The K-Receiver Broadcast Channel with Confidential Messages

1 The K -Recei v er Broadcast Channel with Confidential Messages Li-Chia Choo and Kai-Kit W ong, Senior Member , IEEE Abstract The secrec y capacity region for the K -receiv er degraded broad cast channel (BC) is giv en for confidential messages sent to t h e receivers and to be kept secret from an external wiretapper . Superposition coding and W yner’ s random code partitioning are used to sho w the achiev able rate tuples. Error probability analysis and equi vocation calculation are also provided. In the con ve rse pro of, a new definition for the auxiliary r an dom variab les is u sed, which is d ifferent from either the case of the 2-receiv er BC without common message or the K -receiv er BC with common message, both with an external wiretapper; or the K - rec eiv er BC without a wiretapper . I . I N T RO D U C T I O N The wireless comm unications channel is vulne rable to eavesdropping or wiretap ping due to the o pen natu re of the channel. An important requireme nt fo r wireless systems today is the characte rization of transmission rates that allo w fo r both secure an d reliable communicatio n for the physical layer . Recent studies addressing this issue have included wireless network building block s s uch as the multiple- access channels [1], relay ch annels [2], fading channels [3] , [4] an d m ultiple-input m ultiple-output ( MIMO) ch annels [5]. A suitable model for study ing such simultaneo usly secure and reliable com munication in the wireless b roadcast and commu nications medium is the broadc as t chan nel (BC) with confiden tial messages, wh ich was studied by Csisz ´ ar and K ¨ o rner [6]. It is a generalization of the ch aracterization of the wiretap channel by W yner [7]. In the model of [6], there a re 2 receivers and a common message is sent to both, while a confid ential message is sent to one of the receivers and is to be kept secret from th e other receiv er . The secrecy lev el is d etermined by using the equiv ocation rate, wh ich is the entro py rate of the confidential message conditioned on the channel output at the eavesdropper or wiretapper . The secrecy capacity region is the set o f transmission rates where th e le gitimate receiver decod es its co nfidential me s sage while keep ing the message secret from the wir etapper . In m ore recent studies o n the BC with confiden tial messages, Liu et al. [8] in vestigated the scenario wher e there are 2 receivers and p ri v ate messages are sent to each one and kept secret fro m the un intended recei ver; Xu et al. [9] stud ied the case in [8] b ut with a common message to both recei vers; Bagherik aram et a l. [10] addressed the scenario wher e ther e are 2 receiv ers and one wiretapp er , with confid ential messages sen t to the receivers. The authors are with the Department of Electronic and Electrica l Engine ering, Univ ersity College London, Adastral Park Postgrad uate R esearch Campus, Mart lesham Heat h, IP5 3RE, United Kingdom (email: { l.choo, k.w ong } @adast ral.ucl.ac.uk) . 2 In th is paper, we in vestigate the d e graded K -receiver BC with confiden tial me s sages sent to each r ecei ver in the presence of a wiretapper, from which the messages are kept secre t. Our results are a generalization of ou r work for 3 rec ei vers in [1 1] and ear lier results for 2 receivers in [10]. It is n oted that results similar to ours have been established independen tly in [1 2], where Ekrem and Ulu kus [1 2] examined th e K -rece i ver d e graded BC a nd one wiretapper with confide ntial messages as well as a com mon message sen t to th e recei vers. However , there are some appreciab le differences between o ur app roach and that in [12 ]. In particula r , equiv ocation calculation and proof of the converse in [12] are accom plished from the persp ecti ve of the channel sum rate. In contr ast, we provide the error probability analysis and the equiv ocation calculatio n with respect to th e k th recei ver’ s achiev able rate. Further, we use W yner’ s meth od of random code partition ing instead of Gel’fand-Pinsker b inning which is used in [1 0] to perfor m code partitioning to achie ve p erfect secr ec y . In our proof of the con verse, which we have shown for the k th receiver , we no te that our choice of au xiliary random variables is different f rom that of [ 10] an d [ 12]. Due to the pr esence o f th e wiretapper, it is also different f rom the choice in Bora de et al. [1 3] wher e the capacity region for the degrad ed K -receiver BC using sup erposition coding without confide ntial messages is stu died. The remain der o f this paper is organize d as f ollo ws. In Section II, we in troduce the general K -receiver d e graded BC with confidential me s sages. In Section III, we state o ur main result for the secrecy capacity for the degraded K -receiver BC with wiretapper and show the proo f of achiev ability and e qui vocation calculation in Sections III-A and III- B , respectively . In Section IV, we sho w the converse proo f. Lastly we g i ve co nclusions in Sectio n V. I I . C H A N N E L M O D E L In this pa per , we use the upp ercase letter to denote a ran dom variable (e.g., X ) an d the lo wercase letter fo r its realization (e.g ., x ). T he alp habet set of X is d enoted by X so that x ∈ X . W e can also have a sequ ence o f n random v ariables, denoted by X = ( X 1 , . . . , X n ) with its rea lization x = ( x 1 , . . . , x n ) ∈ X n if x i ∈ X for i = 1 , 2 , . . . , n . Furtherm ore, we define the subsequences of X a s X i , ( X 1 , X 2 , . . . , X i ) and ˜ X i , ( X i , . . . , X n ) . The discrete memoryless K -r ecei ver BC with an external wiretap per consists of a finite input alphabet X and finite output alphab ets Y 1 , . . . , Y K , Z and has cond itional distribution p ( y 1 , . . . , y K , z | x ) . Thus the discrete mem oryless BC with K recei vers and a wiretap per has an inpu t random seq uence X , K output random sequen ces, Y 1 , . . . , Y K , at the intend ed receivers, an d an ou tput r andom seq uence at the wiretapper Z . Likewis e, we have y 1 ∈ Y n 1 , . . . , y K ∈ Y n K and z ∈ Z n . The co nditional d istrib ution for n uses o f th e channel is p ( y 1 , . . . , y K , z | x ) = n Y i =1 p ( y 1 i , . . . , y K i , z i | x i ) . (1) The transmitter has to send ind ependent messages ( W 1 , . . . , W K ) to the recei vers in perfe ct secrecy . This is done using a (2 nR 1 , . . . , 2 nR K , n ) -code for the BC, wh ich co nsists o f th e stochastic encoder f :  1 , . . . , 2 nR 1  ×  1 , . . . , 2 nR 2  × · · · ×  1 , . . . , 2 nR K  7→ X n , (2) and the deco ders g k : Y n k 7→  1 , . . . , 2 nR k  , f or k = 1 , 2 , . . . , K . (3) 3 The pro bability of error is defined as the probab ility that the decode d messages are no t eq ual to the tran smitted messages, i.e., P ( n ) e , Pr ( K [ k =1 { g k ( Y k ) 6 = W k } ) . (4) Perfect secrecy requires that the mutual inform ation of the transmitted messages and the wiretapper goes to zero. Let us illu s trate this fo r m es sage W 1 and receiver 1. T he perfect secrecy requireme nt is I ( W 1 ; Z ) = 0 ⇒ H ( W 1 ) = H ( W 1 | Z ) , (5) where I ( · ; · ) denotes mutu al information and H ( · ) is en tropy . Now , let the infor mation rate for the first r ecei ver be R 1 = 1 n H ( W 1 ) and the eq ui v ocation rate be R e (1) , 1 n H ( W 1 | Z ) . Then, we n eed R e (1) ≥ R 1 − η , for any arbitrary η > 0 . (6) Further to this, we define the following equiv ocation rates for the K -rece i ver degraded BC:              R e ( k ) , 1 n H ( W k | Z ) , for k = 1 , . . . , K , R e ( k,k +1) , 1 n H ( W k , W k +1 | Z ) , for k = 1 , . . . , K , R e (1 ,...,K ) , 1 n H ( W 1 , . . . , W K | Z ) . (7) I I I . T H E S E C R E C Y C A PAC I T Y R E G I O N The secret rate tuple ( R 1 , R 2 , . . . , R K ) is achie vable if f or any arbitrar ily small ǫ ′ > 0 , ǫ k > 0 , k = 1 , . . . , K , ǫ k,k +1 > 0 , k = 1 , 2 , . . . , K − 1 , an d ǫ 1 ,...,K > 0 , there exist (2 nR 1 , . . . , 2 nR K , n ) -codes for which P ( n ) e ≤ ǫ ′ and                R e ( k ) ≥ R k − ǫ k , for k = 1 , . . . , K , R e ( k,k +1) ≥ R k + R k +1 − ǫ k,k +1 , for k = 1 , . . . , K , R e (1 ,...,K ) ≥ K X k =1 R k − ǫ 1 ,...,K . (8) (8) gives the security condition s for the K -receiver BC with a wiretapper under perfect secrecy requirements in (6 ) . Theor em 1: T he secrecy capacity region f or th e K -receiver degraded BC with an external wiretapper is the closure of all rate tuples ( R 1 , . . . , R K ) satisfying R 1 ≤ I ( X ; Y 1 | U 2 ) − I ( X ; Z | U 2 ) , (9a) R k ≤ I ( U k ; Y k | U k +1 ) − I ( U k ; Z | U k +1 ) , fo r k = 2 , . . . , K − 1 , (9b) R K ≤ I ( U K ; Y K ) − I ( U K ; Z ) , (9c) where { U k } K k =2 are auxiliary r andom variables an d will be defined in Section III-A (Random co debook genera tion). Pr oof: T he pro of of achiev ability an d equivocation calculation are given later in this section. The proof of conv erse is giv en separately in Section IV. If we use superposition coding with code pa rtitioning to achie ve the rates in Theorem 1, then the secr ec y ca pacity region m ay be inter preted as the capac ity region for the K -receiver BC u sing superposition co ding with out the 4 wiretapper, with the rates at each rec ei ver each red uced due to the presence of th e wiretapp er . Howev er , we s hall see that the choice of auxiliary ra ndom variables in the proof of con verse for the K -r ecei ver BC will be different from that of [13] , which is withou t the secrecy conditions. This is also in contr ast to the 2- recei ver BC with wiretapp er in [10], where the same definitio n for th e auxiliary rando m v ariables in the co n v erse pro of can b e used for the scenarios with an d w ithout the secrecy conditions. A. Pr oof of Achievability In this paper, we emp lo y superposition coding and W yner’ s random code partitioning to sho w the achiev able rate tuples ( R 1 , . . . , R K ) . For br e vity , we use p Y 1 | X to denote the channel from X to Y 1 , similarly for the channels from X to o utputs Y 2 , . . . , Y K and Z , by p Y 2 | X , . . . , p Y K | X and p Z | X , respectively . The c oding strategy is depicted in Fig. 1. The message W k ∈ { 1 , . . . , L k } with L k , 2 nR k for k = 1 , . . . , K , is sent by a code o f length N k = L k L ′ k . This code is p artitioned into L k subcodes each of size L ′ k , with L ′ k , 2 nR ′ k for some R ′ k . Each of th e L k subcodes is a code fo r the wiretapper p Z | X , while each o f the entire codes of size N k is a code simultaneously f or both the k th r ecei ver p Y k | X and the wiretap per p Z | X . The code s for simultan eous use for p Y k | X and p Z | X have to satisfy the transmission requ irements for the BC [14], so that 1 n log N 1 ≤ I ( X ; Y 1 | U 2 ) , (10a) 1 n log N k ≤ I ( U k ; Y k | U k +1 ) , fo r k = 2 , . . . , K − 1 , (10b) 1 n log N K ≤ I ( U K ; Y K ) . (10c) 1) Rando m code book gen eration: Suppose that we ha ve the probab ili ty density functio ns (p.d .f.s)            p ( u K ) , p ( u k | u k +1 , . . . , u K ) , fo r k = 2 , . . . , K − 1 , p ( x | u 2 , . . . , u K ) . (11) For a giv en rate tuple ( R 1 , . . . , R K , R ′ 1 , . . . , R ′ K ) , in order to send message W K , generate 2 n ( R K + R ′ K ) inde- penden t codew ords u K ( w ′′ K ) , for w ′′ K ∈ { 1 , . . . , 2 n ( R K + R ′ K ) } accordin g to the p.d.f . p ( u K ) = Q n i =1 p ( u K i ) . Then, partition u K ( w ′′ K ) into L K = 2 nR K subcodes, { C ( K ) i } L K i =1 with |C ( K ) i | = L ′ K = 2 nR ′ K ∀ i . The message for the k th receiver , f or k = 2 , 3 , . . . , K − 1 , is sent by generatin g 2 n ( R k + R ′ k ) indepen dent codewords u k ( w ′′ k , . . . , w ′′ K ) , for w ′′ k ∈ { 1 , . . . , 2 n ( R k + R ′ k ) } acco rding to th e co nditional p .d.f. p ( u k | u k +1 , . . . , u K ) = n Y i =1 p ( u ki | u ( k +1) i , . . . , u K i ) . (12) Then, partition u k ( w ′′ k , . . . , w ′′ K ) into L k = 2 nR k subcodes, {C ( k ) i } L k i =1 , with |C ( k ) i | = L ′ k = 2 nR ′ k ∀ i . Finally , to send th e message inten ded for the first r ecei ver , generate 2 n ( R 1 + R ′ 1 ) indepen dent code words x ( w ′′ 1 , . . . , w ′′ K ) , for w ′′ 1 ∈ { 1 , . . . , 2 n ( R 1 + R ′ 1 ) } accordin g to the p. d.f. p ( x | u 2 , . . . , u K ) = Q n i =1 p ( x | u 2 i , . . . , u K i ) . Then , partition x ( w ′′ 1 , . . . , w ′′ K ) into L 1 = 2 nR 1 subcodes, {C (1) i } L 1 i =1 , with |C (1) i | = L ′ 1 = 2 nR ′ 1 ∀ i . 5 1 L K = 2 nR K L' K = 2 nR ' K ...... ...... 1 L k = 2 nR k L' k = 2 nR ' k ...... ...... 1 L 1 = 2 nR 1 L' 1 = 2 nR ' 1 ...... ...... x u k u K ...... ...... Fig. 1. Coding for K recei ver BC wi th wireta pper . Follo wing this code structur e, the codeword indices w ′′ k may be expressed as w ′′ k = ( w k , w ′ k ) , where w k ∈ { 1 , . . . , 2 nR k } is the index of the message transmitted to the k th receiver , and w ′ k ∈ { 1 , . . . , 2 nR ′ k } d enotes the index of the cod e word within the subco des C ( k ) i , selected fo r tran smiss ion alon g with w k to ensur e secrecy . 2) Enc oding : The encod ing is b y sup erposition coding. T o send th e message w K = i K , for 1 ≤ i K ≤ L K , the tran s mitter chooses on e of the u K ( w ′′ K ) cod e words unifor mly and ran domly from {C ( K ) i K } L K i K =1 . Then, to send the message w K − 1 = i K − 1 , for 1 ≤ i K − 1 ≤ L K − 1 , the transmitter selects one of th e u K − 1 ( w ′′ K − 1 , w ′′ K ) uniform ly random ly from {C ( K − 1) i K − 1 } L K − 1 i K − 1 =1 , gi ven u K ( w ′′ K ) . Sequentially , the transmitter sen ds th e message w k = i k , for 1 ≤ i k ≤ L k and k = 2 , . . . , K − 2 , to the k th receiver by choosing o ne of the u k ( w ′′ k , . . . , w ′′ K ) uniformly and random ly from {C ( k ) i k } L k i k =1 , giv en u k +1 ( w ′′ k +1 , . . . , w ′′ K ) . Lastly , to send w 1 = i 1 for 1 ≤ i 1 ≤ L 1 , giv en u 2 ( w ′′ 2 , . . . , w ′′ K ) , the transmitter c hooses one of the x ( w ′′ 1 , . . . , w ′′ K ) un iformly ran domly fro m { C (1) i 1 } L 1 i 1 =1 . 3) Decod ing : W e use the n otation A n ǫ ( P V ) to deno te the set of join tly ty pical n -sequ ences with respect to th e p.d.f. p ( v ) . Also, we use { ˆ w k } K k =1 to den ote th e estimates for the transmitted messages { w k } K k =1 . Then , we have: a) At the K th recei ver , given that y K is r ecei ved, fin d a ˆ w K , such that ( u K ( ˆ w K , w ′ K ) , y K ) ∈ A n ǫ ( P U K Y K ) . b) At th e k th recei ver, for k = 2 , . . . , K − 1 , given th at y k is received, find a ( ˆ w k , . . . , ˆ w K ) such th at ( u K ( ˆ w k , w ′ k ) , . . . , u k ( ˆ w k , w ′ k , . . . , ˆ w K , w ′ K ) , y k ) ∈ A n ǫ ( P U K U K − 1 ··· U k Y k ) . (13) 6 c) Lastly , at th e first receiv er , g i ven that y 1 is received, find a ( ˆ w 1 , . . . , ˆ w K ) such th at ( u K ( ˆ w K , w ′ K ) , . . . , u 2 ( ˆ w 2 , w ′ 2 , . . . , ˆ w K , w ′ K ) , x 1 ( ˆ w 1 , w ′ 1 , . . . , ˆ w K , w ′ K ) , y 1 ) ∈ A n ǫ ( P U K U K − 1 ··· U 2 X Y 1 ) . (14) For each of the above cases, if ther e is non e or mor e than on e possible deco ded message, then an erro r will be declared . Note th at w ′ k is unimp ortant for the decoding of w k at the k th receiver . 4) Obtain ing the sizes of subcodes {C ( k ) i k } : Here, we shall not u s e binnin g but f ollo w the a pproach o f W yn er [ 7], whe re rand om code par titi oning is used. W e shall sho w how to obtain log L ′ k in the encod ing of W k , for k = 2 , ..., K − 1 . Following the same routine, log L ′ 1 and log L ′ K can be o btained ea s ily , and thu s th ese calcu lations will be omitted. T o start with, suppose that we have the messages, w k = i k , . . . , w K = i K . W e now define q ( k ) i k , P r { W k = i k | W k +1 = i k +1 , . . . , W K = i K } = P r  W k = i k | u K ( i K , i ′ K ) , u K − 1 ( i K − 1 , i ′ K − 1 , i K , i ′ K ) , . . . , u k ( i k , i ′ k , . . . , i K , i ′ K )  . (15) The codeword u k ( w ′′ k , . . . , w ′′ K ) is a channel code for p Y k | X and p Z | X simultaneou s ly and is co mprised of L k = 2 nR k subcodes { C ( k ) i k } L k i k =1 . U k is an u niformly ra ndomly ch osen m ember o f { C ( k ) i k } . Th erefore, Pr  U k = u k ( w ′′ k , . . . , w ′′ K ) | u K ( i K , i ′ K ) , . . . , u k +1 ( i k +1 , i ′ k +1 , . . . , i K , i ′ K )  = q ( k ) i k L ′ k . (16) The codew o rd u k ( w ′′ k , . . . , w ′′ K ) is a chann el code f or p Y k | X with prior d is tribution on codewords gi ven by (16). Each of C ( k ) i k is a channel code for the wiretap channel p Z | X with L ′ k codewords and unifo rm prior distribution on the codewords. Let λ ( k ) i k be the err or pro bability for C ( k ) i k with an op timal deco der , when i ′ k is chosen as the index for the codew ord from C ( k ) i k . Then ¯ λ ( k ) is the average error p robability for C ( k ) i k with an o ptimal decoder, a verag ed over the probability that W k = i k is sent given the previous messages were W k +1 = i k +1 , . . . W K = i K . As a result, we have          λ ( k ) i k = P r  X 6 = Z | W k = i k , u K ( i K , i ′ K ) , . . . , u k +1 ( i k +1 , i ′ k +1 , . . . , i K , i ′ K )  , ¯ λ ( k ) = L k X i k =1 q ( k ) i k λ ( k ) i k . (17) By Fano’ s inequality , H ( X | Z , W k = i k , u K ( i K , i ′ K ) , . . . , u k +1 ( i k +1 , i ′ k +1 , . . . , i K , i ′ K )) ≤ 1 + λ ( k ) i k log L ′ k ⇒ H ( U k | Z , U K , . . . , U k +1 , W k = i k ) ≤ 1 + λ ( k ) i k log L ′ k . (18) Since |C ( k ) i k | = L ′ k and has p robability of error λ ( k ) i k , we h a ve I ( U k ; Z | U K , . . . , U k +1 , W k = i k ) = H ( U k | U K , . . . , U k +1 , W k = i k ) − H ( U k | Z , U K , . . . , U k +1 , W k = i k ) = lo g L ′ k − H ( U k | Z , U K , . . . , U k +1 , W k = i k ) ⇒ log L ′ k ≤ I ( U k ; Z | U K , . . . , U k +1 , W k = i k ) + 1 + λ ( k ) i k log L ′ k . (19) 7 A veragin g over i k using { q ( k ) i k } gives log L ′ k ≤ I ( U k ; Z | U K , . . . , U k +1 , W k ) + 1 + ¯ λ ( k ) log L ′ k ( a ) ≤ I ( U k ; Z | U K , . . . , U k +1 ) + 1 + ¯ λ ( k ) log L ′ k ( b ) ≤ nI ( U k ; Z | U K , . . . , U k +1 ) + nδ + 1 + ¯ λ ( k ) log L ′ k , ( c ) = nI ( U k ; Z | U k +1 ) + nδ + 1 + ¯ λ ( k ) log L ′ k , (20) where (a) is by W k → ( U K , . . . , U k +1 ) → U k → Z , (b) results from the fact that (f ollo win g Liu et al. [8]) I ( U k ; Z | U K , . . . , U k +1 ) ≤ nI ( U k ; Z | U K , . . . , U k +1 ) + nδ, (21) with δ → 0 as n → ∞ a nd ( c) is by the Markov chain condition U K → · · · → U k +1 → U k → Z for the degraded BC. Similarly , by su bstituting X fo r U 1 and removing conditionin g fro m (15) fo r k = K , w e have      log L 1 ≤ nI ( X ; Z | U 2 ) + nδ + 1 + ¯ λ (1) log L ′ 1 , log L K ≤ nI ( U K ; Z ) + nδ + 1 + ¯ λ ( K ) log L ′ K . (22) Based on the above, and since R ′ k = 1 n log L ′ k , we let            R ′ 1 , I ( X ; Z | U 2 ) − τ , R ′ k , I ( U k ; Z | U k +1 ) − τ , f or k = 2 , . . . , K − 1 , R ′ K , I ( U K ; Z ) − τ , (23) where τ → 0 for sufficiently large n . 5) Prob ability of er ror analysis : W e fo llo w the method by Co ver and Tho mas in [15], and pr o vide the ana lysis f or the k th receiver . Assume without loss of gener ality that ( W 1 , . . . , W k ) = (1 , . . . , 1) is sent and w ′ k = 1 is sent for the subcod es C ( k ) i k ∀ k . At the k th receiver , define the fo llo wing events ( and their complements denoted by the su perscript c ) :                      E ( Y k ) i K ,i ′ K = n ( U K ( i K , i ′ K ) , Y k ) ∈ A ( n ) ǫ o E ( Y k ) i K ,i ′ K ,i K − 1 ,i ′ K − 1 = n ( U K ( i K , i ′ K ) , U K − 1 ( i K , i ′ K , i K − 1 , i ′ K − 1 ) , Y k ) ∈ A ( n ) ǫ o . . . E ( Y k ) i K ,i ′ K ,...,i k ,i ′ k = n ( U K ( i K , i ′ K ) , . . . , U k ( i K , i ′ K , . . . , i k , i ′ k ) , Y k ) ∈ A ( n ) ǫ o . (24) Then, by the union of ev ents bound , we have P ( n ) e ( k ) ≤ Pr n E ( Y k ) 1 , 1  c o + Pr n E ( Y k ) 1 , 1 , 1 , 1  c o + · · · + Pr n E ( Y k ) 1 , 1 ,..., 1 , 1  c o + X i K ,i ′ K i K 6 =1 Pr n E ( Y k ) i K ,i ′ K  c o + X i K − 1 ,i ′ K − 1 i K − 1 6 =1 Pr n E ( Y k ) 1 , 1 ,i K − 1 ,i ′ K − 1  c o + · · · + X i k ,i ′ k i k 6 =1 Pr n E ( Y k ) 1 , 1 ,...,i k ,i ′ k  c o , (25) where there are ( K − k + 1 ) terms in each of the first and second lines of the inequ ality (25) ab o ve; the last term in the first line of (25) refers to the probability that the compleme nt of the event that the 2 k -leng th 8 vector o f all ones (1 , 1 , . . . , 1 , 1) occurred; and the last ter m in the s econd line o f (25) refers to the probab ility that the event that the 2 k -length vector (1 , 1 , . . . , i k , i ′ k ) occur red. For the first two term s o f ( 25) , we ha ve      Pr n E ( Y 1 ) i K ,i ′ K o ≤ 2 − n ( I ( U K ; Y k ) − 3 ǫ ) , Pr n E ( Y 1 ) 1 , 1 ,i K − 1 ,i ′ K − 1 o ≤ 2 − n ( I ( U K − 1 ; Y k | U K ) − 4 ǫ ) . (26) Denoting the event { ( U K , . . . , U k , Y k ) ∈ A ( n ) ǫ } as ˜ E ( Y k ) , the k th term can be written as Pr n E ( Y k ) 1 , 1 ,...,i k ,i ′ k  c o = Pr n ( U K (1 , 1) , U K − 1 (1 , 1 , 1 , 1) , . . . , U k (1 , 1 , . . . , i k , i ′ k ) , Y k ) ∈ A ( n ) ǫ o = X ˜ E ( Y k ) Pr { ( U K (1 , 1) , U K − 1 (1 , 1 , 1 , 1) , . . . , U k (1 , 1 , . . . , i k , i ′ k ) , Y k ) } = X ˜ E ( Y k )         Pr { U K (1 , 1) } Pr { U K − 1 (1 , 1 , 1 , 1) | U K (1 , 1) } × · · · × Pr  U k (1 , 1 , . . . , i k , i ′ k ) | U K (1 , 1) , U K − 1 (1 , 1 , 1 , 1) , . . . , U k +1 ( 2( k +1) terms z }| { 1 , 1 , . . . , 1 , 1)  × Pr  Y k | U K (1 , 1) , U K − 1 (1 , 1 , 1 , 1) , . . . , U k +1 ( 2( k +1) terms z }| { 1 , 1 , . . . , 1 , 1)          ≤ X ˜ E ( Y k ) 2 − n ( H ( U K ) − ǫ ) 2 − n ( H ( U K − 1 | U K ) − ǫ ) × · · · × 2 − n ( H ( U k | U K ,...,U k +1 ) − ǫ ) 2 − n ( H ( Y k | U K ,...,U k +1 ) − ǫ ) ≤ 2 n ( H ( U K ,...,U k +1 ,U k ,Y k )+ ǫ ) 2 − n ( H ( U K ) − ǫ ) 2 − n ( H ( U K − 1 | U K ) − ǫ ) × · · · × 2 − n ( H ( U k | U K ,...,U k +1 ) − ǫ ) 2 − n ( H ( Y k | U K ,...,U k +1 ) − ǫ ) = 2 − n ( H ( Y k | U K ,...,U k +1 ) − H ( Y k | U K ,...,U k +1 ,U k ) − ( k +2) ǫ ) = 2 − n ( I ( Y k ; U k | U K ,...,U k +1 ) − ( k +2) ǫ ) . As a result, P ( n ) e ( k ) ≤ ( K − k + 1 ) ǫ + 2 n ( R K + R ′ K ) 2 − n ( I ( U K ; Y k ) − 3 ǫ ) + 2 n ( R K − 1 + R ′ K − 1 ) 2 − n ( I ( U K − 1 ; Y k | U K ) − 4 ǫ ) + · · · + 2 n ( R k + R ′ k ) 2 − n ( I ( Y k ; U k | U K ,...,U k +1 ) − ( k +2) ǫ ) ≤ 2 ( K − k + 1) ǫ, for n suf ficien tly large and (27) R K + R ′ K < I ( U K ; Y k ) , (28a) R K − 1 + R ′ K − 1 < I ( U K − 1 ; Y k | U K ) , (28b) . . . R k + R ′ k < I ( U k ; Y k | U K , . . . , U k +1 ) . (28c) Since I ( U K ; Y k ) ≥ I ( U K ; Y k +1 ) ≥ · · · ≥ I ( U K ; Y K ) an d similarly I ( U k +1 ; Y k | U K , . . . , U k +2 ) ≥ · · · ≥ 9 I ( U k +1 ; Y k +1 | U K , . . . , U k +2 ) by the degraded n ature of th e channel, from (28a), we g et R K + R ′ K ≤ I ( U K ; Y K ) , (29a) R K − 1 + R ′ K − 1 ≤ I ( U K − 1 ; Y K − 1 | U K ) , (29b) . . . R k + R ′ k ≤ I ( U k ; Y k | U K , . . . U k +1 ) , (29c) for the second to the last terms in (27) to be ≤ ǫ . Then, as we h a ve the condition U K → U K − 1 → · · · → U k +1 → U k → Y k , we h a ve R K + R ′ K ≤ I ( U K ; Y K ) , (30a) R K − 1 + R ′ K − 1 ≤ I ( U K − 1 ; Y K − 1 | U K ) , (30b) . . . R k + R ′ k ≤ I ( U k ; Y k | U k +1 ) . (30c) Follo wing the same app roach, fo r the first receiver , we can get the ab o ve ineq ualities in (30), as well as R 1 + R ′ 1 ≤ I ( X ; Y 1 | U 2 ) . (31) Therefo re, for all the re cei vers, given the pre vious d efinitions for R ′ 1 , . . . , R ′ K in (23), it can be seen that the probab ility of err or at eac h recei ver satisfies P ( n ) e ( k ) ≤ 2( K − k + 1) ǫ for k = 1 , . . . , K an d fo r any ra te tuple ( R 1 , . . . , R K ) satisfyin g the co nditions in Theorem 1. Thus, the direc t par t of Theorem 1 is p rov ed. B. Eq uivocation Calcu lation W e sho w the calculation f or the k th recei ver R e ( k ) for k = 1 , . . . , K , R e ( k,k +1) for k = 1 , . . . , K − 1 , and the sum rate R e (1 ,...,K ) . W e shall make u se of the relation H ( U, V ) = H ( U ) + H ( V | U ) . (32) For the k th receiv er , k = 2 , . . . , K − 1 , we have nR e ( k ) = H ( W k | Z ) ≥ H ( W k | Z , U K , . . . , U k +1 ) since co nditioning r educes entropy = H ( W k , Z | U K , . . . , U k +1 ) − H ( Z | U K , . . . , U k +1 ) by (32) ( a ) = H ( W k , U k , Z | U K , . . . , U k +1 ) − H ( U k | W k , Z , U K , . . . , U k +1 ) − H ( Z | U K , . . . , U k +1 ) ( b ) = H ( W k , U k | U K , . . . , U k +1 ) + H ( Z | W k , U K , . . . , U k +1 , U k ) − H ( Z | U K , . . . , U k +1 ) − H ( U k | W k , Z , U K , . . . , U k +1 ) 10 ( c ) ≥ H ( U k | U K , . . . , U k +1 ) + H ( Z | U K , . . . , U k +1 , U k ) − H ( Z | U K , . . . , U k +1 ) − H ( U k | W k , Z , U K , . . . , U k +1 ) = H ( U k | U K , . . . , U k +1 ) − I ( U k ; Z | U K , . . . , U k +1 ) − H ( U k | W k , Z , U K , . . . , U k +1 ) , (33) where (a) and (b) hav e the first tw o terms b y (32), and (c) has the first term by (32) an d the second term by the fact tha t W k → ( U K , . . . , U k +1 ) → Z . W e now boun d each o f the terms in the last lin e of (33). For th e first term, gi ven that U K = u K , U K − 1 = u K − 1 , . . . , U k +1 = u k +1 , u k has 2 n ( R k + R ′ k ) possible values with equ al probab ility . As a consequen ce, we have H ( U k | U K , . . . , U k +1 ) = n ( R k + R ′ k ) . (34) For the second term, it can be shown that I ( U k ; Z | U K , . . . , U k +1 ) ≤ nI ( U k ; Z | U k +1 ) + nδ. (35) For the last term, we ha ve by Fano’ s inequality 1 n H ( U k | W k , Z , U K , . . . , U k +1 ) ≤ 1 n  1 + ¯ λ ( k ) log L ′ k  , ǫ ′ k,n (36) where ǫ ′ k,n → 0 for n sufficiently large. T o show that ¯ λ ( k ) → 0 for n sufficiently large so that (36) holds, we consider decoding at the wiretapper and focus o n the co debook with r ate R ′ k to be deco ded at the wire tapper with err or probability ¯ λ ( k ) . L et W k = i k be fixed. The wiretappe r attempts to d ecode u k giv en w k , u K , . . . , u k +1 by finding the estimate for w ′ k , ˆ w ′ k , so th at ( u k ( w k , ˆ w ′ k , w k +1 , w ′ k +1 , . . . , w K , w ′ K ) , z ) ∈ A n ǫ ( P U k Z | U k +1 ...U K ) . (37) where w k , a nd all w k +1 , w ′ k +1 , . . . , w K , w ′ K are kn o wn. If there is no ne or more than one possible codeword, an error is d eclared. Defin ing the e vent E ( Z ) i ′ k , n ( U k ( i k , i ′ k ) , Z ) ∈ A ( n ) ǫ ( P U k Z | U k +1 ...U K ) o , (38) and assuming with out loss of generality that w ′ k = 1 is sent, we then h a ve ¯ λ ( k ) ≤ Pr n E ( Z ) 1  c o + X i ′ k 6 =1 Pr n E ( Z ) i ′ k o ≤ ǫ + 2 nR ′ k 2 − n ( I ( U k ; Z | U k +1 ,...,U K ) − 2 ǫ ) , (39) where ǫ → 0 for n suffi ciently large. Sinc e we have ch osen fro m (23) that R ′ k = I ( U k | Z | U k +1 ) − τ which is = I ( U k | Z | U k +1 , . . . , U K ) − τ by U K → · · · → U k +1 → U k → Z , we hav e ¯ λ ( k ) ≤ 2 ǫ , for τ > 2 ǫ . Thus, ¯ λ ( k ) is small for n suf ficiently large and (36) hold s . Now sub sti tuting (3 4 )–(36) into the last line of (33), we ha ve nR e ( k ) ≥ nR k + nI ( U k ; Z | U k +1 ) − nτ − nI ( U k ; Z | U k +1 ) − nδ − n ǫ ′ k,n = nR k − nǫ k (40) where ǫ k = τ + δ + ǫ ′ k,n . Hence, the security condition in (8) is satisfied for the k th receiver . For the first receiv er , we condition on U K , . . . , U 2 in the second line of the ch ain of inequ alities above, while for the K th receiver , we 11 can omit the second line o f the chain o f inequalities in ( 33) above, while subsequently not p erforming additional condition ing in (33). The equivocation rates for the first rec ei ver a nd the K th receiver will have the same form as (40) above with k = 1 and k = K . W e next show t he equiv oc ation rates for a djacent receivers k , k + 1 for k = 1 , . . . , K − 1 . Du e to the nature of the coding, equiv ocatio n rates for no n-adjacent receivers are not achievable. W e also assume tha t, for the equiv o cation rate for any two ad jacent recei vers k, k + 1 for k = 1 , . . . , K − 1 to be achiev able, the ( k + 1) th receiver m ust ha ve knowledge o f u k +2 , . . . , u K . Then , we have nR e ( k,k +1) = H ( W k , W k +1 | Z ) ≥ H ( W k , W k +1 | Z , U k +2 , . . . , U K ) since co nditioning r educes entropy = H ( W k , W k +1 , Z | U k +2 , . . . , U K ) − H ( Z | U k +2 , . . . , U K ) by (32) ( a ) = H ( W k , W k +1 , U k , U k +1 , Z | U k +2 , . . . , U K ) − H ( U k , U k +1 | W k , W k +1 , U k +2 , . . . , U K , Z ) − H ( Z | U k +2 , . . . , U K ) ( b ) = H ( W k , W k +1 , U k , U k +1 | U k +2 , . . . , U K ) + H ( Z | W k , W k +1 , U k , U k +1 , U k +2 , . . . , U K ) (41) − H ( U k , U k +1 | W k , W k +1 , U k +2 , . . . , U K , Z ) − H ( Z | U k +2 , . . . , U K ) ( c ) ≥ H ( U k , U k +1 | U k +2 , . . . , U K ) + H ( Z | U k , U k +1 , U k +2 , . . . , U K ) − H ( Z | U k +2 , . . . , U K ) − H ( U k , U k +1 | W k , W k +1 , U k +2 , . . . , U K , Z ) ( d ) = H ( U k +1 | U k +2 , . . . , U K ) + H ( U k | U k +1 , U k +2 , . . . , U K ) + [ H ( Z | U k , U k +1 , U k +2 , . . . , U K ) − H ( Z | U k +2 , . . . , U K )] − H ( U k , U k +1 | W k , W k +1 , U k +2 , . . . , U K , Z ) = H ( U k +1 | U k +2 , . . . , U K ) + H ( U k | U k +1 , U k +2 , . . . , U K ) − I ( U k , U k +1 ; Z | U k +2 , . . . , U K ) − H ( U k , U k +1 | W k , W k +1 , U k +2 , . . . , U K , Z ) , where ( a), (b) an d (d) have the first two terms by (32), and (c) has the first term by (32) and the second term by ( W k , W k +1 ) → ( U k U k +1 U k +2 · · · U K ) → Z . W e now bound each of the terms in the last line of (41). For the first term, given U k +2 = u k +2 , . . . , U K = u K , u k +1 has 2 n ( R k +1 + R ′ k +1 ) possible v alues with equ al pr obability . For the second term, gi ven U k +1 = u k +1 , . . . , U K = u K , u k has 2 n ( R k + R ′ k ) possible values with equal pro bability . Therefore, we have      H ( U k +1 | U k +2 , . . . , U K ) = n ( R k +1 + R ′ k +1 ) , H ( U k | U k +1 , U k +2 , . . . , U K ) = n ( R k + R ′ k ) . (42) For the third term , it can be shown th at I ( U k , U k +1 ; Z | U k +2 , . . . , U K ) = I ( U k +1 ; Z | U k +2 , . . . , U K ) + I ( U k ; Z | U k +1 , U k +2 , . . . , U K ) , ≤ nI ( U k +1 ; Z | U k +2 ) + nI ( U k ; Z | U k +1 ) + 2 nδ. (43) 12 For the last term, we ha ve H ( U k , U k +1 | W k , W k +1 , U k +2 , . . . , U K , Z ) = H ( U k +1 | W k +1 , U k +2 , . . . , U K , Z ) + H ( U k | W k , W k +1 , U k +1 , U k +2 , . . . , U K , Z ) ≤ H ( U k +1 | W k +1 , U k +2 , . . . , U K , Z ) + H ( U k | W k , U k +1 , U k +2 , . . . , U K , Z ) , (44) where the first equality is be cause of the fact that W k and U k +1 are in dependent, and th e last line is by conditioning reducing entr opy . Fr om th e last line of (44), b y Fano’ s ineq uality , we have      1 n H ( U k +1 | W k +1 , U k +2 , . . . , U K , Z ) ≤ 1 n  1 + ¯ λ ( k +1) log L ′ k +1  , ǫ ′ k +1 ,n , 1 n H ( U k | W k , U k +1 , U k +2 , . . . , U K , Z ) ≤ 1 n  1 + ¯ λ ( k ) log L ′ k  , ǫ ′ k,n , (45) where ǫ ′ k,n , ǫ ′ k +1 ,n → 0 f or n sufficiently la r ge. W e need to sho w that ¯ λ ( k +1) is small for n suf ficien tly large so that (45) holds, as we already h a ve sho wn that ¯ λ ( k ) is small for n sufficiently large. W e consider the situation where the wire tapper attempts to d ecode U k +1 giv en W k +1 by joint typicality . Then, f ollo win g the same pr ocedure to calculate ¯ λ ( k ) in (39) ab o ve, we hav e ¯ λ ( k +1) ≤ ǫ + 2 nR ′ k +1 2 − n ( I ( U k +1 ; Z | U k +2 ,...,U K ) − 2 ǫ ) . (46) Since we have selected R ′ k +1 = I ( U k +1 ; Z | U k +2 ) − τ = I ( U k +1 ; Z | U k +2 , . . . , U K ) − τ , then ¯ λ ( k +1) ≤ ǫ for τ > 2 ǫ and where ǫ → 0 for n sufficiently large. As a consequence, in (45), bo th ¯ λ ( k ) , ¯ λ ( k +1) are small for n su f ficiently large and (45) hold s and we ha ve 1 n H ( U k +1 | W k +1 , U k +2 , . . . , U K , Z ) ≤ ǫ ′ k +1 ,n + ǫ ′ k,n . (47) Then, substituting (4 2 ), (43) and ( 47) into the last line of (41), and given R ′ k +1 and R ′ k in (23), we ha ve nR e ( k,k +1) ≥ nR k + nR k +1 − ǫ k,k +1 , (48) where ǫ k,k +1 = 2 τ + 2 δ + ǫ ′ k +1 ,n + ǫ ′ k,n , and so security cond ition (8 ) is shown. W e n ote that to show equiv ocation rates for th e pa ir k = 1 , 2 , this can b e d one by following th e pr oof ab o ve, b ut replacing U 1 by X . Lastly , we show the pro of for the equi vocation sum rate R e (1 ,...,K ) . W e ha ve nR e (1 ,...,K ) = H ( W 1 , . . . , W K | Z ) = H ( W 1 , . . . , W K , Z ) − H ( Z ) by (3 2 ) ( a ) = H ( W 1 , . . . , W K , U 2 , . . . , U K , X , Z ) − H ( U 2 , . . . , U K , X | W 1 , . . . , W K , Z ) − H ( Z ) ( b ) = H ( W 1 , . . . , W K , U 2 , . . . , U K , X ) + H ( Z | W 1 , . . . , W K , U 2 , . . . , U K , X ) − H ( U 2 , . . . , U K , X | W 1 , . . . , W K , Z ) − H ( Z ) ( c ) ≥ H ( U 2 , . . . , U K , X ) + H ( Z | U 2 , . . . , U K , X ) − H ( Z ) − H ( U 2 , . . . , U K , X | W 1 , . . . , W K , Z ) ( d ) = H ( U K ) + H ( U K − 1 | U K ) + · · · + H ( X | U 2 , . . . , U K ) + [ H ( Z | U 2 , . . . , U K , X ) − H ( Z )] − H ( U 2 , . . . , U K , X | W 1 , . . . , W K , Z ) 13 = H ( U K ) + H ( U K − 1 | U K ) + · · · + H ( X | U 2 , . . . , U K ) − I ( U 2 , . . . , U K , X ; Z ) − H ( U 2 , . . . , U K , X | W 1 , . . . , W K , Z ) , (49) where ( a), (b) an d (d) have the first two terms by (32), and (c) has the first term by (32) and the second term by the fact that ( W 1 . . . W K ) → ( U K . . . U 2 X ) → Z . W e now boun d each of the terms in the last line of (49). For the first term, u K has 2 n ( R K + R ′ K ) possible values with equal pr obability . For the seco nd to the ( K − 1) th terms, given the preceding cod e words u k +1 , . . . , u K , u k has 2 n ( R k + R ′ k ) possible values with equal prob ability . For the K th term, given all the precedin g codew o rds, x has 2 n ( R 1 + R ′ 1 ) possible values with equal pro bability . As such, we h a ve H ( U K ) = n ( R K + R ′ K ) , (50a) H ( U K − 1 | U K ) = n ( R K − 1 + R ′ K − 1 ) , (50b) . . . H ( X | U 2 , . . . , U K ) = n ( R 1 + R ′ 1 ) . (50c) For the second last term, it can be shown that I ( U 2 , . . . , U K , X ; Z ) = I ( U K ; Z ) + I ( U K − 1 ; Z | U K ) + · · · + I ( X ; Z | U 2 , . . . , U K ) ≤ nI ( U K ; Z ) + nI ( U K − 1 ; Z | U K ) + · · · + nI ( X ; Z | U 2 ) + K nδ. (51) For the last term, we ha ve H ( U 2 , . . . , U K , X | W 1 , . . . , W K , Z ) = H ( U K | W K , Z ) + H ( U K − 1 | W K − 1 , W K , U K , Z ) + · · · + H ( X | W 1 , . . . , W K , U 2 , . . . , U K , Z ) ≤ H ( U K | W K , Z ) + H ( U K − 1 | W K − 1 , U K , Z ) + H ( X | W 1 , U 2 , . . . , U K , Z ) (52) where the first e quality is because of th e fact that successi vely , we ha ve U K and W 1 , . . . , W K − 1 are independe nt, U K − 1 and W 1 , . . . , W K − 2 are independe nt, a nd so on, and the last line is b y cond itioning reducing en tropy . Now by apply ing Fano’ s inequ ality to eac h term in the last line of (52), we have                      1 n H ( U K | W K , Z ) ≤ 1 n  1 + ¯ λ ( K ) log L ′ K  , ǫ ′ K,n , 1 n H ( U K − 1 | W K − 1 , U K , Z ) ≤ 1 n  1 + ¯ λ ( K − 1) log L ′ 2  , ǫ ′ K − 1 ,n , . . . 1 n H ( X | W 1 , U 2 , . . . , U K , Z ) ≤ 1 n  1 + ¯ λ (1) log L ′ 1  , ǫ ′ 1 ,n . (53) where ǫ ′ 1 ,n , . . . , ǫ ′ K,n → 0 for n sufficiently large. It can b e shown tha t ¯ λ (1) , . . . , ¯ λ ( K ) ≤ 2 ǫ where ǫ → 0 for n sufficiently large by co nsidering the wireta pper deco ding X , . . . , U K giv en the respective associated messages W 1 , . . . , W K and the prec eding codewords. This is done using the sam e method of joint typicality an d th e cho ice 14 of rates fo r R ′ k , k = 1 , . . . , K as shown in the ab o ve for th e equiv oc ation rates for R e ( k ) and R e ( k,k +1) . Thu s , 1 n H ( U 2 , . . . , U K , X | W 1 , . . . , W K , Z ) ≤ K X k =1 ǫ ′ k,n . (54 ) Then by substituting (50a)–(50 c ), (51) and (5 4 ) in to the last line of (49), we ha ve, given the definitions for the rate tuple ( R ′ 1 , . . . , R ′ K ) in (2 3), nR e (1 ,...,K ) ≥ n K X k =1 R k − ǫ 1 ,...,K , (55) where ǫ 1 ,...,K = K τ + K δ + P K k =1 ǫ ′ k,n and the secu rity conditions in (8) are satisfied. As a result, we ha ve shown that the equ i vocation rates in (8) a re ach ie vable, and hence the secret ra te tuple ( R 1 , . . . , R K ) . I V . P RO O F O F C O N V E R S E Here, w e show the converse pr oof to The orem 1. Consider a (2 nR 1 , . . . , 2 nR K , n ) code with erro r probability P ( n ) e with the code construction so th at we have the con dition ( W 1 · · · W K ) → X → Y 1 · · · Y K Z . Then, the probab ility distribution on W 1 × · · · × W K × X n × Y n 1 × · · · × Y n K × Z n is given by p ( w 1 ) · · · p ( w 3 ) p ( x | w 1 , . . . , w K ) n Y i =1 p ( y 1 i , . . . , y K i , z i | x i ) . (56) In the fo llo wing , we give the proof for the rate at the k th receiv er . W e shall also show later that the pr oof f or the K th receiver can be easily obtained from this, while the pr oof for th e first recei ver req uires a few a dditional steps. For k = 2 , . . . , K − 1 , the rate R k satisfies nR k = H ( W k ) ≤ H ( W k | Z ) + nǫ k by secrecy cond iti on = H ( W k | Z , W k +1 , . . . , W K ) + I ( W k ; W k +1 , . . . , W K | Z ) + nǫ k = H ( W k | W k +1 , . . . , W K ) − I ( W k ; Z | W k +1 , . . . , W K ) + I ( W k ; W k +1 , . . . , W K | Z ) + nǫ k = I ( W k ; Y k | W k +1 , . . . , W K ) + H ( W k | Y k , W k +1 , . . . , W K ) − I ( W k ; Z | W k +1 , . . . , W K ) (57) + I ( W k ; W k +1 , . . . , W K | Z ) + nǫ k ( a ) ≤ I ( W k ; Y k | W k +1 , . . . , W K ) − I ( W k ; Z | W k +1 , . . . , W K ) + H ( W k | Y k , W k +1 , . . . , W K ) + H ( W k +1 | Z ) + · · · + H ( W K | Z ) + nǫ k ( b ) ≤ I ( W k ; Y k | W k +1 , . . . , W K ) − I ( W k ; Z | W k +1 , . . . , W K ) + n ( δ ′′ k + δ ′ k +1 + · · · + δ ′ K + ǫ k ) , (58) where (a) is by I ( W k ; W k +1 , . . . , W K | Z ) ≤ H ( W k +1 , . . . , W K | Z ) ≤ H ( W k +1 | Z ) + · · · + H ( W K | Z ) , and (b ) is by Fano’ s inequality which g i ves                    H ( W k | Y k , W k +1 , . . . , W K ) ≤ nR k P ( n ) e + 1 , nδ ′′ k , H ( W k +1 | Z ) ≤ nR k +1 P ( n ) e + 1 , nδ ′ k +1 , . . . H ( W K | Z ) ≤ nR K P ( n ) e + 1 , nδ ′ K , (59) 15 where δ ′′ k , δ ′ k +1 , . . . , δ ′ K → 0 if P ( n ) e → 0 . Expand ing the first tw o terms of (57) by the chain r ule g i ves I ( W k ; Y k | W k +1 , . . . , W K ) = n X i =1 I ( W k ; Y k,i | W k +1 , . . . , W K , Y i − 1 k ) , (60a) I ( W k ; Z | W k +1 , . . . , W K ) = n X i =1 I ( W k ; Z i | W k +1 , . . . , W K , ˜ Z i +1 ) . (60b) From (60 a ), by using the iden tity I ( S 1 S 2 ; T | V ) = I ( S 1 ; T | V ) + I ( S 2 ; T | S 1 V ) , w e get I ( W k ; Y k,i | W k +1 , . . . , W K , Y i − 1 k ) = I ( W k , ˜ Z i +1 ; Y k,i | W k +1 , . . . , W K , Y i − 1 k ) − I ( ˜ Z i +1 ; Y k,i | W k , W k +1 , . . . , W K , Y i − 1 1 ) = I ( W k ; Y k,i | W k +1 , . . . , W K , Y i − 1 k , ˜ Z i +1 ) + I ( ˜ Z i +1 ; Y k,i | W k +1 , . . . , W K , Y i − 1 k ) − I ( ˜ Z i +1 ; Y k,i | W k , W k +1 , . . . , W K , Y i − 1 k ) . (61) Substituting this into (60a) we have,                        I ( W k ; Y k | W k +1 , . . . , W K ) = n X i =1 I ( W k ; Y k,i | W k +1 , . . . , W K , Y i − 1 k , ˜ Z i +1 ) + Σ k, 1 − Σ k, 2 Σ k, 1 = n X i =1 I ( ˜ Z i +1 ; Y k,i | W k +1 , . . . , W K , Y i − 1 1 ) , Σ k, 2 = n X i =1 I ( ˜ Z i +1 ; Y k,i | W k , W k +1 , . . . , W K , Y i − 1 k ) . (62) From (60 b ), again by using I ( S 1 S 2 ; T | V ) = I ( S 1 ; T | V ) + I ( S 2 ; T | S 1 V ) , a nd substituting this into (6 0b ) , we get                        I ( W k ; Z | W k +1 , . . . , W K ) = n X i =1 I ( W k ; Z i | W k +1 , . . . , W K , Y i − 1 k , ˜ Z i +1 ) + Σ ∗ k, 1 − Σ ∗ k, 2 Σ ∗ k, 1 = n X i =1 I ( Y i − 1 k ; Z i | W k +1 , . . . , W K , ˜ Z i +1 ) , Σ ∗ k, 2 = n X i =1 I ( Y i − 1 k ; Z i | W k , W k +1 , . . . , W K , ˜ Z i +1 ) . (63) It is kn o wn by Lem ma 7 in [6] that Σ k, 1 = Σ ∗ k, 1 and Σ k, 2 = Σ ∗ k, 2 . Ther efore, nR k ≤ n X i =1 h I ( W k ; Y k,i | W k +1 , . . . , W K , Y i − 1 k , ˜ Z i +1 ) − I ( W k ; Z i | W k +1 , . . . , W K , Y i − 1 k , ˜ Z i +1 ) i + n ( δ ′′ k + δ ′ k +1 + · · · + δ ′ K + ǫ k ) . (64 ) The terms u nder th e summ ation are I ( W k ; Y k,i | W k +1 , . . . , W K , Y i − 1 k , ˜ Z i +1 ) − I ( W k ; Z i | W k +1 , . . . , W K , Y i − 1 k , ˜ Z i +1 ) = H ( W k | W k +1 , . . . , W K , Y i − 1 k , Z i , ˜ Z i +1 ) − H ( W k | W k +1 , . . . , W K , Y i − 1 k , Y k,i , ˜ Z i +1 ) ( a ) ≤ H ( W k | W k +1 , . . . , W K , Y i − 1 k , Z i , ˜ Z i +1 ) − H ( W k | W k +1 , . . . , W K , Y i − 1 k , Y k,i , Z i , ˜ Z i +1 ) = I ( W k ; Y k,i | W k +1 , . . . , W K , Y i − 1 k , Z i , ˜ Z i +1 ) 16 = H ( Y k,i | W k +1 , . . . , W K , Y i − 1 k , Z i , ˜ Z i +1 ) − H ( Y k,i | W k , W k +1 , . . . , W K , Y i − 1 k , Z i , ˜ Z i +1 ) (65) = H ( Y k,i | W k +1 , . . . , W K , Y i − 1 k , Y i − 1 k +1 , . . . , Y i − 1 K , Z i , ˜ Z i +1 ) + I ( Y k,i ; Y i − 1 k +1 , . . . , Y i − 1 K | W k +1 , . . . , W K , Y i − 1 k , Z i , ˜ Z i +1 ) − H ( Y k,i | W k , W k +1 , . . . , W K , Y i − 1 k , Z i , ˜ Z i +1 ) ( b ) = H ( Y k,i | W k +1 , . . . , W K , Y i − 1 k , Y i − 1 k +1 , . . . , Y i − 1 K , Z i , ˜ Z i +1 ) − H ( Y k,i | W k , W k +1 , . . . , W K , Y i − 1 k , Z i , ˜ Z i +1 ) ( c ) ≤ H ( Y k,i | W k +1 , . . . , W K , Y i − 1 k +1 , . . . , Y i − 1 K , Z i , ˜ Z i +1 ) − H ( Y k,i | W k , W k +1 , . . . , W K , Y i − 1 k , Y i − 1 k +1 , . . . , Y i − 1 K , Z i , ˜ Z i +1 ) = I ( W k , Y i − 1 k ; Y k,i | W k +1 , . . . , W K , Y i − 1 k +1 , . . . , Y i − 1 K , Z i , ˜ Z i +1 ) , where (a) and (c) are due to co nditioning r educing entropy , and (b) is due to th e fact th at I ( Y k,i ; Y i − 1 k +1 , . . . , Y i − 1 K | W k +1 , . . . , W K , Y i − 1 k , Z i , ˜ Z i +1 ) = 0 , (66) since Y i − 1 k +1 · · · Y i − 1 K → W k +1 · · · W K → Y i − 1 k Z i ˜ Z i +1 → Y k,i . Now , de fine th e ran dom variables      U K,i , W K Y i − 1 K ˜ Z i +1 , U k,i , W k Y i − 1 k , for k = 2 , . . . , K − 1 , (67) and we h a ve the cond ition U K,i → · · · → U k,i → X i → Y k,i · · · Y K,i → Z i . (68) W e note that ou r ch oice of aux iliary random v ariables is different from Bagherikar am et al. , which d eals with the 2-receiver degraded BC with an external wiretapper [ 10], and from [ 12], which studies the K -receiver degra ded BC with a commo n message and an external wire tapper . The ch oice is also different, due to the presen ce of the wiretapp er , fro m that of Borad e et al. in [1 3] wh ich d eals with the K -receiver degrad ed BC without secrecy condition s. Thus, we have I ( W k , Y i − 1 k ; Y k,i | W k +1 , . . . , W K , Y i − 1 k +1 , . . . , Y i − 1 K , Z i , ˜ Z i +1 ) = I ( U k,i ; Y k,i | U ( k +1) ,i , . . . , U K,i , Z i ) ( a ) = I ( U k,i ; Y k,i | U ( k +1) ,i , Z i ) = I ( U k,i ; Y k,i , Z i | U ( k +1) ,i ) − I ( U k,i ; Z i | U ( k +1) ,i ) = I ( U k,i ; Y k,i | U ( k +1) ,i ) + I ( U k,i ; Z i | U ( k +1) ,i ) − I ( U k,i ; Z i | U ( k +1) ,i ) ( b ) = I ( U k,i ; Y k,i | U ( k +1) ,i ) − I ( U k,i ; Z i | U ( k +1) ,i ) , (69) where (a) is due to the co ndition (68), and (b) is due to I ( U k,i ; Z i | U ( k +1) ,i ) = 0 since we have U k,i → U ( k +1) ,i → Z i . As a result, we have nR k ≤ n X i =1  I ( U k,i ; Y k,i | U ( k +1) ,i ) − I ( U k,i ; Z i | U ( k +1) ,i )  + n ( δ ′′ k + δ ′ k +1 + · · · + δ ′ K + ǫ k ) . (70) 17 T o show the con verse fo r R 1 , we follow the same steps as above, but additionally we use (65) with k = 1 to arrive at the equ i valent chain of equalities (69) above for k = 1 . From the last line o f (65), substituting for the random variables U 2 ,i , . . . , U K,i , we th en h a ve I ( W 1 , Y i − 1 1 ; Y 1 ,i | W 2 , . . . , W K , Y i − 1 2 , . . . , Y i − 1 K , Z i , ˜ Z i +1 ) = I ( W 1 , Y i − 1 1 ; Y 1 ,i | U 2 ,i , . . . , U K,i , Z i ) = I ( W 1 ; Y 1 ,i | U 2 ,i , . . . , U K,i , Z i ) + I ( Y i − 1 1 ; Y 1 ,i | W 1 , U 2 ,i , . . . , U K,i , Z i ) ( a ) = I ( W 1 ; Y 1 ,i | U 2 ,i , . . . , U K,i , Z i ) ( b ) ≤ I ( X i ; Y 1 ,i | U 2 ,i , Z i ) = I ( X i ; Y 1 ,i , Z i | U 2 ,i ) − I ( X i ; Z i | U 2 ,i ) = I ( X i ; Y 1 ,i | U 2 ,i ) + I ( X i ; Z i | U 2 ,i , Y 1 ,i ) − I ( X i ; Z i | U 2 ,i ) ( c ) = I ( X i ; Y 1 ,i | U 2 ,i ) − I ( X i ; Z i | U 2 ,i ) (71) where (a) is by the seco nd term I ( Y i − 1 1 ; Y 1 ,i | W 1 , U 2 ,i , . . . , U K,i , Z i ) = 0 since Y i − 1 1 → W 1 U 2 ,i · · · U K,i Z i → Y 1 ,i , (b) is by Y 1 ,i → X i → W 1 and by the Markov con dition (68), and (c) is by the secon d ter m I ( X i ; Z i | U 2 ,i , Y 1 ,i ) = 0 since X i → U 2 ,i Y 1 ,i → Z i . Thu s , we have nR 1 ≤ n X i =1 [ I ( X i ; Y 1 ,i | U 2 ,i ) − I ( X i ; Z i | U 2 ,i )] + n ( δ ′′ 1 + δ ′ 2 + · · · + δ ′ K + ǫ 1 ) . (72) The pro of f or R K is easily obtained using the above approach, only withou t c onditioning on W k +1 , . . . , W K in the secon d line o f ( 57) . This results in nR K ≤ n X i =1 [ I ( U K,i ; Y K,i ) − I ( U K,i ; Z i )] + n ( δ ′′ K + ǫ K ) . (73) Now , we intr oduce the random v a riable G , which is unifo rmly distributed among the integers { 1 , 2 , . . . , n } an d is indepen dent of all other random variables. De fine the following au xiliary ra ndom v a riables U K = ( G, U K,G ) , (74a) U K − 1 = ( G, U K − 1 ,G ) , (74b) . . . X = X G , (74c) Y 1 = Y 1 ,G , (74d) . . . Y K = Y K,G , (74e) Z = Z G . (74f) 18 Then (70), ( 72 ) , (73) become R K ≤ I ( U K ; Y K ) − I ( U K ; Z ) , (75a) R k ≤ I ( U k ; Y k | U k +1 ) − I ( U k ; Z | U k +1 ) , fo r k = 2 , . . . , K − 1 , (75b) R 1 ≤ I ( X ; Y 1 | U 2 ) − I ( X ; Z | U 2 ) , (75c) and the converse to Theorem 1 is proved. V . C O N C L U S I O N W e have pr esented a ne w secr ec y capacity region for the degraded K -rece i ver BC with priv ate messages in the presence of a w iretapper which gene ralizes previous work which d ealt with 2-rec ei ver BCs. In the direct proof we have used superp osition coding and W yn er’ s random code partition ing instead of binning to show the ac hie vable rate tuples. W e have provid ed e rror prob ability analysis a nd equivocation calculation f or the general k th receiver . In th e converse proof we have used a new definitio n for the au xiliary random variables which is different fro m either the 2 -recei ver BC with a wiretapp er , or the mor e recently studied K -receiver BC with common message and wiretapper, or th e K -re cei ver BC without wiretapper cases. R E F E R E N C E S [1] Y . Liang and H. Poor , “Multipl e-acce ss channels with confidentia l messages, ” IEEE T rans. Info . Theory , vol. 54, no. 3, pp. 976–1002, Mar . 2008. [2] L. Lai and H. E. Gamal, “The relay-ea vesdropper channel: Cooperation for secre cy , ” IEEE T rans. Info. Theory , vol. 54, no. 9, pp. 4005–4019, Sep. 2008. [3] M. Bloch, J. Barros, M. Rodrigue s, and S. McLaughlin, “W ireless infor mation-the oretic securi ty , ” IEEE T rans. Info . Theory , v ol. 54, no. 6, pp. 2 515–2534, Jun. 2008. [4] Y . Liang, H. Poor , and S. Shamai , “Secure communi cation ove r fadi ng channels, ” I EEE T rans. Info. Theory , vol. 54, no. 6, pp. 2470– 2492, Jun. 20 08. [5] R. Liu and H. Poor , “Secrec y ca pacity re gion of a multi-a ntenna Gaussian broadcast channe l with confidential messages, ” IEEE T rans. Info. The ory , submitte d for publ icati on, Sep. 2007. [6] I. Csisz ´ ar and J. K ¨ o rner , “Broadcast channe ls with confiden tial messages, ” IEEE T rans. Info. Theory , vol. 24, no. 3, pp. 339–3 48, Mar . 1978. [7] A. D. W yner , “The wire-t ap channe l, ” Bell Syst. T ech. J . , vol. 54, no. 8, pp. 13 55–1387, 1975. [8] R. Liu, I. Mari ´ c , P . Spasojevi´ c, and R. Y ates, “Discrete m e moryless interferen ce and broadcast channels with confidenti al messages: Secrec y rate re gions, ” IEEE T rans. Info. Theory , vol. 54, no. 6 , pp. 2493–2507, Jun. 2008. [9] J. Xu, Y . Cao, and B. Chen, “Capac ity bounds fo r broadcast channels with confiden tial messages, ” IEE E Tr ans. Info. Theory , submitted for publ icati on, May 2008. [10] G. Bag herikara m, A. S. Motahari, and A. K. Khandani, “Secr ecy rate reg ion of the broadcast channel, ” IEEE T rans. Info. Theory , submitt ed for publ icati on, Jul. 2008. [11] L. C. Choo and K. K. W ong, “Three-recei ver broa dcast channel with confidenti al message s, ” IEEE Info. Theory W orkshop, ITW 2009, V olos, Gr eece , submit ted for p ublicat ion, Dec. 2008. [12] E. Ekrem and S. Ulukus, “Secrec y capaci ty of a class of broadcast channels with an ea vesdropper , ” EUR ASIP J. W irel ess Commun. and Net. , submit ted for publication, Dec. 20 08. [13] S. Bora de, L. Zheng, and M. Trott , “Mu ltile vel broadcast networks, ” in Proc. Int. Sym. Info. Theory , Nice, France, 24–29, 2007 , pp. 1151–1155. 19 [14] P . Ber gmans, “Random codi ng theorem for broadcast cha nnels with degraded compone nts, ” IEE E T rans. In fo. Theory , vol. 19 , no. 2, pp. 197–207, Ma r . 1973. [15] T . Cov er and J. Thomas, Elements of Information Theo ry , 2nd e d., W iley , 2006.