Delay Analysis of Physical Layer Key Generation in Multi-user Dynamic Wireless Networks

Transcription

1 1 Deay Anaysis of Physica Layer Key Generation in Muti-user Dynamic Wireess Networks Rong Jin, Xianru Du, Kai Zeng,Laiyuan Xiao, Jing Xu Department of Computer and Information Science, University of Michigan - Dearborn, MI Schoo of Software Engineering, Huazhong University of Science and Technoogy Department of Eectronic and Information Engineering, Huazhong University of Science and Technoogy Emai: { jinrong, duxianru, kzeng xiao..y@263.net xujing@mai.hust.edu.cn Abstract Secret key generation by extracting the shared randomness in wireess fading channe is a promising way to ensure wireess communication security. Previous works ony consider key generation in static networks, but rea-word key estabishments are usuay dynamic. In this work, for the first time we investigate the pairwise key generation in dynamic wireess networks with a center node (eg. access point(ap)) and random arriva users. We estabish the key generation mode for this kind of networks. We propose a method based on discrete Markov chain to cacuate the average time a user wi spend on waiting and competing the key generation (average key generation deay, AKGD). Our method can tacke both seria and parae key generation scheduing under various conditions. We conduct extensive simuations to show the effectiveness of our mode and method. The anaytica and simuation resuts match to each other. I. INTRODUCTION Estabishing pairwise secret key between two communication parties is crucia to secure wireess communication. Physica ayer key generation mechanisms that expoit reciproca and ocation-specific properties of wireess fading channes, have been proposed [1, 2, 3]. Based on the reciprocity, the bidirectiona channe states shoud be identica between two transceivers at a given instant of time. In a mutipath or mobie environment, the channe states randomy fuctuate due to fading. Therefore, two egitimate parties can take advantage of this natura correated random process to generate a shared key. Furthermore, the channe state observed at an eavesdropper is uncorreated with the egitimate channe if the eavesdropper is more than haf a waveength away from egitimate parties. I The existing research on physica ayer key generation mainy focuses on key generation rate (KGR) in static wireess networks. Most of the works discussed the KGR between two parties, incuding retica KGR anaysis [1, 2, 4] and KGR optimization under practica conditions [5, 6, 7]. Some works discussed mutipe-user case [8, 9], and reticay derived KGR and secrecy capacities for mutipe terminas in static wireess networks. To the best of our knowedge, physica This work is partiay supported by the US Nationa Science Foundation (NSF) CAREER award under Grant No. CNS , Nationa Natura Science Foundation of China ( ), and the Fundamenta Research Funds for the Centra Universities, HUST (2013QN142). Fig. 1. Key generations in a dynamic wireess network ayer key generation in dynamic wireess networks has not been we studied. However, it is a common case in practice. In this paper, for the first time, we consider the key generation probem in dynamic wireess networks. Moreover, we focus on a more reaistic metric: the users key generation deay. Since key generation is a pre-requisite for secure wireess communications, the deays that users suffer are their most concerned issue. Deay anaysis of physica ayer key generation in dynamic networks is different from traditiona deay anaysis for wireess communication [10, 11] in the foowing two aspects. (1) Due to the specific mechanism of key generation (detaied in section II.C), tota service rate (or KGR) depends on the number of users and channe probing scheduing; whie in wireess communication, tota service rates are the same, regardess of scheduing methods. For exampe, using time division mutipexing (TDM) scheme reduces individua user s data transfer rate but wi not affect tota data transfer rate. (2) Key generation has an additiona process: reconciiation, to ensure the keys generated on both sides are identica. In this work, we consider the probem of pairwise key generation in dynamic wireess networks with a center node. We use the the scenario of access point (AP) and egitimate users shown in Fig. 1 to exempify our mode and method. The arriva of users is unknown in prior. We aim at cacuating the average time a user wi spend on waiting and competing the key generation (average key generation deay, AKGD). We propose the key generation mode. Our mode can tacke both seria and parae key generation scheduing (section II). We notice that in key generation mode, each key estabishment period has predictabe time. We make use of such characteristic to deveop a method based on discrete Markov chain for cacuating users AKGD (section III). We conduct simuations to show the effectiveness of our methods (section IV). Finay, we concude this paper (section V).

2 2 Information reconciiation is a discussion process used to correct measurement error between AP and users so as to make identica keys. Privacy ampification is used to discard partia reveaed information due to measurement correation and eavesdropper. Fig. 2. Exampes of channe probing and parae probing Our main contributions are: We formuate the physica ayer key generation probem in dynamic wireess networks and anayse the deay. We propose the key generation mode and deveop a method to cacuate AKGD. We conduct extensive simuation to verify and evauate our mode. II. KEY GENERATION MODEL IN MULTI-USER DYNAMIC A. System Setting WIRELESS NETWORKS We consider a one-hop network with an AP generating pairwise keys for egitimate users (shown in Fig. 1). Users arriva is a stochastic process determined by a Poisson process with the rate of λ. The key generation order foows the rue of first-come, first-serve. Users are numbered according to their arriving order u 1, u 2,. Their key ength requirements are L 1, L 2, taking vaues in the finite discrete set L containing a possibe key ength requirements. For simpicity, we assume users have simiar channe variations, and mainy focus on the dynamic of users arriva. We denote N as the number of users in the system, incuding waiting users and the users currenty conducting key generations (we ca them running users) [12]. Each user arriva increases N by 1; each key generation competion reduces N by 1. We aso assume a passive attacker trying to compromise the shared key by eavesdropping on the communication among AP and users. B. Key Generation Process The key generation process usuay contains four steps: channe probing, quantization, information reconciiation, and privacy ampification [13, 1]. Channe probing is used to coect channe characteristics between AP and egitimate users shown in Fig. 2. The channe characteristics can be RSS (received signa strength), channe impuse response, or phase, etc. In this step, i th user and AP exchange bidirectiona request/repy probing frames to probe the channe characteristics for a duration, say δ seconds. Denote t λ as the interva between any two consecutive request probing frames. The channe probing rate v is thus 1/t λ. At the end of channe probing process, we assume the i th user and AP make N i pairs of channe measurements ĥu ia = [ĥu ia(1)ĥu ia(2) ĥu ia(n i )] and ĥau i within τ seconds. Quantization is used to quantize the measured channe characteristics into bits. C. Key Generation Scheduing In the key generation, quantization and privacy ampification can be done ocay at respective sides. Whie channe probing and information reconciiation need communication between the user and AP. For mutipe users, scheduing is required to avoid interference. In our work, we consider the foowing two scheduing methods. (1) Seria. AP probes the channe with users one by one according to users arriving order. However, this scheduing is not efficient. For seria probing, consecutive channe probing measurements have sma time separation and thus highy correated. The correated part provides itte extra information and shoud be argey discarded in the fina key in privacy ampification [4, 5, 6, 3]. (2) A more efficient optiona key generation scheduing is Parae : mutipe users share the channe using timedivision method. An exampe with three users is shown in Fig. 2, AP probes the channe with mutipe users in a round-robin way. The advantage of parae probing is exempified in Fig. 2: by inserting the channe probing between AP and the 2nd and 3rd users into the channe probing between AP and the 1st user. The time separation between consecutive channe probing measurements increases, thus correation of the measurements reduces. From the point of view of the system, fewer correated bits are discarded and higher tota KGR can be achieved. Our key generation scheduing mode deveoped for parae probing is shown in Fig. 3. We consider the foowing three conditions : a) Light oad: we define a defaut probing rate v d as the maximum probing rate. AP parae probes the channe with users with v d when few users are in the system. Setting v d is to avoid utra ow efficiency. b) Medium oad: AP parae probes the channe with users unti the number of parae probing reaches its maximum N p. c) Heavy oad: when a arge number of users come to the system N > N p, AP parae probes the channe with first N p users; remaining users wait at the service ine. After a period of probing process is finished, AP appies reconciiation for a of the channe measurements with the rate of v recon s/bit. We denote the period from the beginning of channe probing to the end of reconciiation as a key estabishment period. Note that seria probing can be simpy treated as a specia case of parae probing when N p = 1. Our mode tackes both seria probing and parae probing cases. D. User s Key Generation Deay For an individua user, his key generation deay D a is the sum of waiting deay D wait and service deay D service (D a = D wait + D service ). According to the key generation process, D service is divided into four parts: channe probing deay D c, quantization deay D q, information reconciiation

3 3 Fig. 3. u1 u1 ight oad u2 u1 u2 u1 u2 Reconciiation defaut probing interva u2 u3 u4 u1 u2 u3 medium oad u4 heavy oad u1 u2 u3 un0 u1 u2 u3 un0 t Reconciiation t Reconciiation Typica key generation scheduing in a key estabishment period deay D i, and privacy ampification deay D p. D service = D c +D q +D i +D p. D q and D p are very short compared with D c and D i. Moreover, D q and D p are produced at respective oca sides outside of our queueing system. For convenience, we define D service = D c + D i For a constant probing rate v, the channe probing deay D c can be written as D c = /r(v) (1) where is the ength of key generated in bit, and r(v) is the KGR. KGR can be cacuated according to [6]. The maximum bits generated between AP and the i th user can be represented by the conditiona mutua information given an eavesdropper s observation, say Eve: K uia = I(ĥu ia; ĥau i ĥea) (2) where we assume the eavesdropper is near AP, ĥ ea is the channe measurement of user to AP channe by Eve. The other parameters affecting mutua information K uia besides probing rate are: Dopper spread f m (Hz) to represent channe variation in time, signa variance σs, 2 noise variance σn, 2 eavesdropper channe variance σe 2 (channes are assumed zero mean Gaussian distribution), the distance between AP and the eavesdropper. The computationa method for K uia can be found in [6] (In that paper, it uses reative veocity to waveength v/λ to express f m ). Due to space imit, we do not repeat it here. The upper-bound of KGR is ri max (v) = K uia/τ. In practice, due to the non-perfect reciprocity and different efficiency of quantization, reconciiation, and privacy ampification, KGR is ower than the upper-bound: we use η(v) < 1 to denote the ratio between practica KGR and the upper-bound of KGR. r i = ηr max i (3) For a given key generation protoco, η(v) is determined. Information reconciiation deay can be written as D i = raw v recon (4) where v recon is the reconciiation rate; raw is the amount of raw channe probing measurements within t seconds. Under a constant probing rate v, raw = vt III. DELAY ANALYSIS OF KEY GENERATION IN DYNAMIC WIRELESS NETWORKS Now we introduce our methodoogy in detai. We notice that each key estabishment period has predictabe time determined by the number of running users. We make use of such characteristic and focus on the discrete time points at the end of each key estabishment period; so that the key generations t Fig. 4. Fig. 5. Yn waiting ine k Xn - m Xn-m-k Xn Xn - k Xn +Yn - k Queueing mode for key generation in service m m tn Tn + reconciiation tn+1 Time ine K L0 (K-1) L0 (K-2) L0 2 L0 1 L0 Phase decomposition can be treated as a discrete-time Markov process. We then appy phase decomposition to buid state transition matrix. We cacuate the steady-state. We derive mean number of users and timespan between states to get AKGD. In our deduction, the foowing parameters are used: t n : the nth moment, n = 0, 1, 2, t 0 = 0. The time point is just after a key estabishment period. It is the moment when some users may compete their key generation and eave the queue. X n : the number of users in the system at t n. L 0 : during t n to t n+1, AP generates L 0 bit key for individua user. T n : the time of a key estabishment period from t n to t n+1, incuding the time of channe probing and reconciiation. Y n : the number of new coming users during T n. k : the number of key generation competion during T n. N p : the maximum number of parae probing. m: the number of running users at t n. When X n > N p, m = N p ; when X n N p, m = X n. M: the maximum number of users the system can hod. We use Fig. 4 to show the queueing system. From the figure, we can have { Y n X n = 0 X n+1 = (5) X n + Y n k X n > 0 X n forms a discrete-time Markov chain. We denote the transition probabiity p ij = P (X n+1 = j X n = i). A. Estabish One-step State Transition Matrix To get k, we make the foowing phase decomposition: Suppose we need K key estabishment periods to satisfy i th user s key ength requirement K = L i /L 0. We treat the shared key between i th user and AP as K phases. From each t n to t n+1, the i th user competes one phase (shown in Fig. 5). When a K phases are competed, the key generation of the i th user is competed. We use X n (c 1, c 2.., c Xn ) to express the state at t n ; where X n is the number of users in the system and c 1, c 2.., c Xn are their remaining phases arranged in user arriva order. We denote p Xn(c 1,c 2..,c Xn ),X n+1(c 1,c 2..,c Xn+1 ) as the transition probabiity from state X n (c 1, c 2.., c Xn ) to state X n+1 (c 1, c 2.., c Xn+1 ) during T n. We write out one-step state transition matrix (Fig. 6).

4 4 Fig. 6. P= state 0 1(1) 1(2) 2(1,1) 2(1,2) 2(2,1) 2(2,2) state State transition matrix 0 1(1) 1(2) 2(1,1) 2(1,2) 2(2,1) 2(2,2) p00 p0,1(1) p0,1(2) λ/ p1(1),0 p1(1),1(1) p1(1),1(2) p1(1),2(11) p1(1),2(12) p1(1),2(21) p1(1),2(22) 0 p1(2),1(1) 0 p1(2),2(11) p1(2),2(12) 0 0 p2(11),0 p2(11),1(1) p2(11),1(2) p2(11),2(11) p2(11),2(12) p2(11),2(21) p2(11),2(22) 0 P2(12),1(1) 0 p2(12),2(11) p2(12),2(12) P2(21),1(1) 0 p2(21),2(11) p2(21),2(12) p2(22),2(11) B. Probabiities of State Transition We first consider the condition that the state at t n is 0 (X n = 0). We assume if there is no user in the system, the next coming user woud get an instant start of key generation. We use a sufficient short period of time t to denote T n, so that ony two types of transitions are possibe: one come or no come. Thus, the transition probabiity at state 0 is: p 01 = P (Y n = 1 T n = t) = (λ t)e λ t (6) The transition probabiity that the coming user has the initia phase of q is p 0,1(q) = p 01 p(q) (7) In the given exampe, users key ength requirements take the vaue of 64 and 128 bits (corresponding to initia phase of 1 and 2) with the same probabiity. Therefore, p 0,1(1) = p 0,1(2) = p 01 /2. Next, we consider the condition of X n 1 at t n. In this case, T n is the sum of channe probing time and reconciiation time T n (X n ) = T P rob + T recon (8) where the probing time equas to T P rob = L 0 /r(v m ) (9) where r(v m ) is the KGR during T n. Its probing rate v m ony reates to the number of running users m: (m = X n, X n N p ; m = N p, X n > N p ). { v m = v d m 1/v d δ (10) v m = 1/mδ m > 1/v d δ r(v m ) can be cacuated by (3). T recon is the reconciiation time to estabish L 0 bit keys for a running users, which can be written as T recon = ml r nv recon (11) where v recon is the reconciiation rate; L r n = v m T P rob is the amount of raw channe probing measurements. Since users arriva foows Poisson distribution, the probabiity of j users coming during T n is P (Y n = j T n = t) = (λt)j e λt (12) j! We can write out the probabiities of state transition in matrix P: (1) no user coming during T n : p Xn(c 1,c 2..,c Xn ),X n+1(c 1,c 2..,c Xn+1 ) = P (Y n = 0 T n ) (13) (2) more than 1 user coming during T n : p Xn(c 1,c 2..,c Xn ),X n+1(c 1,c 2..,c Xn+1 ) = P (Y n = j T n )p(c Xn+1 j+1)p(c Xn+1 j+2)...p(c Xn+1 ) (14) where j = X n+1 X n + k is the number of coming users. In the given exampe, p 3(122),2(11) = P (Y n = 0 T n (3)); p 3(122),3(112) = P (Y n = 1 T n (3))(1/2); where underined part denotes the initia phase of new coming users during T n. C. Stationary Distribution The steady state π is a row vector, whose entries π Xn(c 1,c 2..,c Xn ) are non-negative and sum to 1, that satisfies the equation π = πp (15) Thus π is a normaized (meaning that the sum of its entries is 1) eft eigenvector of P associated with the eigenvaue 1. Then we derive mean number of users between states φ Xn(c 1,c 2..,c Xn ),X n+1(c 1,c 2..,c Xn+1 ). Since users arriva foows Poisson distribution, the time points of user arriva u t (if there is any) are uniform distributed between t n and t n+1. φ Xn(c 1,c 2..,c Xn ),X n+1(c 1,c 2..,c Xn+1 ) = Tn X n + Y n p(u t = t)dt = N + Y n /2; 0 The mean number of users at state X n (c 1, c 2.., c Xn ) is φ Xn(c 1,c 2..,c Xn ) = φ Xn(c 1,c 2..,c Xn ),X n+1(c 1,c 2..,c Xn+1 ) p Xn(c 1,c 2..,c Xn ),X n+1(c 1,c 2..,c Xn+1 ) D. Average Key Generation Deay (16) (17) The steady state probabiities of continuous-time Markov process can be computed by mutipying the steady state probabiities of the discrete-time version of Markov chain by the mean times spent in the various states [14]. This eads to p Xn(c 1,c 2..,c Xn ) = Cπ Xn(c 1,c 2..,c Xn )T n (N) (18) where T n (0) = T ; C is normaization constant making pxn(c 1,c 2..,c Xn ) = 1; The average number of users in the system is L s = p Xn(c 1,c 2..,c Xn )φ Xn(c 1,c 2..,c Xn ) (19) From Litte s aw, we can get the AKGD D a = L s /λ (20) IV. SIMULATIONS In the foowing, we conduct numerica simuations to investigate our mode and method. We set Dopper spread f m = 5Hz, the variance of signa, noise and eavesdropper are σ 2 s = σ 2 n = σ 2 e = 1. We assume an adversary is cose to AP with the distance of 1m. We aso assume bi-directiona probing duration δ = s; This makes the highest probing rate v = 1/δ = 500Hz. The users have different key ength requirements which takes the vaues of 64 and 128 bits with the same probabiity P (L(u k ) = 64) = P (L(u k ) = 128) = 1/2. We set L 0 = 64, reconciiation rate v recon = 64 1 s/bit, η = 0.4 and defaut probing rate v d = 250Hz.

5 5 Fig. 7. (eft) Users key generation deay distribution (10000 users) (right) Number of users distribution Fig. 9. Fig. 8. Comparison of seria and parae probing A. Methodoogy Verification We randomy generate users coming to the system (same for a other simuations) foowing Poisson distribution with the rate of λ = 6 1 (that is on average 6s a user). We set M = 14. We use the key generation rue and AP scheduing method introduced in section II to simuate the whoe key generation procedure. We assume the maximum number of parae probing Np = 4. We record each user s arriva time as we as key generation competion time, so as to cacuate each user s key generation deay. Fig. 7(eft) shows the user s key generation deay distribution with the interva of 1s. AKGD is sim Ds = 13.42s. We aso use our proposed method in section III to reticay get AKGD Ds = 13.46s. It can be seen that simuation and retica AKGD are cose. We then observe the whoe key generation procedure by counting the number of users in the system at every second (from the beginning of the first user arriva to the end of ast user key generation competion). Fig. 7(right) shows the number of the observations fas into the case of k users in the system (k from 0 to 12). We aso use equation (19) to reticay derive the number of users distribution. Simuation and retica resuts are in good agreement. B. Comparison between Seria and Parae We then compare seria probing with parae probing. We vary users arriva rate λ from 8 1 to 14 1 (that is from on average 8s a user to 14s a user). For parae probing, we set the maximum number of parae probing Np = 4. Fig. 8 shows AKGD curves of seria and parae probing. It can be seen that parae probing has smaer AKGD than seria probing in the condition of same users arriva rate. It conforms with the anaysis in section II.C. Next, we investigate various parameters impact on AKGD and compare simuation and retica resuts. C. Impact of Parae on Key Generation We set λ = 6 1 to see the impact of the number of parae probing on key generation. We vary the maximum number of Impact of (eft) parae probing (right) user arriva rate on AKGD parae probing Np from 2 to 9. Fig. 9(eft) shows simuation and retica AKGD. When Np increases from 2 to 5, AKGD decreases from 16.14s to 13.44s (when Np = 1 system diverges because users arriva rate is arger than tota KGR; converge region wi be deduced ater). When Np increases from 5 to 9, AKGD stops decreasing. This is because efficiency is aready high, increasing Np wi gain itte KGR increment. To get the difference between simuation and retica q resuts, we compute mean absoute deviation Ds = sim E(Ds q Ds )2 = 0.064s and reative deviation sim e(ds ) = E(Ds Ds )2 /(Ds )2 = 0.44%. D. Impact of Users Arriva Rate on Key Generation We then fix Np = 4 to see the impact of users arriva rate on key generation. We vary λ from 5 1 to 20 1 (that is from on average 5s a user to 20s a user). Fig. 9(right) shows AKGD with respect to λ. AKGD decreases from 15.86s to 8.84s as users arriva rate decreases from λ = 5 1 to λ = We compute the differences between simuation and retica resuts: Ds = 0.069s, e(ds ) = 0.60%. Note that when users arriva rate approaches to 0, AKGD approaches to its ower bound: whenever a new user comes, he aways face an empty system and get an instant start. Its corresponding AKGD can be written as min Ds = X P (L(uk ) = )[ L + ] r(vd ) vrecon (21) min In the exampe, the AKGD ower bound is Ds = 7.54s. Fig. 10 shows the average number of users under different users arriva rates. When λ = 5 1, the probabiities of 0, 1, 2, 3, 4 and 5 users in the system are 5%, 19%, 17%, 18%, 16%. When users arriva rate decreases, the number of users in the system decreases. When λ = 20 1, the probabiities of 0, 1, 2 and 3 users in the system become 60%, 33%, 5%, 1%; the probabiity that more than 4 users is ess than 1%. Suppose the system can hod infinite users M =. Note that the system is diverge if users arriva rate is too fast and the number of users in the system cumuate to infinite. According to section II.C, the system has the highest tota KGR when AP parae probing the channe with Np users. Users average arriva rate shoud be ower than the highest tota KGR. Therefore, we can cacuate converge region: λ 1 > X L P (L(uk ) = )[ ] max + v ra recon (22)

6 6 Fig. 10. Impact of users arriva rate on number of users in the system cacuated mean number of users and timespan between states to get AKGD. Our method can dea with both seria and parae probing scheduing. We conducted simuations to evauate the effectiveness of our mode and method. The simuated AKGD and number of users distribution consist with our retica deduction. Simuation resuts show that parae probing has shorter AKGD than seria probing due to its advantage on tota KGR. We aso compared simuation and retica AKGDs under various conditions. The resuts show that there are ony sight differences between them. Our mode is usefu in predicting AKGD under various conditions with different key generation scheduing. Extending our mode to muti-hop wireess networks with heterogeneous channe conditions wi be our future work. Fig. 11. Impact of (eft) channe variation (right) SNR on AKGD where the maximum overa KGR is r max a = N p K(vN p ), v Np = 1/N p δ. In the given exampe, converge region is λ < E. Impact of Dopper Spread on Key Generation Dopper spread f m is aso an important parameter in key generation. Larger f m impies more drastic channe variation and eads to arger KGR. Indoor environment usuay has arger Dopper spread than outdoor environment. We set λ = 6 1, N p = 4 to evauate the impact of channe variation on key generation. We vary Dopper spread f m from 3 to 10Hz. Fig. 11(eft) shows f m with AKGD. AKGD decreases as f m increases due to the increase of KGR. We compute the differences between simuation and retica resuts: D s = 0.108s, e(d s ) = 0.86% F. Impact of SNR on Key Generation Finay, we investigate the impact of signa to noise ratio on key generation. We set λ = 6 1, N p = 4, f m = 5 and vary SNR (σ 2 s/σ 2 n) from 0dB to 30dB. Fig. 11(right) shows SNR with AKGD. AKGD decreases as SNR increases due to the improvement of channe quaity. We compute the differences between simuation and retica resuts: D s = 0.029s, e(d s ) = 0.63% V. CONCLUSION In this paper, we considered the pairwise key generation in dynamic wireess networks with a center node (exempified by the scenario of AP with random arriva users). We aim at deriving users average key generation deay. We estabished the key generation mode. We observed that in key generation mode, each key estabishment period has predictabe time. We made use of such characteristic to treat the key generations as a discrete Markov process. We constructed state transition matrix. We deduced the probabiity of state transition and compute stationary distribution. We REFERENCES [1] Christian Cachin and UeiM. Maurer. Linking information reconciiation and privacy ampification. Journa of Cryptoogy, 10:97 110, [2] Gies Brassard and Louis Savai. Secret-key reconciiation by pubic discussion. In Workshop on the ry and appication of cryptographic techniques on Advances in cryptoogy, EUROCRYPT 93, pages , Secaucus, NJ, USA, Springer-Verag New York, Inc. [3] A. Khisti, S.N. Diggavi, and G.W. Worne. Secret-key generation using correated sources and channes. Information Theory, IEEE Transactions on, 58(2): , feb [4] U.M. Maurer. Secret key agreement by pubic discussion from common information. Information Theory, IEEE Transactions on, 39(3): , may [5] Kai Zeng, D. Wu, An Chan, and P. Mohapatra. Expoiting mutipe-antenna diversity for shared secret key generation in wireess networks. In INFOCOM, 2010 Proceedings IEEE, pages 1 9, march [6] Y. Wei, K. Zeng, and P. Mohapatra. Adaptive wireess channe probing for shared key generation based on pid controer. Mobie Computing, IEEE Transactions on, PP(99):1, [7] Chunxuan Ye, S. Mathur, A. Reznik, Y. Shah, W. Trappe, and N.B. Mandayam. Information-reticay secret key generation for fading wireess channes. Information Forensics and Security, IEEE Transactions on, 5(2): , june [8] I. Csiszar and P. Narayan. Secrecy capacities for mutipe terminas. Information Theory, IEEE Transactions on, 50(12): , dec [9] Lifeng Lai and Siu-Wai Ho. Simutaneousy generating mutipe keys and muti-commodity fow in networks. In Proc. IEEE Inform. Theory Workshop, (Lausanne, Switzerand), Sept [10] Chee-Hock Ng and Soong Boon-Hee. Queueing Modeing Fundamentas: With Appications in Communication Networks. Wiey Pubishing, 2 edition, [11] M.J. Neey. Deay anaysis for maxima scheduing in wireess networks with bursty traffic. In INFOCOM The 27th Conference on Computer Communications. IEEE, [12] V Sundarapandian. Probabiity, statistics and queuing ry. PHI Learning, New Dehi, [13] Suman Jana, Sriram Nandha Premnath, and Mike etc. Cark. On the effectiveness of secret key extraction from wireess signa strength in rea environments. In MobiCom 09. [14] W.C. Krumbein and MichaeF. Dacey. Markov chains and embedded markov chains in geoogy. Journa of the Internationa Association for Mathematica Geoogy, 1(1):79 96, 1969.