depend ent chan nel. T his app roach is detailed in th e next section. B. Secure r eﬁnement Consider all in put distributions p ( u, x | s ) satisfying I ( U ; Y ) ≥ I ( U ; Z ) . For such distributions, it is possible to send reﬁnement information securely over the chan nel. Th is way , the leakage increase due to reﬁnement index is decreased as the security of the index will lo wer th e cor respond ing leakag e rate achieved at Eve compa red to the non-secu red case. I n th e following, we ﬁrst f ocus on tr ansmission of secu re messages over the state dep endent channel, and th en detail the p roposed secure r eﬁnement appr oach. 1) Secure message transmission over state d epende nt channels: Consider that a transmitter wants to sen d a secure message M over the state dependen t chann el in the pr esence of eavesdropper . This pro blem is studie d in [5], and, we will r evisit it h ere. In particular, we giv e a c odebo ok co nstruction and provide a lemma that upp er bound s I ( M ; Z n ) , the leakag e to th e eav esdropp er 2 . This result is th en utilized in the f ollowing part, in showing the propo sed secure reﬁnement appro ach. 2 The codebook we provide here is a s pecia l case of the one proposed in [5 ], which considers an exten ded version for an equiv ocati on rate analysi s. Novem ber 19, 2018 DRAFT 13 Codeboo k Gen eration: W e divide the co deboo k con struction in two pa rts, depend ing on whether I ( U ; Z ) > I ( U ; S ) or not. If I ( U ; Z ) > I ( U ; S ) , generate 2 nR u codewords U n ( M , T , K ) , wh ere M ∈ [1 : 2 nR m ] , K ∈ [1 : 2 n ( I ( U ; S )+ δ ) ] , an d T ∈ [1 : 2 n ( R u − R m − I ( U ; S ) − δ ) ] . Here , T is rand omly selected. W e set R m = I ( U ; Y ) − I ( U ; Z ) and R u = I ( U ; Y ) − δ , which imply H ( T ) = n ( I ( U ; Z ) − I ( U ; S ) − 2 δ ) . If I ( U ; Z ) ≤ I ( U ; S ) , g enerate 2 nR u codewords U n ( M , K ) , where M ∈ [1 : 2 nR m ] , K ∈ [1 : 2 n ( I ( U ; S )+ δ ) ] . W e set R m = I ( U ; Y ) − I ( U ; S ) − 2 δ an d R u = I ( U ; Y ) − δ . In both c ases, M is the secu re message index, an d K is used as a covering index ( similar to the pr evious section). The above codebook constructio n is the same as that of Gel’fand- Pinsker co deboo k (describ ed in the previous section), with the only d ifference b eing that, for pro bability distributions satisfying I ( U ; Z ) > I ( U ; S ) , we select a part of the message as rando m (r epresented as T in the paragrap h ab ove). This enab les to argue that the remaining part of the Gel’fand- Pinsker m essage (represente d as M in th e parag raph above) is secure again st th e eavesdropper (in terms o f the le akage rate). W e ha ve th e following. Lemma 3: For the co deboo k gene ration g i ven ab ove, for so me ǫ → 0 as n → ∞ , I ( M ; Z n ) ≤ n I ( S ; Z , U ) + H ( S n | U n , Z n ) − H ( S n | M , T ) + nǫ, where T is a random variable un iformly d istributed on the set [1 : 2 n ( I ( U ; Z ) − I ( U ; S ) − 2 δ ) ] for I ( U ; Z ) > I ( U ; S ) , and T = ∅ for I ( U ; Z ) ≤ I ( U ; S ) . Pr oof: See Appendix C. This result is u tilized in the next section for the secur e reﬁneme nt approach . Here, we no te the following corollary , which is an alternate pro of of security in state depen dent wiretap channels. Cor o llary 4: For the co deboo k gene ration given above, if M is ind ependen t of S n , then 1 n I ( M ; Z n ) ≤ ǫ , for some ǫ → 0 as n → ∞ . Pr oof: See Appendix D. 2) Secure reﬁnement via secur e and state dependen t message transmission: In the previous sectio n, a reﬁnem ent approa ch is pr oposed wher e a Gel’fand-Pinsker cod ed message is utilized to resolve som e am biguity regarding the channel state at Bob. In such an approach , the message rate is utilized as the b in index of a covering ( V n ) codebo ok. (See proof of Theorem 1.) Here, we consider transmission of this reﬁnement me ssage secu rely to Bob by u tilizing the c odebo ok construction g iv en above. As the message rate is modiﬁed from I ( U ; Y ) − I ( U ; S ) to secure message rate I ( U ; Y ) − max { I ( U ; S ) , I ( U ; Z ) } , this mo diﬁcation results in a reﬁnement rate of R r ≤ min  I ( U ; Y ) − ma x { I ( U ; S ) , I ( U ; Z ) } , H ( S | Y , U )  . Howe ver , the leakage rate due reﬁnement is now independent of R r . The co rrespon ding region is given b y the following. Novem ber 19, 2018 DRAFT 14 Theor em 5: Let R 3 be the clo sure of the union of all ( R a , R l ) pairs satisfying R a ≤ I ( S ; Y , U ) + R r R l ≥ min { I ( U, S ; Z ) , I ( S ; Z, U ) } R r ≤ min  I ( U ; Y ) − max { I ( U ; S ) , I ( U ; Z ) } , H ( S | Y , U )  over all distributions p ( u, x | s ) satisfying I ( U ; Y ) ≥ I ( U ; S ) Th en, R 3 ⊆ C . Note th at, the a mpliﬁcation r ate th at can b e o btained with such an app roach is lower than the previous case (Theor em 1), as the message rate R r ≤ I ( U ; Y ) − I ( U ; Z ) < I ( U ; Y ) − I ( U ; S ) if I ( U ; Z ) > I ( U ; S ) . Therefore, the improvement on the le akage expression co mpared to Theorem 1 is obtained with a degrad ation o n the ampliﬁcation rate. Pr oof: W e use the codeboo k g eneration gi ven ab ove. Th e only difference c ompared to the codebook construction utilized in the proof of Theorem 1 is t hat part of the message (i.e., T ) is selected as random when I ( U ; Z ) > I ( U ; S ) . Therefo re, the ampliﬁcation rate analysis is the same as that of the one given in th e pr oof o f Theorem 1 with R r bound ed b y R r ≤ I ( U ; Y ) − max { I ( U ; S ) , I ( U ; Z ) } in stead of R r ≤ I ( U ; Y ) − I ( U ; S ) . T he proo f of R l ≥ I ( U, S ; Z ) fo llows from the same steps giv en in th e pro of of Theorem 1 . Here, we sh ow that R l ≥ I ( S ; Z , U ) holds as well. Consider the fo llowing. I ( S n ; Z n ) ≤ I ( M , S n ; Z n ) = I ( M ; Z n ) + I ( S n ; Z n | M ) ( a ) ≤ [ nI ( S ; Z , U ) + H ( S n | U n , Z n ) − H ( S n | M , T ) + nǫ ] + H ( S n | M ) − H ( S n | M , Z n ) = nI ( S ; Z , U ) + nǫ + H ( S n | U n , Z n ) − H ( S n | M , Z n ) + H ( S n | M ) − H ( S n | M , T ) ( b ) ≤ nI ( S ; Z , U ) + n ǫ + I ( S n ; T | M ) ( c ) = nI ( S ; Z , U ) + nǫ, where (a) follows fr om Lemma 3, (b) follows as H ( S n | U n , Z n ) = H ( S n | U n , M , Z n ) ≤ H ( S n | M , Z n ) and H ( S n | M ) − H ( S n | M , T ) = I ( S n ; T | M ) , a nd (c) follows by observin g I ( S n ; T | M ) = 0 , where T = ∅ for I ( U ; Z ) ≤ I ( U ; S ) ; an d, T is indepen dently gene rated random variable f or I ( U ; Z ) > I ( U ; S ) , imply ing that H ( T | M ) = H ( T | M , S n ) = H ( T ) . As ǫ ca n b e made arbitrarily small, we conclud e from last expression that any R l ≥ I ( S ; Z , U ) is achiev able. I I I . O U T E R B O U N D S In this sectio n, we provid e outer bound s on the achievable a mpliﬁcation-le akage rate region. In particular, we derive regions d enoted by R o , to which any ach iev able ( R a , R l ) must b elong. Novem ber 19, 2018 DRAFT 15 Pr oposition 6 : If ( R a , R l ) is achiev able, then ( R a , R l ) ∈ R 1 o , where R 1 o is the closure of the unio n of all ( R a , R l ) pairs satisfying R a ≤ min { H ( S ) , I ( X , S ; Y ) } R l ≥ I ( S ; Z, U ) over all p ( u , x | s ) d istributions satisfyin g I ( U ; Z ) ≥ I ( U ; S ) . Pr oof: See Appendix E. If the c hannel is d egraded, wher ein p ( y , z | x, s ) = p ( y | x, s ) p ( z | y ) , the following o uter bou nd can be obtained . Pr oposition 7 : If the channel satis ﬁes p ( y , z | x, s ) = p ( y | x, s ) p ( z | y ) and if ( R a , R l ) is achie vable, th en ( R a , R l ) ∈ R 2 o , where R 2 o is the clo sure of the u nion o f all ( R a , R l ) pairs sarisfying R a ≤ min { H ( S ) , I ( X , S ; Y ) } R l ≥ I ( S ; Z, U ) R a − R l ≤ I ( X , S ; Y | Z ) over all p ( u , x | s ) d istributions satisfyin g I ( U ; Y ) ≥ I ( U ; S ) . Pr oof: See Appendix F . These outer b ound regions are used in the fo llowing to establish special case results. I V . S P E C I A L D I S C R E T E M E M O RY L E S S C H A N N E L M O D E L S A. Reversely degraded DMCs W e say that the ch annel is reversely d egraded if ( X, S ) → Z → Y forms a Mar kov Chain. Note that, this correspo nds to a stronger channel seen by Eve compar ed to that of Bob. There fore, re versely degra ded scen arios imply C d ≤ 0 , mean ing that the state kn owledge at Bob is not highe r than that of at E ve. W e h av e the following result fo r th is set of channels. Theor em 8: The op timal differential am pliﬁcation rate for r ev ersely degraded DMCs is given by C d = max p ( x | s ) I ( S ; Y ) − I ( S ; Z ) Pr oof: Achiev ability of the stated difference f ollows f rom Theorem 1 by sub stituting U = ∅ . W e provide the conv erse in Ap pendix G. Note that co ding can not improve this difference as th e chan nel is reversely degraded. Thus, cod ing might help to increase R a at the expen se of po ssibly de creasing R a − R l for the r ev ersely degrad ed scenario. This is also the case fo r the covering sch eme given by Corollary 2. That is, R a vs. R a − R l can be tr aded-o ff u sing different input distributions. ( U = ∅ case will co rrespon d to the m aximum R a − R l , and achie ve C d .) Novem ber 19, 2018 DRAFT 16 B. Modulo additive b inary mode l Consider the ch annels given by Y i = X i ⊕ S i ⊕ N i Z i = X i ⊕ S i ⊕ ˜ N i , where th e state and noise distributions are gener ated i.i.d. as S i ∼ Bern ( p s ) , N i ∼ Bern ( p n ) , ˜ N i ∼ Bern ( p n z ) . ( All p k s satisfy p k ∈ [0 , 0 . 5 ] for k ∈ { s, n, n z } .) In this section , we use the fo llowing notation for the b inary co n volution p ⊗ q , p (1 − q ) + q (1 − p ) . 1) State can cellation scheme: T o can cel the state fr om the chan nel, we send X i = U i ⊕ S i , where U i ∼ Bern ( p u ) and the co dewords U n carry a description o f the state sequ ence S n . This way , we achieve the fo llowing inner-bound. Cor o llary 9: Th e state ca ncellation scheme, which sends Bern ( p u ) distributed signal XORed with state sequenc e at each time instant, achieves the set o f ( R a , R l ) pairs den oted by the region R SC , wher e R SC = Con vex Hull    [ p u ∈ [0 , 0 . 5] ,p u ⊗ p s ≤ 0 . 5 ( R a ( p u ) , R l ( p u ))    ⊆ C , with R a ( p u ) ≤ min { H ( p s ) , H ( p u ⊗ p n ) − H ( p n ) } R l ( p u ) ≥ H ( p u ⊗ p n z ) − H ( p n z ) . Pr oof: Ac hiev ability fo llows from Corollary 2. 2) Optimal differ ential a mpliﬁcation rate: Cor o llary 10 : If p n ≤ p n z and H ( p s ) ≥ 1 − H ( p n ) for a bin ary mod el, the optim al ampliﬁcatio n an d leaka ge rate difference is given by C d = H ( p n z ) − H ( p n ) . Pr oof: Fr om Propo sition 7 , we o btain the following. If p n ≤ p n z , any given ( R a , R l ) ∈ C satisﬁes R a − R l ≤ H ( p n z ) − H ( p n ) + max p ( x | s ) { H ( X ⊕ S ⊕ N ) − H ( X ⊕ S ⊕ N z ) } . Note that, th is upp er-bound can be evaluated by o bserving max p ( x | s ) { H ( X ⊕ S ⊕ N ) − H ( X ⊕ S ⊕ N z ) } = max p ( x | s ) { H ( X ⊕ S ⊕ N ) − H ( X ⊕ S ⊕ N ⊕ N ∗ z ) } ≤ max p ( x | s ) { H ( X ⊕ S ⊕ N ) − H ( X ⊕ S ⊕ N ⊕ N ∗ z | N ∗ z ) } = 0 , Novem ber 19, 2018 DRAFT 17 0 1 1 0 0 0 1 1 0 0 1 1 S i = 0 w.p. p S i = 1 w.p. q S i = 2 w .p. r X i Y i Y i N i Z i X i Y i X i Y i Fig. 2. Channel model of memory with defecti v e cells. p = Pr { S = 0 } (probability of being stuck at 0 ), q = Pr { S = 1 } (probability of being stuck at 1 ), r = Pr { S = 2 } (probability of being in a noiseless state), and N ∼ B er n ( n ) , where n ∈ [0 , 0 . 5] is the cross ove r probabilit y of the BSC from Y to Z . where th e equ ality is due to the chann el degradedn ess co ndition with appro priate noise term N ∗ z indepen dent of N such that N ⊕ N ∗ z = N z , and the inequality is due to the fact that co nditionin g d oes not increa se th e e ntropy . Using this we observe that the outer-bound is maximiz ed with a cho ice of p ( x ) = 0 . 5 , which evaluates to R a − R l ≤ H ( p n z ) − H ( p n ) . This expression is ach iev ed by Corollary 9, wh en we ch oose p ( u ) = 0 . 5 , if H ( p s ) ≥ 1 − H ( p n ) . C. Memory with defe ctive cells mo del W e co nsider the mo del of infor mation transmission over write-once me mory d evice with stuck -at def ectiv e cells [ 43], [ 44], [ 45]. In this channe l model, ea ch m emory cell cor respond s to a chann el state instan t with card inality |S | = 3 , where the bina ry c hannel outpu t is determ ined from the b inary chann el inp ut and the ch annel state a s: p ( y = 0 | x, s = 0) = 1 p ( y = 1 | x, s = 1) = 1 p ( y = x | x, s = 2 ) = 1 , where Pr { S = 0 } = p is th e prob ability that the cha nnel is stuck at 0 , Pr { S = 1 } = q is the probab ility that the ch annel is stuck at 1 , and Pr { S = 2 } = r is the proba bility of having a goo d chan nel where y = x with p + q + r = 1 . W e con sider a binary symmetric c hannel (BSC) from Y to Z , where Z i = Y i ⊕ N i , with N i ∼ B ern ( n ) for some n ∈ [0 , 0 . 5] . This co rrespon ds to a degraded DM C model. ( See Fig. 2 .) W e p resent numerical results for this channel m odel with three regions: Unc oded region, coded region, and an outer-bound region. The un coded r egion is obtained by setting U = ∅ in The orem 1, where we have the set of Novem ber 19, 2018 DRAFT 18 ( R a , R l ) pairs satisfying R a ≤ I ( S ; Y ) R l ≥ I ( S ; Z ) over all possible p ( x | s ) . For the coded r egion, we simulate a sub-r egion of the one given in Theo rem 1, where we set U = Y and achie ve the set of ( R a , R l ) pairs satisfying R a ≤ min { H ( S ) , H ( Y ) } R l ≥ I ( Y , S ; Z ) = H ( Z ) − H ( N ) over all p ossible p ( x | s ) . For con verse a rguments, we co nsider the o uter-bound region given by the set of ( R a , R l ) pairs satisfying R a ≤ min { H ( S ) , I ( X , S ; Y ) = H ( Y ) } R l ≥ I ( S ; Z ) over all possible p ( x | s ) . This outer-bound r egion follows fr om Proposition 6 . W e evaluate th e regions ab ove in terms of the ch annel par ameters as follows. L et Pr { X = 1 } = α . Then, H ( S ) = H ( p , q , r ) , H ( Y | S ) = r H ( α ) H ( Y ) = H ( q + r α ) H ( Z | S ) = ( p + q ) H ( n ) + rH ( α ⊗ n ) H ( Z ) = H (( q + r α ) ⊗ n ) , where H ( · , · , · ) is the ter nary entropy fu nction, H ( · ) is the binary en tropy f unction , and ⊗ is the binar y co n volution giv en b y p ⊗ q = p (1 − q ) + q (1 − p ) . The numerical results are given in Fig. 3. The regions are truncated with R l ≤ H ( S ) as any R l > H ( S ) is trivially ach iev able. W e n ote th at, the co ded region is potentially larger than its uncod ed counte rparts even when we o nly compute a subset of th e c oded achiev able region. T his sh ows the gains that can b e lev eraged by th e propo sed sch eme, i.e., send ing a reﬁnem ent of the state sequence over th e channel. V . G AU S S I A N S C E N A R I O Consider the ch annels given by Y i = X i + S i + N i Z i = X i + S i + ˜ N i , where th e state and noise distributions are gener ated i.i.d. as S i ∼ N (0 , σ 2 s ) , N i ∼ N (0 , σ 2 n ) , ˜ N i ∼ N (0 , σ 2 n z ) , and the cost co nstraint on th e channel in put is given by c ( x ) = x 2 and C = P , i.e., 1 n n P i =1 E { X 2 i } ≤ P . (See Fig. 4.) Novem ber 19, 2018 DRAFT 19 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 R a R l p=1/3, q=1/3, r=1/3, n=0.1 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 R a R l p=1/12, q=1/12, r=5/6, n=0.1 Uncoded Scheme Coded Scheme Converse Region Fig. 3. Simulation results for memory with defecti ve cells model. S i N i ˜ N i X i Y i Z i Fig. 4. The channel model for the Gaussian setting. S i ∼ N (0 , σ 2 s ) , N i ∼ N (0 , σ 2 n ) , and ˜ N i ∼ N (0 , σ 2 n z ) . A. An inner bound fo r C using an uncod ed scheme The inner bou nd is based on sending an amp liﬁed version o f S together with some addition al G aussian noise. This un coded sign al is co nstructed as fo llows. X i = ρ σ x σ s S i + p (1 − ρ 2 ) σ x T i , (11) where T i ∼ N (0 , 1 ) ind epende nt o f S i , ρ ∈ [ − 1 , 1] , and σ 2 x ≤ P . Here, ρ 2 is the fraction of th e power allocated to S i . This scheme achieves the following region. Theor em 11: The u ncoded scheme , which f orwards S i at ea ch time step toge ther with so me i.i.d . Gaussian noise as given in ( 11), achieves the set of ( R a , R l ) pairs denoted by the region R uncoded , wher e R uncoded = Conve x Hu ll    [ ρ ∈ [ − 1 , 1] ,σ 2 x ∈ [0 ,P ] ( R a ( ρ, σ x ) , R l ( ρ, σ x ))    ⊂ C , Novem ber 19, 2018 DRAFT 20 with R a ( ρ, σ x ) = 1 2 log  1 + σ 2 s + 2 ρσ s σ x + ρ 2 σ 2 x σ 2 n + (1 − ρ 2 ) σ 2 x  (12) R l ( ρ, σ x ) = 1 2 log  1 + σ 2 s + 2 ρσ s σ x + ρ 2 σ 2 x σ 2 n z + (1 − ρ 2 ) σ 2 x  . (13) The expr essions above are obtained b y evaluating R a = I ( S ; Y ) and R l = I ( S ; Z ) on account of u ncode d transmission in (1 1). Examples: • If P ≥ σ 2 s , on e can set X = − S and ach iev e the pair ( R a = 0 , R l = 0) . • Another trivial po int is obtain ed by setting X = 0 , which a chieves  R a = 1 2 log  1 + σ 2 s σ 2 n  , R l = 1 2 log  1 + σ 2 s σ 2 n z  . B. Outer-bounds o n C Cor o llary 12 : Let ρ denote the co rrelation coefﬁcient between X and S . The set of all achiev able rate pairs ( R a , R l ) , satisfy R a ≤ 1 2 log  1 + σ 2 s + σ 2 x + 2 ρσ s σ x σ 2 n  R l ≥ 1 2 log  1 + σ 2 s + ρ 2 σ 2 x + 2 ρσ s σ x σ 2 n z + σ 2 x (1 − ρ 2 )  for − 1 ≤ ρ ≤ 1 an d σ 2 x ≤ P . Pr oof: Usin g Pro position 7 , we ha ve R a ≤ I ( X, S ; Y ) = h ( Y ) − h ( Y | X , S ) = h ( Y ) − h ( N ) ≤ 1 2 log  1 + σ 2 s + 2 ρσ s σ x + σ 2 x σ 2 n  . (14) Using Proposition 7, the linear estimate ˆ S ( Z ) = E [ S Z ] E [ Z 2 ] Z , and the fact the conditioning doe s not increase entropy , we get R l ≥ I ( S ; Z , U ) ≥ I ( S ; Z ) = h ( S ) − h ( S | Z ) = h ( S ) − h ( S − ˆ S ( Z ) | Z ) ≥ h ( S ) − h ( S − ˆ S ( Z )) . Since the en tropy maximizing distribution for a gi ven secon d momen t is a Gaussian, we h av e h ( S − ˆ S ( Z )) ≤ 1 2 log 2 π e   σ 2 s 1 + σ 2 s +2 ρσ s σ x + ρ 2 σ 2 x σ 2 n z + σ 2 x (1 − ρ 2 )   , leading to R l ≥ 1 2 log  1 + σ 2 s + 2 ρσ s σ x + ρ 2 σ 2 x σ 2 n z + σ 2 x (1 − ρ 2 )  . Novem ber 19, 2018 DRAFT 21 Cor o llary 13 : Let ρ den ote the cor relation coefﬁcient between X and S . If σ 2 n ≤ σ 2 n z , then the set of all achiev able r ate pairs ( R a , R l ) satisfy R a − R l ≤ 1 2 log  1 + σ 2 s + 2 ρσ s σ x + σ 2 x σ 2 n  − 1 2 log  1 + σ 2 s + 2 ρσ s σ x + σ 2 x σ 2 n z  , (15) for − 1 ≤ ρ ≤ 1 an d σ 2 x ≤ P . Pr oof: By Proposition 7, we hav e R a − R l ≤ I ( X , S ; Y | Z ) . W ithout loss o f gener ality , we consider ˜ N = N + N ′ with σ 2 n z = σ 2 n + σ 2 n ′ where N ′ is in depend ent of N . Noting that, I ( X, S ; Y | Z ) = h ( Y | Z ) − h ( Y | X, S, Z ) = h ( Y | Z ) − h ( N | ˜ N ) , we upp er bound h ( Y | Z ) using the f ollowing. Consider two ze ro-mean correlated ran dom variables A and B . h ( A | B ) ( a ) = h ( A − ˆ A ( B ) | B ) ≤ h ( A − ˆ A ( B )) ( b ) ≤ 1 2 log(2 π e σ 2 e ) , where in (a) we used ˆ A ( B ) as the estimate of A given B , and ( b) follows by deﬁnin g the estimation er ror variance σ 2 e , E h ( A − ˆ A ( B )) 2 i and the fact tha t Gaussian distribution maximizes en tropy given the variance. W e then upper bou nd the optimal estimato r error variance by the lin ear MMSE variance. Ther efore, h ( A | B ) ≤ 1 2 log 2 π e var ( A ) − E  ( AB ) 2  var ( B ) !! . Using the a bove, we ob tain R a − R l ≤ 1 2 log  2 π e  σ 2 s + 2 ρσ s σ x + σ 2 x + σ 2 n − ( σ 2 s + 2 ρσ s σ x + σ 2 x + σ 2 n ) 2 σ 2 s + 2 ρσ s σ x + σ 2 x + σ 2 n + σ 2 n ′  − 1 2 log  2 π e  σ 2 n − ( σ 2 n ) 2 σ 2 n + σ 2 n ′  = 1 2 log  1 + σ 2 s + 2 ρσ s σ x + σ 2 x σ 2 n  − 1 2 log  1 + σ 2 s + 2 ρσ s σ x + σ 2 x σ 2 n + σ 2 n ′  . (16) This com pletes th e pro of. C. Comparison of in ner and o uter boun ds for the de graded G aussian channel W e n ow compar e the unco ded scheme and the outer bou nd presented above. In particu lar , we show that the uncod ed transmission scheme achieves cer tain corner points o f the am pliﬁcation-m asking region a nd that the gap between the inne r and outer boun ds o n the region is within 1 / 2 bits for a wide set of chann el p arameters. W e also show that the u ncode d sche me achieves the op timal difference R a − R l . Novem ber 19, 2018 DRAFT 22 1) Characterization of the g ap b etween a chievable and conver se re gio ns: W e show that given any po int ( R a , R l ) in the conv erse region correspo nding to a giv en ( ρ, σ x ) , un coded transm ission a chieves within 1 / 2 bits of the con- verse region u nder certain condition s on ch annel param eters. In particu lar , for any giv en R a , unco ded tr ansmission achieves that R a and within 1 / 2 bits of the boun d on R l if P σ 2 n z ≤ 1 . Similarly f or any given R l , un coded transmission achieves the given R l and within 1 / 2 bits of th e boun d on R a if P σ 2 n ≤ 1 . W e p rove the se as f ollows. Using Corollary 12, any po int in th e outer bo und r egion is described as R a = 1 2 log  1 + σ 2 s + σ 2 x + 2 ρσ s σ x σ 2 n  R l = 1 2 log  1 + σ 2 s + ρ 2 σ 2 x + 2 ρσ s σ x σ 2 n z + σ 2 x (1 − ρ 2 )  (17) for − 1 ≤ ρ ≤ 1 and σ 2 x ≤ P . Now le t us show that unco ded transmission achieves a ny R l in the region above an d the gap from R a as ab ove is within 1 / 2 bits. Le t the u ncoded schem e be desig ned such that X i = σ x σ s ρS i + T i , where T i ∼ N (0 , σ 2 x (1 − ρ 2 )) and independ ent of S i . Now , by (13) an d (1 2) th is in put achieves a lea kage, I ( S ; Z ) = 1 2 log  1 + σ 2 s + ρ 2 σ 2 x +2 ρσ s σ x σ 2 n z + σ 2 x (1 − ρ 2 )  and R a giv en b y I ( S ; Y ) = 1 2 log  1 + σ 2 s + ρ 2 σ 2 x +2 ρσ s σ x σ 2 n + σ 2 x (1 − ρ 2 )  , which implies that the g ap is given by I ( X, S ; Y ) − I ( S ; Y ) = I ( X ; Y | S ) = 1 2 log  1 + σ 2 x (1 − ρ 2 ) σ 2 n  ≤ 1 2 , (18) for P σ 2 n ≤ 1 . Now , in ord er to prove th e other claim, that uncod ed ach iev es a ny g iv en R a and the gap with R l is within 1 / 2 bits, we pro ceed as follows. Given R a = 1 2 log  1 + σ 2 s + σ 2 x +2 ρσ s σ x σ 2 n  , we achieve this by choo sing an uncod ed scheme such that X i = σ x ′ σ s S i if ρ ≥ 0 o r X i = − σ x ′ σ s S i if ρ < 0 . W e choose σ x ′ such that σ 2 x ′ + 2 ρ | ρ | σ s σ x ′ = σ 2 x + 2 ρσ s σ x , if ρ 6 = 0 , σ 2 x ′ + 2 σ s σ x ′ = σ 2 x , if ρ = 0 , where 0 ≤ σ x ′ ≤ √ P . By the intermediate value theor em for continuo us fun ctions, it is clear th at ther e exists σ x ′ ≤ √ P such that the cond itions above are satisﬁed. Further, th e unc oded scheme ach iev es the desired R a . For ρ 6 = 0 , the scheme ach iev es an R l giv en by 1 2 log  1 + σ 2 s + σ 2 x ′ +2 ρ | ρ | σ s σ x ′ σ 2 n z  leading to a gap 1 2 log 1 + σ 2 s + σ 2 x ′ + 2 ρ | ρ | σ s σ x ′ σ 2 n z ! − 1 2 log  1 + σ 2 s + ρ 2 σ 2 x + 2 ρσ s σ x σ 2 n z + σ 2 x (1 − ρ 2 )  = 1 2 log  1 + σ 2 s + σ 2 x + 2 ρσ s σ x σ 2 n z  − 1 2 log  1 + σ 2 s + ρ 2 σ 2 x + 2 ρσ s σ x σ 2 n z + σ 2 x (1 − ρ 2 )  = 1 2 log  σ 2 n z + σ 2 s + σ 2 x + 2 ρσ s σ x σ 2 n z  − 1 2 log  σ 2 n z + σ 2 s + σ 2 x + 2 ρσ s σ x σ 2 n z + σ 2 x (1 − ρ 2 )  = 1 2 log  σ 2 n z + σ 2 x (1 − ρ 2 ) σ 2 n z  = 1 2 log  1 + σ 2 x (1 − ρ 2 ) σ 2 n z  ≤ 1 2 log 2 = 1 2 , (19) when P σ 2 n z ≤ 1 . Follo wing the same steps, ρ = 0 case can be shown as well, implying the ch aracterizatio n of the trade-off region within 1 / 2 b its for P σ 2 n z ≤ 1 and P σ 2 n ≤ 1 . Novem ber 19, 2018 DRAFT 23 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5 3 3.5 R a R l P=2, σ s 2 =10, σ n 2 =1, σ n z 2 =5 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5 3 R a R l P=4, σ s 2 =5, σ n 2 =1, σ n z 2 =0.5 Uncoded Scheme Converse Region Fig. 5. Simulation results for the Gaussian scenario. 2) Differ ential amp liﬁcation capacity: Note th at the uncode d transmission ac hieves th e maximu m R a − R l . The upper bou nd on R a − R l in (1 5) is ma ximized for σ 2 x = P and ρ = 1 . Th is m aximum difference between R a and R l is achieved by un coded transmission cor respond ing to X = √ P σ s S in Th eorem 11, and is given by C d = 1 2 log 1 + ( σ s + √ P ) 2 σ 2 n ! − 1 2 log 1 + ( σ s + √ P ) 2 σ 2 n z ! . 3) Corner points of the trade-o ff r e g ion: Consider the corne r points of the ampliﬁcation -masking region . Inspect- ing (14), we obser ve that the point in the outer bound region correspo nding to max imum am pliﬁcation is given by ρ = 1 . Clearly , from (18) and (1 9), we see that the gap is zer o fo r ρ = 1 . Similarly , consider the p oint cor respond ing to min imum leakag e R l in the weak and mo derate interference regimes as in [9]. T hese poin ts correspon d to ρ = − 1 and we have I ( X , S ; Y ) = I ( S ; Y ) and I ( X , S ; Z ) = I ( S ; Z ) , leading to the ga p b eing zero. This is also veriﬁed by setting ρ = − 1 in (18) and (19 ). 4) Numerical r esults: W e compare the unco ded region with an outer-bound region (g i ven in Proposition 12) in Fig. 5 . Th e ﬁrst case co rrespon ds to a degraded scenar io, where the g ap between the r egions is fairly sm all as expected from th e an alysis given ab ove. Howev er , for the reversely degrad ed scenar io with larger power constrain t P comp ared to th e state power σ 2 s , the g ap is larger . In Fig. 6, we plo t the differential ampliﬁcatio n capacity for a degraded chann el ( σ 2 n = 1 , σ 2 n z = 5 ) for a rang e of p ower co nstraints P and different values o f σ 2 s . Note that the differential amp liﬁcation capacity saturates in the high SNR regime, and the effect of encoder in incr easing C d is decreasing as the power of the additive state increases. V I . C O N C L U S I O N W e study th e problem of state ampliﬁcation under the masking constraints, where the encoder (with the knowledge of non-ca usal state S n ) facilitates the ampliﬁcatio n rate ( 1 n I ( S n ; Y n ) ) at Bob, wh ich observes Y n , while minim izing Novem ber 19, 2018 DRAFT 24 −50 −40 −30 −20 −10 0 10 20 30 40 50 0 0.2 0.4 0.6 0.8 1 1.2 1.4 P (dB) C d C d vs. P (dB) for σ n 2 =1, σ n z 2 =5 σ s 2 =20dB σ s 2 =10dB σ s 2 =1dB σ s 2 =−10dB σ s 2 =−20dB Fig. 6. Dif ferenti al ampliﬁcatio n capacit y C d vs. po wer P (dB). T he dashed curv e with diamond marke rs correspond to the same scenario gi ven in the degraded setting of Fig. 5 ( σ 2 s = 10 ) for diff erent power le vel s. the leak age rate ( 1 n I ( S n ; Z n ) ) as muc h as p ossible at Eve, which observes Z n . Ou r codin g schemes ar e ba sed on tr ansmission of state d epend ent messages over the state d epende nt channel to Bob . The achiev able region correspo nding to th is reﬁn ement stra tegy is deriv ed by calculating bounds on ampliﬁcation and m asking r ates. W e also show that f or the input distributions enabling Bob to b e a “stronger” receiver than Eve, th e reﬁneme nt informa tion can be sent securely over the channel. This secure r eﬁnemen t app roach is shown to lead to non - trivial achiev able regions. W e also p rovided outer bou nds, using which we showed that the scheme withou t secure reﬁnement achieves the o ptimal R a − R l over the r egion in the reversely degraded DMCs, the degraded binary channels, a nd Gaussian chann els. For the d egraded Gaussian model, we also c haracterized the optim al cor ner points, and the gap between the outer bound an d achievable regions. Sev eral inter esting pr oblems can be con sidered as future directions. First, the chan nel cost may be introdu ced for th e DMC mod el as well, and th e cost may have some depend ence on the state sequenc e or vary accordin g to a stochastic model. Second, cau sal ch annel state k nowledge can be con sidered. Th ird, in addition to the task of state ampliﬁcation and masking , transm ission of messages to the rece iv ers can be co nsidered . T owards th is end , signaling techniqu es fo r (secure) broa dcast ch annel models can b e u tilized. Anoth er extension direction is the coded state sequen ce setting [45], wh ich is a scenario that is mo re r elev ant to the b roadcast and cogn iti ve radio systems, where the co ded signal (that carries a message from a codebo ok) correspon ds to the chann el state sequence that is non-ca usally kn own at the enco der . Finally , a sou rce coding extension of the cur rent model can be stud ied. In such a problem setup , the trade-off betwe en th e distor tions a chieved at both users can be analyzed . A r elev ant work f or such an extension is [46], where a so urce coding setup is considered with two sources (one has to b e ampliﬁed an d the other has to be masked) with a single receiver . W e rema rk that the ampliﬁcation an d leak age ra tes an alyzed in this paper can be utilized to provide a lower b ound o n the d istortion that can be achieved at Bob an d E ve, Novem ber 19, 2018 DRAFT 25 respectively , by evaluating the correspond ing distortion- rate fun ctions. AC K N O W L E D G E M E N T The auth ors are thankful to Y anling Chen of Ruhr-Univ ersity Boch um, Y eow-Khiang Chia o f In stitute for Infoco mm Research , V incen t Y . F . T an of National University of Singapor e, and anonymo us revie wers fo r their valuable feed back. A P P E N D I X A Pr {E c 3 ∩ E 4 } → 0 A S n → ∞ W e use the arguments gi ven in [4 0] in ord er to show this. Similar to the joint source-c hannel coding scenario studied in [4 0], we have a cod ew ord U n with a covering ind ex K , which makes th e cod ebook and covering index K depend ent to each oth er . In other words, S n sequence r ealization together with th e cod ebook determ ines the index K , from which ( S n , U n ) gen erates Y n throug h an i.i.d. generatio n pr ocess. Under such a scenario, as rep orted in [40], deco ding th e index K with Y n at receiver is successfu l if the numb er o f K ind ices is less than 2 nI ( U ; Y ) . W e now provide this a nalysis her e for completeness. Using the steps g i ven in [4 0] for the scena rio here, we have the following: For a given M , we have 2 nR k number of U n ( M , k ) c odewords ( k ∈ [1 : 2 nR k ] ), and the bo und below . Novem ber 19, 2018 DRAFT 26 Pr {E c 3 ∩ E 4 } = P r { ( U n ( M , k ) , Y n ) ∈ T ( n ) ǫ for some k 6 = K } ( a ) ≤ 2 nR k X k =1 Pr { ( U n ( M , k ) , Y n ) ∈ T ( n ) ǫ , K 6 = k } = 2 nR k X k =1 X s n p ( s n ) Pr { ( U n ( M , k ) , Y n ) ∈ T ( n ) ǫ , K 6 = k | S n = s n } ( b ) = 2 nR k X s n p ( s n ) Pr { ( U n ( M , 1) , Y n ) ∈ T ( n ) ǫ , K 6 = 1 | S n = s n } ≤ 2 nR k X s n p ( s n ) Pr { ( U n ( M , 1) , Y n ) ∈ T ( n ) ǫ | K 6 = 1 , S n = s n } = 2 nR k X s n p ( s n ) X ( u n ,y n ) ∈T ( n ) ǫ Pr { U n ( M , 1) = u n , Y n = y n | K 6 = 1 , S n = s n } ( c ) = 2 nR k X s n p ( s n ) X ( u n ,y n ) ∈T ( n ) ǫ X ¯ C Pr { U n ( M , 1) = u n , Y n = y n | K 6 = 1 , S n = s n , ¯ C = ¯ C } × Pr { ¯ C = ¯ C | K 6 = 1 , S n = s n } ( d ) = 2 nR k X s n p ( s n ) X ( u n ,y n ) ∈T ( n ) ǫ X ¯ C Pr { U n ( M , 1) = u n | K 6 = 1 , S n = s n , ¯ C = ¯ C } × Pr { Y n = y n | K 6 = 1 , S n = s n , ¯ C = ¯ C } Pr { ¯ C = ¯ C | K 6 = 1 , S n = s n } ( e ) ≤ 2 nR k X s n p ( s n ) X ( u n ,y n ) ∈T ( n ) ǫ X ¯ C 2 P r { U n ( M , 1) = u n } × Pr { Y n = y n | K 6 = 1 , S n = s n , ¯ C = ¯ C } Pr { ¯ C = ¯ C | K 6 = 1 , S n = s n } = 2 nR k X s n p ( s n ) X ( u n ,y n ) ∈T ( n ) ǫ 2 P r { U n ( M , 1) = u n } P r { Y n = y n | K 6 = 1 , S n = s n } ( f ) ≤ 2 nR k X s n p ( s n ) X ( u n ,y n ) ∈T ( n ) ǫ 4 P r { U n ( M , 1) = u n } P r { Y n = y n | S n = s n } ( g ) = 2 nR k +2 X ( u n ,y n ) ∈T ( n ) ǫ n Y i =1 p U ( u i ) Pr { Y n = y n } ( h ) ≤ 2 n ( R k − I ( U ; Y )+ δ ) where (a) is due to the union of e vents b ound , and (b) follows b y the symmetry of the codebook generation and coding, (c) is by deﬁning ¯ C = { U n ( M , k ) , k 6 = 1 } , (d) is due to th e fact that given K 6 = 1 , U n ( M , 1) → ( ¯ C , S n ) → Y n forms a Markov chain, (e) is du e to Lemma 14 given at the end of this section, (f) is due to Lemma 1 5 g iv en at the end of this section, (g) is due to h aving i.i.d . gener ation for U n , i.e., Pr { U n ( M , 1) = u n } = n Q i =1 p U ( u i ) , (h ) is due to joint typicality lemm a [39]. Novem ber 19, 2018 DRAFT 27 From the last expression, we o btain that P r {E c 3 ∩ E 4 } → 0 as n → ∞ , if R k < I ( U ; Y ) − δ . Th is implies the existence of sequen ce of codes imply ing the desired result as we set R k = I ( U ; S ) + δ < I ( U ; Y ) − δ . Lemma 14 (Lemma 1 in [40]): For sufﬁciently large n , Pr { U n ( M , 1) = u n | K 6 = 1 , S n = s n , ¯ C = ¯ C } ≤ 2 P r { U n ( M , 1) = u n } Pr oof: W e hav e Pr { U n ( M , 1) = u n | K 6 = 1 , S n = s n , ¯ C = ¯ C } = Pr { U n ( M , 1) = u n | S n = s n , ¯ C = ¯ C } × Pr { K 6 = 1 | U n ( M , 1) = u n , S n = s n , ¯ C = ¯ C } Pr { K 6 = 1 | S n = s n , ¯ C = ¯ C } ( a ) ≤ Pr { U n ( M , 1) = u n } Pr { K 6 = 1 | S n = s n , ¯ C = ¯ C } ( b ) ≤ 2 Pr { U n ( M , 1) = u n } where (a) is due to in depend ence o f U n ( M , 1) and ( S n , ¯ C ) to gether with b oundin g P r { K 6 = 1 | U n ( M , 1) = u n , S n = s n , ¯ C = ¯ C } ≤ 1 , a nd (b) f ollows fro m P r { K 6 = 1 | S n = s n , ¯ C = ¯ C } ≥ 1 2 , as shown below . Consider t = t ( ¯ C , s n ) = |{ u n ( M , k ) ∈ ¯ C : ( u n ( M , k ) , s n ) ∈ T ( n ) ǫ ′ }| . Then, if t ≥ 1 , by the symmetry of the codebo ok gen eration a nd codin g, Pr { K = 1 | S n = s n , ¯ C = ¯ C } = Pr { ( U n ( M , 1) , s n ) ∈ T ( n ) ǫ ′ } t + 1 ≤ 1 t + 1 ≤ 1 2 , where we upper bound th e probab ility with 1 and used t ≥ 1 . On the o ther han d, if t = 0 , for sufﬁciently large n , and due to the symmetry of the codeb ook g eneration and cod ing, we have Pr { K = 1 | S n = s n , ¯ C = ¯ C } ≤ Pr { ( U n ( M , 1) , s n ) ∈ T ( n ) ǫ ′ } + Pr { ( U n ( M , 1) , s n ) / ∈ T ( n ) ǫ ′ } 2 nR k ≤ Pr { ( U n ( M , 1) , s n ) ∈ T ( n ) ǫ ′ } + 1 2 nR k ≤ 2 − n ( I ( U ; S ) − δ ( ǫ ′ )) + 1 2 nR k ≤ 1 2 , where b ound the probab ility with 1 and u tilized the joint ty picality lem ma [3 9]. The last inequ ality holds in th e limit of large n as R k > 0 and I ( U ; S ) > δ ( ǫ ′ ) , where δ ( ǫ ′ ) → 0 as ǫ ′ → 0 . Lemma 15 (Lemma 2 in [40]): For sufﬁciently large n , Pr { Y n = y n | K 6 = 1 , S n = s n } ≤ 2 Pr { Y n = y n | S n = s n } Pr oof: W e hav e Pr { Y n = y n | K 6 = 1 , S n = s n } = Pr { Y n = y n | S n = s n } P r { K 6 = 1 | S n = s n , Y n = y n } Pr { K 6 = 1 | S n = s n } ( a ) ≤ Pr { Y n = y n | S n = s n } Pr { K 6 = 1 | S n = s n } ( b ) ≤ 2 P r { Y n = y n | S n = s n } , Novem ber 19, 2018 DRAFT 28 where (a) follows as P r { K 6 = 1 | S n = s n , Y n = y n } ≤ 1 , an d (b) is due to having Pr { K 6 = 1 | S n = s n } ≥ 1 / 2 for sufﬁciently large n due to the symmetry of the codebook generatio n and co ding. A P P E N D I X B I N D I C A T O R E V E N T C O N D I T I O N I N G L E M M A Lemma 16: Conside r an indicator rando m variable E , where E = 1 for E , an d E = 0 fo r E c , for an event E . Then, for any A, B , H ( A | B ) ≤ H ( A | B , E ) + 1 I ( A ; B ) ≥ I ( A ; B | E ) − 1 I ( A ; B ) ≤ I ( A ; B | E ) + 1 Pr oof: H ( A | B ) ( a ) ≤ H ( A | B ) + H ( E | B , A ) = H ( E | B ) + H ( A | B , E ) ( b ) ≤ H ( A | B , E ) + 1 (20) I ( A ; B ) ( c ) ≥ H ( A | E ) − H ( A | B ) ( d ) ≥ I ( A ; B | E ) − 1 I ( A ; B ) ( c ) ≤ H ( A ) − H ( A | B , E ) ( e ) ≤ I ( A ; B | E ) + 1 , where ( a) is due to H ( E | B , E ) ≥ 0 , (b) is d ue to upper b ound o n the entropy of a bin ary rand om variable, (c) follows as conditio ning does not increase en tropy , ( d) f ollows by (20 ), and (e) f ollows by con sidering B = ∅ in (20) to u pper boun d H ( A ) . A P P E N D I X C P R O O F O F L E M M A 3 Pr oof: W e ﬁrst consider I ( U ; Z ) > I ( U ; S ) case, for which the codewords are rep resented by U n ( M , T , K ) . I ( M ; Z n ) = H ( M ) − H ( M | Z n ) = H ( M ) − H ( T , K , Z n , M ) + H ( Z n ) + H ( T , K | Z n , M ) = H ( Z n ) + H ( T , K | Z n , M ) − H ( T , K | M ) − H ( Z n | M , T , K ) ( a ) ≤ n ( H ( Z ) + ǫ 1 ) − H ( T | M ) − H ( U n | M , T ) − H ( Z n | M , T , U n ) ( b ) = n ( H ( Z | U ) + I ( U ; S ) + ǫ 2 ) − H ( U n , Z n | M , T ) = n ( H ( Z | U ) + I ( U ; S ) + ǫ 2 ) − H ( U n , Z n | M , T , S n ) − I ( S n ; U n , Z n | M , T ) ( c ) ≤ n ( H ( Z | U ) + I ( U ; S ) + ǫ 2 ) − H ( Z n | M , T , S n , U n ) − H ( S n | M , T ) + H ( S n | U n , Z n ) ( d ) ≤ n ( I ( S ; Z , U ) + ǫ 2 ) − H ( S n | M , T ) + H ( S n | U n , Z n ) , where (a) follows by H ( Z n ) = n P i =1 H ( Z i | Z i − 1 1 ) ≤ n P i =1 H ( Z i ) = n H ( Z ) , H ( T , K | Z n , M ) ≤ nǫ 1 for some ǫ 1 → 0 as n → ∞ (this is deco ding of ( T , K ) at eavesdropper h aving ( Z n , M ) , following fro m steps similar to the Novem ber 19, 2018 DRAFT 29 proof of Th eorem 1 replacing Bob with Eve, as number of ( T , K ) ind ices is 2 n ( I ( U ; Z ) − δ ) , see a lso Appe ndix A), H ( K | M , T ) = H ( U n | M , T ) , and having H ( Z n | M , T , K ) = H ( Z n | M , T , U n ) as ( M , T , K ) is a o ne-to- one function of ( M , T , U n ( M , T , K )) , (b ) follows by H ( T | M ) = H ( T ) = n ( I ( U ; Z ) − I ( U ; S ) − 2 δ ) as T is indepen dent of M and unifo rmly random and takin g ǫ 2 = ǫ 1 + 2 δ , (c) follows as H ( U n | M , T , S n ) ≥ 0 and H ( S n | M , T , U n , Z n ) = H ( S n | U n , Z n ) as ( M , T ) un iquely d etermined given U n ( M , T , K ) , (d) is by having H ( Z n | M , T , S n , U n ) = n X i =1 H ( Z i | Z i − 1 1 , M , T , S n , U n ) = n X i =1 H ( Z i | S i , U i ) = n H ( Z | S , U ) , which is du e to the Markov c hain ( Z i − 1 1 , M , T , S i − 1 1 , S n i +1 , U i − 1 1 , U n i +1 ) → ( S i , U i ) → Z i as ( U i , S i ) gen erates ( X i , S i ) wh ich g enerates Z i i.i.d. due to the memor yless channel. Secondly , consider I ( U ; Z ) ≤ I ( U ; S ) case, for wh ich the c odewords are represen ted by U n ( M , K ) . W e consider K = [ K 1 , K 2 ] with K 1 = [1 : 2 n ( I ( U ; S ) − I ( U ; Z )+2 δ ) ] and K 2 = [1 : 2 n ( I ( U ; Z ) − δ ) ] , wh ich together represen t the covering index K . (This can be o btained via rando m binning of 2 n ( I ( U ; S )+ δ ) number o f co dew ords U n ( M , k ) , k ∈ [1 : 2 n ( I ( U ; S )+ δ ) ] , into bins represented b y k 1 , where the co deword index per bin represented by k 2 .) W e continue as fo llows. I ( M ; Z n ) = H ( M ) − H ( M | Z n ) = H ( M ) − H ( M , K 1 , K 2 , Z n ) + H ( Z n ) + H ( K 1 , K 2 | Z n , M ) = H ( Z n ) + H ( K 1 | Z n , M ) + H ( K 2 | Z n , M , K 1 ) − H ( K 1 , K 2 | M ) − H ( Z n | M , K 1 , K 2 ) ( a ) ≤ n ( H ( Z | U ) + I ( U ; S ) + ǫ 2 ) − H ( U n | M ) − H ( Z n | M , U n ) = n ( H ( Z | U ) + I ( U ; S ) + ǫ 2 ) − H ( U n , Z n | M ) = n ( H ( Z | U ) + I ( U ; S ) + ǫ 2 ) − H ( U n , Z n | M , S n ) − I ( S n ; U n , Z n | M ) ( b ) ≤ n ( H ( Z | U ) + I ( U ; S ) + ǫ 2 ) − H ( Z n | M , S n , U n ) − H ( S n | M ) + H ( S n | U n , Z n ) ( c ) ≤ n ( I ( S ; Z , U ) + ǫ 2 ) − H ( S n | M ) + H ( S n | U n , Z n ) , where (a) fo llows by having H ( Z n ) ≤ nH ( Z ) as d escribed above, H ( K 1 | Z n , M ) ≤ H ( K 1 ) ≤ n ( I ( U ; S ) − I ( U ; Z ) + 2 δ ) , H ( K 2 | Z n , M , K 1 ) ≤ n ǫ 1 for so me ǫ 1 → 0 as n → ∞ (this is d ecoding of K 2 at eavesdropper having ( Z n , M , K 1 ) , following from steps similar to the proo f of Theore m 1 re placing Bob with Eve, as number of K 2 indices is 2 n ( I ( U ; Z ) − δ ) , see also Ap pendix A), H ( K 1 , K 2 | M ) = H ( U n | M ) , having H ( Z n | M , K 1 , K 2 ) = H ( Z n | M , K 1 , K 2 , U n ) as U n ( M , K 1 , K 2 ) is determined by ( M , K 1 , K 2 ) , and deﬁnin g ǫ 2 = 2 δ + ǫ 1 , (b) is by H ( U n | M , S n ) ≥ 0 , and (c) is same as that of the step (d) of th e previous par agraph . Novem ber 19, 2018 DRAFT 30 A P P E N D I X D P R O O F O F C O RO L L A RY 4 From Lemm a 3, we h av e I ( M ; Z n ) ≤ n I ( S ; Z , U ) + H ( S n | U n , Z n ) − H ( S n | M , T ) + nǫ. Here, H ( S n | U n , Z n ) = n P i =1 H ( S i | S i − 1 1 , U n , Z n ) ≤ n P i =1 H ( S i | U i , Z i ) = nH ( S | U, Z ) . In ad dition, as S n is generated i.i.d ., and also indepen dent o f ( M , T ) , we hav e H ( S n | M , T ) = H ( S n ) ≥ n ( H ( S ) − ǫ 1 ) w ith som e ǫ 1 → 0 as n → ∞ . (The latter exp ression follows as we can bound H ( S n ) ≥ H ( S n | E ) where E is an indicato r random variable with E = 1 if S n is ty pical. The n, H ( S n | E ) = Pr { E = 0 } H ( S n | E = 0) + Pr { E = 1 } H ( S n | E = 1) ≥ Pr { E = 1 } H ( S n | E = 1 ) ≥ (1 − ǫ 0 ) n ( H ( S ) − ǫ 0 ) for som e arbitrarily small ǫ 0 , fro m which the assertion follows by tak ing ǫ 1 = ǫ 0 (1 + H ( S ) − ǫ 0 ) .) The n, u sing these two o bservations in the equation ab ove, we obtain I ( M ; Z n ) ≤ n I ( S ; Z , U ) + nH ( S | U, Z ) − nH ( S ) + nǫ 1 + nǫ = n ( ǫ 1 + ǫ ) , which co ncludes the p roof. A P P E N D I X E P R O O F O F P RO P O S I T I O N 6 Pr oof: De ﬁne a random variable Q unif orm over { 1 , · · · , n } and inde penden t o f everything e lse. (A standa rd technique o f usin g Q as a tim e sh aring par ameter will be utilized in the following.) Also deﬁne U i = ( S i − 1 1 , Z n i +1 ) . W e have the following bound s. I ( S n ; Y n ) ≤ I ( X n , S n ; Y n ) ( a ) ≤ n X i =1 I ( X i , S i ; Y i ) ( b ) = nI ( X Q , S Q ; Y Q | Q ) ( c ) ≤ nI ( U Q , Q, X Q , S Q ; Y Q ) (21) where (a) is due to the memory less chann el p ( y i | x i , s i ) and the fact tha t co ndition ing d oes not in crease the e ntropy , (b) f ollows from the distribution of Q , (c) follows as the added ter m I ( Q ; Y Q ) + I ( U Q ; Y Q | Q, X Q , S Q ) ≥ 0 . In addition, I ( S n ; Y n ) ≤ H ( S n ) = n X i =1 H ( S i | S i − 1 1 ) ( a ) = n X i =1 H ( S i ) = nH ( S Q | Q ) , (22) Novem ber 19, 2018 DRAFT 31 where (a) h olds as S n has an i.i.d . distribution. Moreover , I ( S n ; Z n ) = H ( S n ) − H ( S n | Z n ) = n X i =1 H ( S i ) − H ( S i | S i − 1 1 , Z n ) ( a ) ≥ n X i =1 H ( S i ) − H ( S i | Z i , S i − 1 1 , Z n i +1 ) ( b ) = n X i =1 H ( S i ) − H ( S i | Z i , U i ) ( c ) = nI ( S Q ; Z Q , U Q | Q ) , (23) where (a) is d ue the fact that con ditioning does n ot inc rease the en tropy , (b) f ollows fr om the d eﬁnition o f U i , an d (c) is the stan dard time shar ing argumen t. Finally , we have 0 ≤ n X i =1 I ( Z n i +1 ; Z i ) = n X i =1 I ( S i − 1 1 , Z n i +1 ; Z i ) − I ( S i − 1 1 ; Z i | Z n i +1 ) ( a ) = n X i =1 I ( S i − 1 1 , Z n i +1 ; Z i ) − I ( Z n i +1 ; S i | S i − 1 1 ) ( b ) = n X i =1 I ( S i − 1 1 , Z n i +1 ; Z i ) − I ( S i − 1 1 , Z n i +1 ; S i ) ( c ) = n X i =1 I ( U i ; Z i ) − I ( U i ; S i ) = n ( I ( U Q ; Z Q | Q ) − I ( U Q ; S Q | Q )) , (24) where ( a) follows by Csisz ´ ar’ s sum lemma [39] (note th at th is is similar to th e c onv erse result of the Gel’fand- Pinsker problem giv en in [44], whe re the indices of S an d Z are r ev ersed), (b) is due to I ( S i − 1 1 ; S i ) = 0 as S n has an i.i.d . distribution, (c) follows fr om deﬁnitio n o f U i , (d) is the standard time sh aring argum ent. Now , deﬁn e U = ( U Q , Q ) , S = S Q indepen dent o f Q (no te th at this argumen t is also u sed in [9] in the correspo nding single-letter ization argumen ts), X = X Q , Y = Y Q , and Z = Z Q . Then, (21) im plies I ( S n ; Y n ) ≤ nI ( U Q , Q, X Q , S Q ; Y Q ) ≤ nI ( U, X , S ; Y ) = nI ( S, X ; Y ) as U → ( X , S ) → Y due to the memory less channel p ( y | x, s ) . ( 22) red uces to I ( S n ; Y n ) ≤ nH ( S Q | Q ) = nH ( S ) , as S Q = S is independ ent o f Q . (23) implies I ( S n ; Z n ) ≥ nI ( S Q ; Z Q , U Q , Q ) = nI ( S ; Z , U ) as I ( S Q ; Z Q , U Q | Q ) = H ( S Q | Q ) − H ( S Q | Z Q , U Q , Q ) = H ( S Q ) − H ( S Q | Z Q , U Q , Q ) = I ( S Q ; Z Q , U Q , Q ) due to the indepen dence of S Q = S and Q . (24) is 0 ≤ n ( I ( U Q ; Z Q | Q ) − I ( U Q ; S Q | Q )) . Consider add ing n ( I ( Q ; Z Q ) − I ( Q ; S Q )) to th e r ight h and side of th is inequ ality . W e h ave, as S Q = S and Q ar e indepen dent, I ( Q ; S Q ) = 0 , and hen ce n ( I ( Q ; Z Q ) − I ( Q ; S Q )) ≥ 0 . Then, 0 ≤ n ( I ( U Q ; Z Q | Q ) − I ( U Q ; S Q | Q )) ≤ n ( I ( U Q , Q ; Z Q ) − I ( U Q , Q ; S Q )) = n ( I ( U ; Z ) − I ( U ; S )) , which implies 0 ≤ I ( U ; Z ) − I ( U ; S ) as a necessary conditio n. Novem ber 19, 2018 DRAFT 32 This conclud es the proo f, as comb ining the bo unds above with the fact that any ac hiev able ( R a , R l ) for the giv en chann el p ( y , z | x, s ) and p ( s ) shou ld satisfy 1 n I ( S n ; Y n ) ≥ R a − ǫ and 1 n I ( S n ; Z n ) ≤ R l + ǫ imp lies th e inequalities stated in th e pro position. A P P E N D I X F P R O O F O F P RO P O S I T I O N 7 Pr oof: L et P 1 denote th e set of p ( u, x | s ) satisfying I ( U ; Y ) ≥ I ( U ; S ) , and den ote P 2 denote the set of p ( u, x | s ) satisfying I ( U ; Z ) ≥ I ( U ; S ) . For the ch annel p ( y , z | x, s ) = p ( y | x, s ) p ( z | y ) , any p ∈ P 2 also satisﬁes p ∈ P 1 . Ther efore, using the Proposition 6, we h av e that if ( R a , R l ) is achie vable, then ( R a , R l ) ∈ R 3 o , where R 3 o = [ p ( u,x | s ) ( R a , R l ) satisfying R a ≤ min { H ( S ) , I ( X , S ; Y ) } R l ≥ I ( S ; Z, U ) 0 ≤ I ( U ; Y ) − I ( U ; S ) . It remain s to show R a − R l bound . W e add the f ollowing b ound to the ones stated ab ove. (No te that, we use the same U i and Q -depen dent deﬁnitio ns stated in Appendix E, and henc e the following bound can b e add ed to the on es stated in App endix E.) n ( R a − R l ) = I ( S n ; Y n ) − I ( S n ; Z n ) = I ( X n , S n ; Y n ) − I ( X n , S n ; Z n ) − ( I ( X n ; Y n | S n ) − I ( X n ; Z n | S n )) ( a ) ≤ I ( X n , S n ; Y n ) − I ( X n , S n ; Z n ) ( b ) = I ( X n , S n ; Y n | Z n ) = n X i =1 H ( Y i | Y i − 1 1 , Z n ) − n X i =1 H ( Y i | Y i − 1 1 , Z n , X n , S n ) ( c ) ≤ n X i =1 H ( Y i | Z i ) − n X i =1 H ( Y i | Z i , X i , S i ) = n X i =1 I ( X i , S i ; Y i | Z i ) ( d ) = n I ( X Q , S Q ; Y Q | Z Q , Q ) ( e ) ≤ n I ( U Q , Q, X Q , S Q ; Y Q | Z Q ) ( f ) = n I ( U, X , S ; Y | Z ) ( g ) = n I ( X, S ; Y | Z ) , Novem ber 19, 2018 DRAFT 33 where (a) and (b) ar e du e to the degraded ness conditio n, (c) is d ue to the fact th at con ditioning does not increase entropy and the Markov chain ( Y i − 1 1 , Z i − 1 1 , Z n i +1 , S i − 1 1 , S n i +1 , X i − 1 1 , X n i +1 ) → ( X i , S i , Z i ) → Y i as ( X i , S i ) generates Y i , (d) follows fro m the memoryless pro perty of the ch annel, (d) and (f ) are due to deﬁnition s giv en in Append ix E, (e ) is by adding the n on-negative term I ( Q ; Y Q | Z Q ) + I ( U Q ; Y Q | Z Q , Q, X Q , S Q ) , an d (g) is due to the Markov chain U → ( X , S ) → ( Y , Z ) as o utputs ( y i , z i ) are generated i.i.d. using ( x i , s i ) over the memo ryless channel p ( y , z | x, s ) = p ( y | x, s ) p ( z | y ) . Th e result then follows by taking the union over all joint distributions p ( u, x | s ) . A P P E N D I X G P R O O F O F T H E C O N V E R S E F O R T H E O R E M 8 Pr oof: W e bound the rate d ifference as follows. n ( R a − R l ) ≤ I ( S n ; Y n ) − I ( S n ; Z n ) = n X i =1 I ( S n ; Y i | Y i − 1 1 ) − I ( S n ; Z i | Z n i +1 ) ( a ) = n X i =1 I ( S n ; Y i | Y i − 1 1 , Z n i +1 ) + I ( Z n i +1 ; Y i | Y i − 1 1 ) − I ( Z n i +1 ; Y i | Y i − 1 1 , S n ) −  I ( S n ; Z i | Y i − 1 1 , Z n i +1 ) + I ( Y i − 1 1 ; Z i | Z n i +1 ) − I ( Y i − 1 1 ; Z i | Z n i +1 , S n )  ( b ) = n X i =1 I ( S n ; Y i | Y i − 1 1 , Z n i +1 ) − I ( S n ; Z i | Y i − 1 1 , Z n i +1 ) = n X i =1 I ( S i ; Y i | Y i − 1 1 , Z n i +1 , S i − 1 1 , S n i +1 ) + I ( S i − 1 1 , S n i +1 ; Y i | Y i − 1 1 , Z n i +1 ) − I ( S i ; Z i | Y i − 1 1 , Z n i +1 , S i − 1 1 , S n i +1 ) − I ( S i − 1 1 , S n i +1 ; Z i | Y i − 1 1 , Z n i +1 ) ( c ) ≤ n X i =1 I ( S i ; Y i | U i ) − I ( S i ; Z i | U i ) ( d ) = n [ I ( S ; Y | U ) − I ( S ; Z | U )] , where, in (a ), we used the equalities I ( S n ; Y i | Y i − 1 1 ) + I ( Z n i +1 ; Y i | Y i − 1 1 , S n ) = I ( Z n i +1 ; Y i | Y i − 1 1 ) + I ( S n ; Y i | Y i − 1 1 , Z n i +1 ) and I ( S n ; Z i | Z n i +1 ) + I ( Y i − 1 1 ; Z i | Z n i +1 , S n ) = I ( Y i − 1 1 ; Z i | Z n i +1 ) + I ( S n ; Z i | Z n i +1 , Y i − 1 1 ); in (b ), we used the Csiszar’ s sum lemm a [4] (an d a co nditional f orm of it) to obtain th e equa lities n X i =1 I ( Z n i +1 ; Y i | Y i − 1 1 ) = n X i =1 I ( Y i − 1 1 ; Z i | Z n i +1 ) and n X i =1 I ( Z n i +1 ; Y i | Y i − 1 1 , S n ) = n X i =1 I ( Y i − 1 1 ; Z i | Z n i +1 , S n ); Novem ber 19, 2018 DRAFT 34 in (c), we d eﬁne U i , ( Y i − 1 1 , Z n i +1 , S i − 1 1 , S n i +1 ) ; and use the rev ersely d egradedne ss of the chan nel re sulting in I ( S i − 1 1 , S n i +1 ; Z i | Y i − 1 1 , Z n i +1 ) ≥ I ( S i − 1 1 , S n i +1 ; Y i | Y i − 1 1 , Z n i +1 ) , in (d) , we ob tain the single-letter expression (b y deﬁning U = ( U i , i ) , etc., see, e.g., [47]). Note that, with the d eﬁnition of U i in (c), X i is gene rated from S n , an d hence fr om ( U i , S i ) . In addition, given ( X i , S i ) , Z i is indep endent of ( U i , S i ) . Thu s, ( U, S ) → ( X , S ) → Z forms a Markov chain, and the upper boun d is gi ven b y R a − R l ≤ max p ( u,x | s ) s.t. ( U,S ) → ( X ,S ) → Z I ( S ; Y | U ) − I ( S ; Z | U ) = max p ( x | u ∗ ,s ) , u ∗ ∈U I ( S ; Y | U = u ∗ ) − I ( S ; Z | U = u ∗ ) = max p ( x | s ) I ( S ; Y ) − I ( S ; Z ) , where th e equalities follow due to the following: First, the conditio nal mutual inform ation expression is max imized with a par ticular inpu t u ∗ (as ran domizing over different u values will not incr ease the sum P u ( I ( S ; Y | U = u ) − I ( S ; Z | U = u )) Pr { U = u } ) ) and a probability distribution p ∗ ( x | u ∗ , s ) . Second, th e op timal p ∗ ( x | u ∗ , s ) will correspo nd to a p ( x | s ) . 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State Amplification Subject To Masking Constraints

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