Closed-Form Expressions for Secrecy Capacity over Correlated Rayleigh Fading Channels

1 Closed-F orm Expressions for Secrec y Capacity o v er Correlate d Rayleigh Fading Channels Xiaojun Sun, Chunming Zhao , Member IEEE, and Ming Jian g, Member IEEE, Abstract —W e in vestigate the se cure communications o ver cor - related wiretap Rayleigh fading channels assumin g the full chan- nel state informa tion (CSI) av ailable. Based on the information theoretic formulation, we deri ve closed-f orm expre ssions for t he a verag e secre cy capacity and the outage p robability . Simulation results confirm our analytical expressions. Index T erms —Infor mation-theoretic security , wiretap channel, secrecy capacity , corre lated Rayleigh fading. I . I N T R O D U C T I O N Because wireless communica tions are susceptible to ea ves- dropp ing, traditional security mechanisms mainly rely on cryp- tograph ic protoco ls. Recently , poten tial benefits of deriving secure information from physical lay er ha ve bee n reported in [1]. In [2 ], the infor mation-th eoretic secrecy capacity was introdu ced b y using the p hysical properties of chann els. The basic pr inciple of informa tion-theo retic security has been widely accepted as the strictest notion of security , which guaran tees that th e sent massages can not be decoded by a malicious eavesdropper [1]. W yn er introd uced wiretap channel model to ev alu ate secure transmissions at the physical layer [2], where Alice transmits confiden tial data to Bob and Eve eav esdrop s the data. Csiszar et. al. and Leung-Y a n-Cheong et. al. gen eralized it to br oadcast chan nels and basic Gau ssian channels, respecti vely in [3] and [ 4]. W ei Kang e t. al. stud- ied secure comm unication s over two-user semi-d eterministic broadc ast channels [5 ]. T he secre cy capacity is defined as th e difference between the main ch annel capacity (Alice to Bob) and the eaves drop ping’ s chan nel capacity (Alice to Eve) [4]. Barros et. al. and Gop ala et. al. generalized this Gaussian wiretap channel model to wireless quasi-static fading channels [6]-[8]. The secur e MIMO systems are studied in [9]-[10]. Motiv ated by emerging wireless a pplications, there is a grow- ing in terest in exploiting th e be nefits of relay an d co operative strategies in order to guar antee secure transmissions [11]-[13]. In this paper, we consider the secure communicatio ns within W y ner’ s corr elated wiretap chann el by b uilding on th e detailed technique in [7]. Similar w o rk studied in [14] gives the limiting value of the average secrecy capacity , which only converges into the secrecy cap acity at th e h igh signal-to-n oise ratio (SNR). W e der iv e the clo sed-form expressions of the a verag e secure communicatio n capacity and the o utage probab ility un- der the assum ption of the full chann el state inf ormation (CSI) av ailab le. Simulation results verify our analytical expr essions. Nationa l Mobile Communicat ions Research Laboratory , Southeast Uni versi ty , Na njing 210096, CHIN A. E-m ail: { sunxiaojun, cmzha o, jiang ming } @seu.edu.c n Fig. 1. Fading wir etap cha nnel model. I I . S Y S T E M M O D E L Fig. 1 sho ws a fadin g wiretap channel mod el considered in this paper . Here, the source Alice transmits confidential informa tion to the destination Bob . A third party (Eve) is able of eav esdrop ping on th e transmissions. The received signals at Bob and Eve are gi ven by y = h sd x + n d (1a) z = h se x + n e (1b) where h se and h sd are co mplex Gaussian random variables (R V) with zero -mean. N d and N e denote zero-m ean comp lex Gaussian noise R Vs with unit-variance. The instantan eous SNRs α = | h sd | 2 and β = | h se | 2 are expone ntially distrib ute d. The average value is λ 1 and λ 2 , respectively . Since α and β are correlated variables, the joint PDF is expressed as [14][1 5] f ( α, β ) = I 0  2 1 − ρ q αβ ρ λ 1 λ 2  (1 − ρ ) λ 1 λ 2 exp  − α/λ 1 + β /λ 2 1 − ρ  (2) where ρ is the correlation between h se and h sd . I 0 ( · ) is the zeroth-o rder modified Bessel fun ction of the first kind. Using the infinite-series repr esentation of I 0 ( x ) [ 16] I 0 ( x ) = 1 + ∞ X k =1 x 2 k 4 k ( k !) 2 W e can re write the joint PDF as f ( α, β ) = ∞ X k =0 c k exp  − α/λ 1 + β /λ 2 1 − ρ  λ 1 λ 2  α λ 1  k  β λ 2  k = ∞ X k =0 c k 1 λ 1 λ 2 exp  − α/λ 1 + β /λ 2 1 − ρ   α λ 1  k  β λ 2  k = ∞ X k =0 c k f k ( α, β ) (3) 2 where c k = ρ k ( k !) 2 (1 − ρ ) 2 k +1 . In this study , we assume that Alice has access to CSI on both the main channel and th e eavesdropper’ s ch annel. For instance, Eve may b e not a covert eavesdropper, but simply another user [6][7 ]. Alice wants to transmit some con fidential data to Bob which Alice d oes not wish Eve kn ow . So, Alice can estimate the CSI of the eavesdropper’ s channel [6][7]. I I I . S E C U R E C O M M U N I C AT I O N S OV E R C O R R E L A T E D R AY L E I G H FA D I N G C H A N N E L Start with the technique detailed in [7], which introduces the secrecy capacity over fading chann el. A similar introd uction was presented in [8]. Recalling the results of [7] fo r the Rayleigh fading wiretap channel, the secrecy capacity fo r one realization can be wr itten as C s ( α, β ) = ( ln ( 1 + α ) − ln (1 + β ) , if α > β 0 , if α 6 β (4) where ln (1 + α ) is the rate of the m ain ch annel, and ln (1 + β ) den otes the rate of th e eav esdropp er’ s channel. A. avera ge secr ecy cap acity The average secrecy cap acity over co rrelated chann els is derived as fo llows. Theor em 1: The a verage secrecy c apacity is av eraged over all chann el realizations C s = Z ∞ 0 Z ∞ 0 C s ( α, β ) f ( α, β ) dαdβ = ∞ X k =0 c k k ! (1 − ρ ) k +1 F  λ 1 , k , 1 1 − ρ  − k ! k X m =0 ( λ 1 /λ 2 ) m m ! (1 − ρ ) k +1 − m F  λ 1 , k + m, 1 + λ 1 /λ 2 1 − ρ  − k ! k X m =0 ( λ 2 /λ 1 ) m m ! (1 − ρ ) k +1 − m F  λ 2 , k + m, 1 + λ 2 /λ 1 1 − ρ  (5) where fu nction F ( λ, k , µ ) can be recur si vely e valuated or computed by using popular symbolic software like MA TLAB. After integration by parts, we get F ( λ, k , µ ) = Z ∞ 0 ln ( 1 + λx ) exp ( − µx ) x k dx = λ µ F k + k µ F ( λ, k − 1 , µ ) (6) where F k is defin ed by [16, 3.3 53.5] F k = Z ∞ 0 x k 1 + λx e − µx dx = 1 λ "  − 1 λ  k exp  µ λ  E 1  µ λ  + k X m =1 Γ ( m ) ( − λ ) m − k µ m # . Function F ( λ, 0 , µ ) = 1 µ E 1  µ λ  exp  µ λ  . E 1 ( x ) = R ∞ 1 e − xt t dt is the expon ential-integral functio n [16]. Γ ( · ) is the gam ma function [16]. Pr oof: Th e integral in (5) is re-expr essed as C s = ∞ X k =0 c k Z ∞ 0 Z α 0 ln (1 + α ) f k ( α, β ) dαdβ − Z ∞ 0 Z ∞ β ln ( 1 + β ) f k ( α, β ) dαdβ = ∞ X k =0 c k Z ∞ 0 Z λ 1 u/λ 2 0 ln (1 + λ 1 u ) f k ( u, v ) du dv − Z ∞ 0 Z ∞ λ 2 v/λ 1 ln ( 1 + λ 2 v ) f k ( u, v ) du dv = ∞ X k =0 c k  R 1 k − R 2 k  The integral R 1 k can be evaluated b y[16, 3.38 1.1] R 1 k = Z ∞ 0 ln (1 + λ 1 u ) exp  − u 1 − ρ  u k du × Z λ 1 u/λ 2 0 exp  − v 1 − ρ  v k dv = Z ∞ 0 (1 − ρ ) k +1 ln (1 + λ 1 u ) exp  − u 1 − ρ  u k × γ  k + 1 , λ 1 u λ 2 (1 − ρ )  du (7) where γ ( · , · ) is th e inco mplete gamma functio n d efined by [16] γ ( n + 1 , x ) = n ! − n ! e − x n X m =0 x m m ! Further with some ma nipulation s, R 1 k can be rewritten as R 1 k = Z ∞ 0 k ! (1 − ρ ) k +1 ln (1 + λ 1 u ) exp  − u 1 − ρ  u k du − k ! k X m =0 ( λ 1 /λ 2 ) m m ! Z ∞ 0 (1 − ρ ) k +1 − m ln (1 + λ 1 u ) × exp  − 1 + λ 1 /λ 2 1 − ρ u  u k + m du (8) Using (6), we can e valuate the integral R 1 k = k ! (1 − ρ ) k +1 F  λ 1 , k , 1 1 − ρ  − k ! k X m =0 ( λ 1 /λ 2 ) m m ! (1 − ρ ) k +1 − m F  λ 1 , k + m, λ 1 /λ 2 1 − ρ  (9) Similarly , we e valuate R 2 k by using the complem entary incom- plete gamma fu nction Γ ( n + 1 , x ) = n ! e − x n P m =0 x m m ! [16]. It yields R 2 k = k ! k X m =0 ( λ 2 /λ 1 ) m m ! (1 − ρ ) k +1 − m F  λ 2 , k + m, 1 + λ 2 /λ 1 1 − ρ  (10) 3 B. outage pr o bability of secr ecy cap acity The secrecy capa city can also be char acterized in ter ms of the outage pro bability for a target secrecy rate. The o utage probab ility can be calculated accor ding to P out ( R ) = 1 − P r ( C s ( α, β ) > R ) = 1 − P r  α > e R (1 + β ) − 1  Theor em 2: Similarly in voking again the in finite-series rep - resentation of I 0 ( x ) , the ou tage proba bility is equiv a lent to P out ( R ) = 1 − exp  − y 1 − ρ  ∞ X k =0 c k k ! k X m =0 1 m !  µ 1 − ρ  m × m X n =0 n m !  y µ  m − n  1 − ρ 1 + µ  k + n +1 Γ ( k + n + 1) (11) where y =  e R − 1  /λ 1 and µ = e R λ 2 /λ 1 . At this point in this study , the closed-fo rm expressions of secrecy capacity have been d erived f or a c orrelated Rayleig h fading channel. In all cases of practical significa nce, the infi- nite series representation s c an be truncated without sacrificing numerical accuracy . W e also note that th e results in [7] are a special case o f our result by assum ing indepen dent channels. I V . S I M U L A T I O N R E S U LT S In Fig. 2, we plot the secr ecy capacity versus SNR over correlated fading chann els in the case of the scenar io that the SNR of ma in channel and eavesdropper’ s channel are equal. It is c learly shown that th e co rrelation b etween the m ain and the eavesdropper’ s chann el reduce the secrecy capacity . For co mparison , th e limiting value given in [14 ] also is depicted in Fig 2 . The secrecy capacity converges into the limit of the secrecy capacity [14] at h igh SNRs. However , the limiting value is far away form the secrecy capacity at low and mod erate SNRs. V . C O N C L U S I O N In this paper, we in vestigate the secure co mmunica tion over co rrelated Rayleigh fadin g W yner’ s wiretap channel. The closed form expressions for av erage secrecy cap acity an d outage probability are derived assuming th e full chann el state informa tion available. A C K N O W L E D G M E N T This work is supported by Natio nal Science Found ation of China under Gran t 6 08020 07 and supported by the Research Fund of National Mob ile Communications Research Labora- torySouth east Uni versity 200 9A10. R E F E R E N C E S [1] U. Maure r , “Sec ret k ey agreement by public disc ussion from common informati on, ” IEEE T rans. Info.Theory . , vol.39 , pp. 733-7 42, May . 1993. [2] A. D.W yner , “The wire-t ap channel , ” Bell Syst. T ec h. J . , vol .54, no.5, pp.1355-1367, Oct . 1975. [3] I. Csisz ´ ar and J. K ¨ orn er , “Broa dcast channels with confide ntial mes- sages, ” IEEE T rans. Inf o. Theory . , vol .24, no.3, pp. 339-348, May 1978. [4] S. K. Leung-Y an-Ch eong and M. E. Hellman, “The Gaussian wire-tap channe l, ” IEEE T rans. Inf . Theory . , v ol.24, no.4, pp. 451-456, Jul 1978. 5 10 15 20 25 30 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 SNR(dB) Secrecy Capacity (in nats) ρ =0 ρ =0.1 ρ =0.5 ρ =0.9 Fig. 2. 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