Localization-based secret key agreement for wireless network

Transcription

1 The University of Toleo The University of Toleo Digital Repository Theses an Dissertations 2015 Localization-base secret key agreement for wireless network Qiang Wu University of Toleo Follow this an aitional works at: Recommene Citation Wu, Qiang, "Localization-base secret key agreement for wireless network" (2015). Theses an Dissertations This Thesis is brought to you for free an open access by The University of Toleo Digital Repository. It has been accepte for inclusion in Theses an Dissertations by an authorize aministrator of The University of Toleo Digital Repository. For more information, please see the repository's About page.

2 A Thesis entitle Localization-base Secret Key Agreement for Wireless Network by Qiang Wu Submitte to the Grauate Faculty as partial fulfillment of the requirements for the Master of Science Degree in Electrical Engineering Dr. Junghwan Kim, Committee Chair Dr. Richar G. Molyet, Committee Member Dr. Ezzatollah Salari, Committee Member Dr. Patricia R. Komuniecki, Dean College of Grauate Stuies The University of Toleo May 2015

3 Copyright 2015, Qiang Wu This ocument is copyrighte material. Uner copyright law, no parts of this ocument may be reprouce without the expresse permission of the author.

4 An Abstract of Localization-base Secret Key Agreement for Wireless Network by Qiang Wu Submitte to the Grauate Faculty as partial fulfillment of the requirements for the Master of Science Degree in Electrical Engineering The University of Toleo May 2015 Due to the share nature of wireless meium, generating secret key between legitimate noes uner the presence of eavesroppers remains challenging in wireless network environment. In this research, a framework of secret key agreement utilizing observations of noes relative location is consiere. While many current works concern secret key generation only for 3-noes (2 legitimate noes an 1 eavesropper), this research proposes an approach an analysis for a wireless network environment with m (m 3) legitimate noes an 1 eavesropper. The propose algorithm uses the istance between a ranomly selecte noe 1 an noe 2 as the Reference Distance (RD) to establish the secret key. In orer to secure the elivery of RD to other m-2 noes, an Aitive Distance Value (ADV) is use for public iscussion. Further, ifferent types of topologies are evelope to accomplish secret key agreement for m noe wireless network incluing star, chain an hybri topologies. After the ADV istribution an RD estimation, the secret key is generate through secret bit extraction. The Maximum achievable Secret key generation Rate (MSR) base on the above network topologies is stuie through theoretical analysis an mathematical estimation. Base on these analyze, the feasibility of propose secret key iii

5 generation algorithm is valiate with a comparable secret key generation rate. The relationship between secret key, wireless network size an signal-noise ratio has been ientifie. Moreover, a comparison between star topology, chain topology an hybri topology has been iscusse. After that, several wireless network moels incluing ranom patterne moving wireless noes are stuie to show the feasibility an performance of our propose secret key generation algorithm. A chain topology improving metho is propose base on the wireless network moel simulation. Last but not the lease, in orer to reuce the bit mismatch rate, an optional secret key agreement proceure is also propose. In all, this research stuies the secret key agreement base on localization information in an m noe wireless network uner the presence of eavesroppers. iv

6 Acknowlegements First of all I woul like to give my highest gratitue to my granma, my mom an my a. It s their altruistic love an encouragement which mae me go this far. I am like a ki sitting on their shoulers all the time an I hope one ay I woul be able to protect my family with my knowlege an work. Also, I want to give my greatest appreciation to my avisor Doctor Junghwan Kim. This thesis woul never been one without the help of Doctor Kim. The patience an kinness of him have assiste me to get through rough times. His iligent attitue an extensive knowlege of communication research have stimulate my esire of stuying. v

7 Table of Contents Abstract... iii Acknowlegements...v Table of Contents... vi List of Tables..... viii List of Figures... ix List of Abbreviations... xi List of Symbols... xii 1 Introuction Wireless Network an Secret Key Generation Localization Base Secret Key an Motivation of the Thesis Summary an Structure of Thesis Relate Work Mathematical Moel of Wireless Network Parameters of Wireless Network Moel Attacker Moel Basic Mathematical Moel of Wireless Network Calculation Rules for Entropy an Mutual Information Gaussian an Multivariate Gaussian Distribution...15 vi

8 4 Secret Key Generation Algorithm Basic Protocols between Noes Secret Key Generation Algorithm Theoretical Analysis on Secret Key Generation Rate Star Topology-base Maximum Secret Key Generation Rate Chain Topology-base Maximum Secret Key Generation Rate Hybri Topology-base Maximum Secret Key Generation Rate Mathematical Analysis for Maximum Secret Key Generation Rate Star Topology-base Maximum Secret Key Generation Rate Chain Topology-base Maximum Secret Key Generation Rate Hybri Topology-base Maximum Secret Key Generation Rate Discussions on the Star, Chain, Hybri Topologies Simulation of Wireless Network Moels Ranom Patterne Wireless Network Moel Simulation Ranom Patterne Wireless Network Moel Discussions on chain topology improvement Secret Key Agreement Algorithm Conclusions References...86 Appenix A...90 vii

9 List of Tables 6.1 Performance comparison between star, chain an hybri topology Ranom patterne performance comparison between star, chain an hybri Chain topology improving metho performance simulation...79 viii

10 List of Figures 4-1 Illustration of the 3-noe network of basic protocol Illustration for an m noe wireless network with star topology Illustration for an m noe wireless network with chain topology Illustration for wireless network with hybri topology Brief on star topology Schematic iagram of chain topology Scenario of hybri topology Star topology base MSR vs. SNR Star topology base MSR vs. wireless network size Chain topology base MSR vs. SNR Chain topology base MSR vs. wireless network size Hybri topology base MSR vs. SNR Hybri topology base MSR vs. wireless network size M Hybri topology base MSR vs. smallest star size ma Star topology vs. Chain Topology vs. Hybri Topology for SNR Star topology vs. Chain Topology vs. Hybri Topology for network size Star topology wireless network moel with an eavesropper Secret Bit Extraction Algorithm Generate secret key for noes an eavesropper when SNR = ix

11 6-13 Generate secret key for noes an eavesropper when SNR = 9.35B Chain topology wireless network moel Generate secret key for noes an eavesropper when SNR = 20.46B Generate secret key for noes an eavesropper when SNR = 7.45B Larger scale wireless network moel Star topology for the larger scale wireless network moel Chain topology for the larger scale wireless network moel Hybri topology for the larger scale wireless network moel Ranom patterne wireless network moel Ranom patterne wireless network moel with star topology Ranom patterne wireless network moel with chain topology Ranom patterne wireless network moel with chain topology Improve chain topology performance with hybri topology Wireless network with chain topology an improve hybri topology Illustration of secret key agreement for star topology Illustration of secret key agreement for chain topology Improve chain topology performance with hybri topology...85 x

12 List of Abbreviations ADV...Aitive Distance Value AOA...Angle Of Arrival AWGN...Aitive White Gaussian Noise BMR...Bit Mismatch Rate CDF...Cumulative Distribution Function ESPAR...Electronically Steerable Parasitic Array Raiator i.i....inepenent an ientically istribute Liar...Light etection an ranging LPS...Local Positioning System MSR...Maximum achievable Secret key generation Rate NLOS...Non-Line-Of Sight RD...Reference Distance SS...Signal Strength SNR...Signal-to-Noise Ratio TOA...Time of arrival UAV...Unmanne Aerial Vehicle xi

13 List of Symbols...istance between noe i an noe j i, j E...public iscussion pub h, H...entropy calculation i, j... i th/ j th wireless noe I...mutual information calculation k... k th time slot l...location for noe i i m...number of total wireless noes M...number of total wireless noes specifically in hybri topology mi...number of noes in the i th star in the hybri topology ma...number of noes in the smallest star in the hybri topology r...length of generate secret key bits for a single time slot R...secret key generation rate S i ()...generate secret key v...size of strings in the secret key agreement algorithm 2 2 β...signal to noise ratio, / w βb...signal to noise ratio in B, B 10log...aitive istance value ADV ε...any small positive number μ...mean value base on Gaussian assumption...variance of localization information base on Gaussian assumption 2...variance of noise base on Gaussian assumption 2 w xii

14 Chapter 1 Introuction In this chapter, the wireless network popularity an its challenges are initially iscusse. Later, the avantages of secret key generation using localization information is escribe as well as the propose algorithm an topology. To that en, a summary of the thesis is given. 1.1 Wireless Network an Secret Key Generation Wireless network has been wiely use throughout the worl nowaays. Laptops, tablets, smartphones an other forms of wireless evices have become an important part of our moern life, not to mention the comprehensive use of wireless network in meical, military an many other areas in the society. However, with the explosive growth of wireless communication network, security has become a critical issue because of the open nature of wireless meium an mobility of the wireless noes [5]. For example, a group of stuents want to share their project results with laptops or tablets among themselves, a group of tourists want to share their photos with smartphones to each other or a group of soliers 1

15 want to report the battle situation to the team. In such ynamic environments, the wireless parties nee to form their connection on-the-fly. Secret key generation has rawn more attention than the traitional cryptography methos for secure wireless communication in above scenarios [5-8]. In secret key generation, legitimate noes coul agree on a synchronize secret key, while eavesroppers can only overhear limite information through the wireless channel. In here, eavesroppers are malicious wireless noes which overhear the wireless channel communication an try to ecipher the secret key. The secret key is generate base on a ranom sequence extracte from certain characteristic of the legitimate noes, such as noes relative istance, wireless channel reciprocity an so on, therefore the legitimate noes woul have privacy privilege over the eavesroppers. In orer to achieve a higher secret key generation rate, the entropy of the extracte ranom sequence shoul be maximize, while the amount of information transmitte on the public channel shoul be minimize [22]. Therefore the maximum secret key generation rate (MSR) is frequently use to estimate the performance of secret key generation algorithm [6-12]. Security in wireless network has several challenges [5]: (1) Wireless nature of communication. The open nature of the wireless meium makes the secret key establishment process easy to be eavesroppe by the opponent. As a result, the secret key generation rate shoul be analyze uner the presence of eavesropping-aversaries. (2) Resource limitation on sensor noes (processor spee, memory storage an power supply, etc). (3) Lack of fixe infrastructure (ue to the highly ynamic mobile wireless environment). Many traitional cryptography methos such as authentication an key 2

16 exchange base on public-key cryptography [1-4] may not be feasible in many situation because of the limite resource on wireless noes an the lack of fixe key management infrastructure. (4) Unknown network topology prior to eployment. 1.2 Localization Base Secret Key an Motivation of the Thesis Recently, there have been overwhelming stuy that focus on secret key generation base on the wireless channel reciprocity (such as impulse response, signal envelopes, signal phases an receive signal strength) [9-14]. Unlike these research, the possibility of utilizing relative wireless noe localization to generate secret key is iscusse in this thesis. The avantage of using the relative noe istance is ue to the variety of technologies that can be use for localization, such as infrare, raio, ultrasoun, Liar (Light Detection An Ranging), Raar, an so on. This versatility of the localization technology makes the key generation system more capable in many circumstances an more powerful than just using wireless raio channel reciprocity. For example, Liar or narrow beam-with infrare system can enhance the ifficulty for eavesropping from ifferent angles. Furthermore, in many applications the localization information is reay to use which makes the propose secret key generation algorithm easy to hook up with the existing wireless network system. Plus, regarless of wireless noe the localization measurement initiates, the istance measure between two wireless noes is always ientical in certain time interval, even when ifferent frequency bans are use or it is non-line-of sight (NLOS) for the wireless noe pairs. 3

17 There are several motivations for this thesis. i) Most of the present secret key generation algorithm an its analysis are only base on a simple 3-noe wireless network moel (Noe A, Noe B an Eavesropper E). In orer to consier the secret key generation for a larger scale wireless network, a wireless network moel of m (m 3) legitimate noes an 1 eavesropper is consiere in this thesis. Different types of network topologies incluing star topology, chain topology an hybri topology are propose to moel the wireless communication in such a large scale wireless network. Furthermore, the propose secret key generation algorithm base on these topologies are analyze in etail. ii) The possibility of secret key generation using localization is stuie instea of the overwhelming research base on wireless channel reciprocity. Its avantage has been mentione. iii) Although there are some current research working on secret key generation topologies [12], an improvement has been mae by using propose secret key generation algorithms. In the propose algorithm, the noise accumulation along the chain is reuce, which results higher MSR. Further, the theoretical an mathematical analysis on the hybri topology is also stuie in this research. iv) In orer to reuce the bit mismatch rate, an optional secret key agreement algorithm is propose base on ifferent types of topologies. 1.3 Summary an Structure of Thesis This thesis is organize as following: A iscussion of the relate work an the tren of secret key generation is provie in Chapter 2. The system moel incluing the attacker moel is escribe in Chapter 3. The basic mathematical moel of secret key generation an its calculation rules are also iscusse in Chapter 3 an will be use further in the 4

18 secret key generation rate erivation. In Chapter 4, the framework of secret key generation algorithm is propose. The propose secret key generation algorithm inclues quantization for localization, public iscussion consiering network topology (star, chain an hybri topologies) an bit extraction. In Chapter 5, a theoretic analysis for the Maximum achievable Secret key generation Rate (MSR) is conucte base on the previously iscusse network topologies (star, chain an hybri) respectively. In Chapter 6, with the help of mathematic analyzer, the relationship between the maximum secret key generation rate, the wireless network scale an the signal-to-noise ratio (SNR) is analyze with intuitive chart view. In Chapter 7, wireless network moels with ranom patterne moving wireless noes is simulate to further stuy the feasibility an performance of our propose algorithm. An a metho of improving the chain topology performance is suggeste. In Chapter 8, an optional secret key agreement proceure is propose towar reucing bit mismatch rate. Finally, the conclusions of the research is given in Chapter 9. 5

19 Chapter 2 Relate Work With the robust popularization of wireless network, how to secure the wireless communication within the authorize wireless noes has become a critical issue. In such scenario, traitional authentication an key exchange methos base on public-key cryptography [1-4] may not be applicable because of its fixe key infrastructure. Meanwhile, secret key agreement algorithms have recently rawn more attention ue to its synchronize key generation scheme which gives the legitimate noes more privacy privilege over eavesroppers. Multiple key istribution mechanisms are iscusse by Seyit [5]. In [5], Seyit iscusse about pair-wise, group-wise an network-wise key istribution schemes, which provie a reference for esigning secret key generation algorithm. In [6], Maurer has stuie the lower an upper limit of secret key generation rate for a 3-noe wireless network moel (incluing two legitimate noes A, B an one eavesropper E). Maurer provie the upper boun with the assumption that the eavesropper is receiving a very small amount of information an the lower boun given by the eavesropper can only access the public channel information. Later, the maximum secret key generation rate uner such 3-noe moel was improve by consiering ifferent wireless communication 6

20 scenarios in Ruolph [7] an Maurer [8]. However, the maximum secret key generation rate of a propose algorithm still nees to be stuie further towar improvement. Recently, an overwhelming amount of stuies are focuse on generating the secret key for wireless network by exploiting reciprocal properties of the wireless channel [9-14]. [9] an [10] stuie the secret key generation utilizing channel response. Detaile secret key generation algorithms also have been propose in this regar an its performance is evaluate through secret key generation rate an secret bit mismatch rate analysis. Example stuies in [11-14] are base on the signal envelop an receive signal strength. [11] i an experimental setup of 3-noe wireless network to test the propose algorithm. [12] gave an initial research on large scale wireless network an its topology. However, the hybri topology is not stuie an the algorithm coul be further improve. Further research [13] uses the electronically steerable parasitic array raiator (ESPAR) antenna an [14] concerns on the multiple antenna evices in signal reception. Different from the previous stuies base on channel reciprocity, some recent researches [15-20] are utilizing localization information for wireless network secret key generation. Due to the variable technology that can be use for wireless localization, the localizationbase secret key generation algorithm [15] is applicable to more iverse circumstances. [15] iscusse on the localization using ultra-wieban raios. Time of arrival (TOA), angle of arrival (AOA) an signal strength (SS) base wireless positioning techniques are introuce in this research. [16] gave an overview of localization techniques via wieban raios an iscusse its funamental limits. [17] presente stuy on wireless localization leveraging ultrasoun technology. The feasibility of ultrasonic localization for local positioning system (LPS) is prove through both theoretical an experimental stuy. [18] 7

21 introuce an infrare local positioning system (LPS) esigne for inoor unmanne aerial vehicle (UAV) use, which emonstrate the possibility of wireless localization base on infrare technology. Due to this reainess of localization technology in most wireless systems, the propose localization-base secret key generation algorithm can be easily integrate. There are also some comparable researches about secret key generation via localization [19-20]. However, [19] is base on pre-istribute personal secret information. Also the sensors within the network are consiere as either low mobility or fixe. [20] stuie the secret key generation algorithm only base on a 3-noe wireless network moel. Actually, most of the previously mentione researches are conucte only for the simplest 3-noe wireless network moel. In orer to further stuy the probability of secret key establishment via localization in a larger scale real wireless network, the research for an m (m 3) legitimate noes an 1 eavesropper wireless network moel is conucte in this thesis. [21] is consiering high secret key generation rate for the secret key generation algorithm base on wireless channel reciprocity. [22] is research for faing wireless channel. Increasing the secret key generation rate an analyzing the algorithm with ifferent kins of wireless channel coul be a future work. In this thesis, the analysis is base on Gaussian istribution assumption of the sample localization information for the noes an the noise is also consiere as Aitive White Gaussian noise (AWGN). In all, this thesis proposes an algorithm of secret key generation using localization information for multiple noe wireless network. A theoretical stuy of the Maximum Secret key generation Rate (MSR) is erive an the MSR is further examine through mathematical analysis. Aitionally, an optional secret key agreement proceure is presente to reuce the bit mismatch rate. 8

22 Chapter 3 Mathematical Moel of Wireless Network 3.1 Parameters of Wireless Network Moel Secret key agreement is essential for securing wireless communication. However, most of the previous localization-base secret key generation research only consier a simple network of 3-noes (Noe A, Noe B an Attacker E). In orer to stuy the secret key generation algorithm that works for a real wireless network, a wireless network moel consisting of a large scale area of m (m 3) legitimate noes an 1 malicious eavesropper must be consiere. Towar this goal, we efine that the istance between two ranomly selecte noe 1 an noe 2 is terme as the Reference Distance (RD) an use for secret key agreement. Furthermore, instea of transmitting the RD itself, an Aitive Distance Value (ADV) is publishe uring public iscussion to secure the secret key agreement. Since an m noe wireless network is consiere in this research, ifferent types of topologies (star, chain an hybri) are iscusse, as the group of wireless evices in the secret key establishment may or may not within the communication range of each other. For the circumstance that 9

23 each wireless evice is within the communication range of another wireless evice, a star topology is employe, while a chain topology is use for the scenarios in that not all wireless evices are within the communication range of others, but all noes are interconnecte. Other than the star an chain topologies, a hybri topology coul be utilize uner other circumstances. The hybri topology is a combination of star an chain topology. After the istribution of ADV, RD is estimate an the secret key is generate through bit extraction. Further, the maximum achievable secret key generation rate is erive an analyze base on the star, chain an hybri topology. After that, a secret key agreement proceure is propose to reuce the bit mismatch rate. Finally, a conclusion is mae for the propose framework of secret key generation utilizing localization for wireless network. More etaile system escription is presente as follow: i) To quantize the localization information for the m noe 1 eavesropper wireless network, the time is ivie into n iscrete slots. Let l i (k) be the location for the noe i at time slot k, where i{1,2,, m, e} an k {1,2,, n}. Then the istance between noe i an noe j at slot k can be presente as i,j (k) li(k) l j(k), while the noes exchange their localization information in public iscussion. ii) Two ranom wireless noes are selecte as Noe 1 an Noe 2. The istance between them, 1,2 (k) l1(k) l2(k), is terme as the Reference Distance (RD) to generate the wireless network secret key. Noe 1 an noe 2 estimates the RD as 1,2(k) an 2,1(k) respectively. 10

24 iii) Instea of transmitting the RD irectly, an Aitive Distance Value (ADV) is use in the public iscussion to better ensure the wireless network security. Above assumptions an efinitions are use for the following respective network moels: a) In the star topology, noe 1 will be selecte as the central noe of star topology. It will publish ADV to all other noes to perform the secret key generation. All the other noes observe the istance between themselves an noe 1, where (k) (k) (k), i {3,4,, m}. 1,i l1 l i b) In the chain topology, noe 1 will be selecte as the hea noe of the chain an ADV will be passe by each noe throughout the chain topology. All the other noes i observe the istance between themselves an the other two ajacent noes, where i 1,i (k) li 1(k) li (k) an i,i 1(k) li (k) li 1(k), i {3,4,, m}. c) In the hybri topology, noe 0 an noe 1 will be ranomly selecte an the istance between noe 0 an noe 1 is terme as RD. Noe 1 will be selecte as the central noe of the 1st star. It publishes the ADV to all noes in the 1st star through the star topology an forwar the ADV to the 2n star central noe through the chain topology. 2n star central noe publishes the ADV to all noes in 2n star an forwar the ADV to the next star central noe through the chain. iv) When all noes receive the ADV in any topologies, they are able to calculate RD through ADV. The secret key is then generate through secret bit extraction base on RD. 11

25 3.2 Attacker Moel In this thesis, a passive eavesropper noe e is consiere. This passive aversary noe e oes not have transmitting beacons, however, it eavesrops all the public iscussion uring the secret key generation. Noe e also observes the istance between e an any other noe i. We efine its relative istance as e, i(k) le(k) li (k). 3.3 Basic Mathematical Moel of Wireless Network Maurer [6], Ahlswee an Csisz ar [7] performe the initial research on secret key generation using correlate information. In [7], the theoretical bouns of secret key generation rate for a simple 3-noe wireless network with two legitimate noes A, B an one eavesropper E has been ientifie. In their work, iscrete ranom variables X an Y respectively represent the information observe an sample by noe A an noe B in n iscrete time slots, where X an Y are inepenent an ientically istribute (i.i.) ranom variables such that X [X(1), X(2), X(n)] an Y [ Y(1), Y(2), Y(n)]. In any given time instance k, k {1,2,, n}, the observe information pair (X, Y) is statistically highly epenent so that noe A an B are able to extract synchronize secrete key. Noe A an noe B then generate the secret key by communicating over a public error-free channel, an the public communication between A an B is represente collectively by Z. Let the ranom variable S with finite range s be the secret key generate by noe A an noe B, if there exist two functions f A an f B so that S f (X, Z), S f ( Y, Z), an for any A A B B small positive number >0, following limitations must be met: 12

26 P ( S S S ) 1 (3-1) r A B I( S; Z) (3-2) H( S) log s (3-3) Here, Pr ( S ) enotes the probability mass function of S, I(S; Z) enotes the mutual information between S an Z, H(S) enotes the entropy of S. For these quantities, there are certain conitions attache to the Eq.(4-1)-Eq.(4-3) such that: Conition (1): noe A an noe B generate the same secret key with high probability. Conition (2): the generate secret key is well encrypte from the aversary noe E observing the public communication Z. Conition (3): the generate secret key is nearly uniformly istribute in entropy sense. 3.4 Calculation Rules for Entropy an Mutual Information The entropy of an arbitrary ranom variable X is efine as H(X) P(x )log P(x ) (3-4) i i i The mutual information for correlate ranom variable X an Y is terme as p(x i, y j) p(x, y) I(X;Y) p(x i, y j)log( ) p(x, y)log( ) xy p(x )p( y ) Y X p(x)p( y) (3-5) i j i j Conitional mutual information can be terme as 13

27 p (z )p (x, y,z ) (3-6) Z k X, Y, Z i j k I(X;Y Z) p X, Y, Z (x i, y j,z k )log( ) k j i p X, Z (x i,z k )p Y, Z (y j,z k ) The following aitional calculation rules between entropy, mutual information an conitional mutual information will be use in future theoretical erivation. (a) Relationship between mutual information an entropy: I(X;Y) H (X) H(X Y) H ( Y ) H( Y X ) H (X) H(Y) H(X, Y) H(X,Y) H(X Y) H (Y X) (3-7) (b) Relationship between conitional mutual information an conitional entropy: I(X;Y Z) H(X Z) H(X Y,Z) (3-8) (c) Chain rule for mutual information I(X;Y,Z) I(X;Z) I(X;Y Z) (3-9) () Bayes rule for conitional entropy H(Y X ) H(X Y) H(X) H(Y) (3-10) These efinitions an rules will be further utilize in Chapter 5 for the theoretical analysis of the MSR. 14

28 3.5 Gaussian an Multivariate Gaussian Distribution Gaussian istribution assumption for the signals is wiely use in recent theoretical analysis for the secret key generation. Here are some basic calculation rules for Gaussian an Multivariate Gaussian istribution which will be use later. 2 2 A Gaussian istribution can be presente as N ~ (, ), where is the mean an is the variance. The entropy for such Gaussian istribution is [24]: 1 ln(2 2 e ) h (3-11) 2 A multivariate Gaussian istribution of an m ranom vector X X X X can be,? m presente as N ~ (, ), where the mean vector {E[ X1], E[ X 2],, E[ X m ]} an the covariance matrix [Cov [ X, X ]], i 1,2,, m, j 1,2,, m. i j 1 2 The entropy for such multivariate Gaussian istribution is [24]: hm 1 ln{(2 ) m e } (3-12) 2 where is the eterminant of the covariance matrix. Here are some matrix eterminant calculation rules for certain matrix. For an mm matrix with iagonal elements equal to a an all other numbers equal to b, the eterminant woul be: et( ) [ ( 1) ]( ) m ( 1) a m b a b (3-13) 15

29 Laplace Expansion: Suppose B = { b ij } is an n n matrix, i, j {1, 2,..., n}. Then its eterminant B is given by: B b C b C b C i1 i1 i2 i2 in in b C b C b C 1 j 1 j 2 j 2 j nj nj n b C ij ij ij ij j1 i1 n b C. (3-14) where C ( 1) i j M an M ij is the eterminant of i, j minor matrix of B which is the ij ij eterminant of an (n-1) (n-1) matrix that results from eleting the i-th row an the j-th column of B. 16

30 Chapter 4 Secret Key Generation Algorithm The framework of propose secret key generation algorithm is introuce in this chapter. First of all, a basic protocol of secret key generation is presente. In this basic protocol, a simple network incluing 3 legitimate noes (noe 1, 2, 3) is consiere. Further, the propose secret key generation algorithm for the m ( m 3 ) noe wireless network is introuce, incluing the proceure of quantizing the noes position for localization information, public iscussion consiering network topology (star, chain an hybri topology), bit extraction an secret key agreement. 4.1 Basic Protocols between Noes In the basic protocol, 3 legitimate noes are consiere, incluing noe 1, noe 2 an noe 3. Fig.4-1 shows an illustration of the 3-noe network. Step 1: In a certain time instance k, noe 1, 2 an 3 will observe an sample its localization information li ( k ). Further, the noes will calculate the istance between each other through public beacon exchange accoring to Eq.(4-1). 17

31 ( ) ( ) ( ) k k k ADV 1,2 1,3 Noe 1 Noe 2 RD ( k) 1,2 ( k ADV ) Noe 3 ( k) ( k) ( k) 1,2 ADV 3,1 Figure 4-1: Illustration of the 3-noe network of basic protocol ( k) ( k) l ( k) l ( k) 1,2 2,1 1 2 ( k) ( k) l ( k) l ( k), k(1,2,, n) 1,3 3,1 1 3 (4-1) Here l i (k) is the quantize position for noe i in time slot k. (k) is efine as the 1,2 Reference Distance (RD). An note that istance between noe 1 an 2 is ientical no matter how it is measure from noe 1 or noe 2 uring the same time interval. Step 2: Noe 1 will calculate the Aitive Distance Value (ADV) as ( k ) ADV 1,2( k ) 1,3( k ) an then forwar the ADV to noe 3, while noe 2 alreay know RD as (k). 2,1 Step 3: When noe 3 receives the ADV from noe 1, it is able to estimate the RD (istance between noe 1 an noe 2), which is 1,2 ( k) ADV ( k) 3,1( k). Since noe 2 can measure the RD from its en, 2,1( k) 1,2 ( k), the noe 1, noe 2 an noe 3 have obtaine the same localization information RD. Therefore, the secret key can be generate collaboratively 18

32 through secret bit extraction (the secret bit extraction algorithm will be explaine later) base on RD. 4.2 Secret Key Generation Algorithm Phase 1. Quantization as ( ) i First, the noes 1,2,,m quantize the fiel Φ an estimate their localization information l k, i (1,2,,m), k (1,2,, n), at time slot k an store them in their buffers. In this phase, a variety of technologies for localization estimation coul be utilize such as infrare, wireless raios, ultrasoun, Liar, Raar, an so on, which make the applicability of secret key generation over localization very robust. Phase 2. Public Discussion In this phase, a public iscussion phase is conucte for the noes to calculate their relative istance as i,j(k) j,i(k) li(k) l j(k),, (1,2,,m), k (1, 2,, n) i j at time slot k. The istance between two ranomly selecte noes 1 an 2 (k) 1,2 is terme as RD. Then ADV is calculate an istribute for secret key generation base on the topology of the wireless network. Following is the etaile protocol of public iscussion base on ifferent topologies. A. Star Topology Uner the circumstance that every noe is within the communication range of another wireless noe, a star topology is usually forme. In this case, we can easily exten the basic 19

33 protocol of 3-noe network to an m noe wireless network. See Fig.4-2 for an illustration of the star topology network. 1) In the star topology, noe 1 an noe 2 are ranomly selecte, while the istance between noe 1 an noe 2 is terme as R RD ( k). 1,2 2) Noe 1 is selecte as the central noe an it calculates the ADV for every noe i other than noe 1 an noe 2, where ADV,i( k) 1,2 ( k) 1, i( k), i (3,4,,m). Noe 2 estimates the RD as ( ) 2,1 k. An all other noe i estimates its relative istance from noe 1 as (k). i,1 3) Noe 1 istributes the ADV ( ), k for each noe i through the public iscussion. ADV i After noe i receives the ADV ( ),i k, it can calculate the RD as 1,2 ( k) ADV, i( k) ADV ( ) i,1 k. As a result, all the m noes woul obtain the RD after the public iscussion. 4) Meanwhile, note that the eavesropper E woul overhear all the public iscussion. In the star topology, the public iscussion is a set of ADV value for noe 3, 4 m. Let E ( ) pub k be the public iscussion overhear by eavesropper E. E ( k ) can be presente as: pub E ( k) [ ( k), ( k),, ( k)] pub ADV,3 ADV,4 ADV,m [( ( k) ( k)),( ( k) ( k)),,( ( k) ( k))] 1,2 1,3 1,2 1,4 1,2 1,m (4-2) 20

34 Noe 2 Noe E RD ( k) 1,2 Noe 1 ( k) ( k) ( k) ADV, i 1,2 1, i i(3,4,, m) ( ) ADV,3 k ( ) ADV,4 k ( ) ADV,m k Noe 3 Noe 4 Noe m ( k) ( k) (k) 1,2 ADV,3 3,1 ( k) ( k) ( k) 1,2 ADV,4 4,1 ( k) ( k) ( k) 1,2 ADV, m m,1 Figure 4-2: Illustration for an m noe wireless network with star topology B. Chain Topology For the situation that not every noe is within the communication range of other wireless noes, however they are interconnecte, a chain topology is suggeste. See Fig.4-3 for an illustration of the chain topology network. Noe E ( ) ( ) ( ) ADV,3 k 2,1 k 2,3 k ( k) ( k) ( k) ( k) ADV,i( k) ADV, i1 ( k) i, i1 ( k) i, i1( k) ADV,4 ADV,3 3,2 3,4 Noe 1 Noe 2 Noe 3 Noe 4 ( ) ( ) ADV,3 k ADV,4 k RD ( k) 1,2 ( k) ( k) ( k) 1,2 ADV,3 3,2 ( k) ( k) ( k) 1,2 ADV,4 4,3 ADV, i ( k) Noe m ( k) ( k) ( k) 1,2 ADV, m m,m1 Figure 4-3: Illustration for an m noe wireless network with chain topology 21

35 1) Noe 1 an noe 2 are ranomly selecte, while the istance between noe 1 an noe 2 is terme as RD ( k). 1,2 2) Noe 1 is selecte as the hea noe an noe m is selecte as the tail noe of the chain topology. Noe 2 calculates the ADV for noe 3 as ADV,3( k) 2,1( k) 2,3( k). Noe 2 estimates the RD as ( k ). 2,1 3) Upon the reception of ( ),3 k, noe 3 calculates the RD as 1,2 ( k) ADV,3( k) ( k) 3,2 ADV an also passes the ADV for next noe 4 as ADV,4( k) ADV,3( k) 3,2( k) 3,4( k). Similarly, upon the reception of ( ) ADV,4 k, noe 4 calculates RD as 1,2 ( k) ADV,4( k) ( k) an also passes ADV for next noe 5 as 4,3 ADV,5( k) ADV,4( k) 4,3( k) (k). 4,5 4) In summary, noe i i(3,4,,m 1) estimates its relative istance from its neighbor noe i-1 an i+1 as ( ) ii, 1 k an ( ) ii, 1 k. Then, upon the reception of its ADV ( ) ADV, i k, noe i calculates the RD as 1,2 ( k) ADV, i( k) i, i1( k) an also passes the ADV for next noe i+1 as ADV, i 1 ( k) ADV, i( k) i, i 1 ( k) i, i 1( k). An for noe m, it only nees to calculate the RD as 1,2 ( k) ADV, m( k) m,m 1( k). 5) Also, the eavesropper E woul overhear all the public iscussion through all the public iscussion. In the chain topology, the public iscussion is also a set of ADV value for noe 3, 4 m. The public iscussion overhear by eavesropper E E ( k) can be presente as: 22 pub

36 E ( k) [ ( k), ( k),, ( k)] pub ADV,3 ADV,4 ADV,m [( ( k) ( k)),( ( k) ( k)),,( ( k) ( k))] 1,2 2,3 1,2 3,4 1,2 m1,m (4-3) C. Hybri Topology For certain circumstances like large scale wireless network, only the star an chain topology may not be aequate for the wireless network. Uner such situation, a hybri topology is suggeste as a combination of star an chain topology. Fig.4-4 illustrates the hybri topology. Noe E Noe 1 1 Noe 2 1 Noe m 1 Noe 0 Noe 1 Noe 2 Noe m Chain Topology Star Topology Noe 1 2 Noe 2 2 Noe m 2 Figure 4-4: Illustration for wireless network with hybri topology In the hybri topology shown in Fig.4-4, noe 0 is the hea noe of the hybri topology. Noe 1, 1 1, 1 2 form a star topology, as well as noe 2, 2 1, 2 2 through noe m, m 1, m 2. Meanwhile, noe 0, noe 1 through noe m form a chain topology. In such wireless network, some noes are outsie the communication range of other noes so that the star topology is not aequate. Further, a chain cannot form a Hamiltonian path [24] in the wireless network so that the chain topology is not enough as well. Here the Hamiltonian 23

37 path is a path that visits each noe in the wireless network exactly once. Therefore, a combination of star an chain topology the hybri topology is suggeste in such situation. 1) In the hybri topology, noe 0 an noe 1 is ranomly selecte while the istance between noe 0 an noe 1 is terme as RD ( k). 0,1 2) The noes on the chain except hea noe 0 (noe 1,2,,m) is selecte as the central noe for each star. Further, central noe 1 nees to calculate an forwar two ADV. One ADV is for every other noes in the 1 st star (noe 1 1, 1 2, 1 m ), assuming there 1 are m noes in the 1 st star. Here 1 j( k),1 1,0( k) j( k), j ( 1,2,, m ADV 1,1 1). Noe 1 j then coul calculate the RD as 0,1 ( k) j( k) j ( k). The other ADV is for the next,1 1,1 central noe in the chain, noe 2. Here ADV,2( k) 1,0 ( k) 1,2 ( k). ADV 3) Upon the reception of ( ),2 k at noe 2. Noe 2 can calculate the RD as ADV ( k) ( k) ( k). Noe 2 also nees to calculate an forwar two ADV. One 0,1 ADV,2 2,1 ADV is for every other noes in the 2 n star (noe 2 1,2 2, 2 m ), assuming there are 2 m noes in the 2 n star. Here 2 j( k),2 ADV,2( k) 2,1( k) j( k), j ( 1,2,, m 2,2 2). Noe ADV 2 j then coul calculate the RD as 0,1 ( k) j( k) j ( k). The other ADV is for the,2 2,2 next central noe in the chain, noe 3. Here ADV,3( k) ADV,2( k) 2,1( k) 2,3( k). ADV 4) Accoringly, noe i, i (2,3,, m) calculates the RD as 0,1 ( k) ADV,i( k) i,i 1( k) upon its ADV reception of ( ), k. Noe i also nees to calculate an forwar two ADV. ADV i One ADV is for every other noes in the i th star (noe i 1, i 2 m, i i ), assuming there 24

38 are m i noes in the i th star. Here j( k) ADV,i( k) i,i 1( k) j( k), j ( 1,2,, mi ). ADV, i i, i An noe i j then coul calculate the RD as 0,1 ( k ) ( k j ) j ( k ). The other ADV is,i,i for the next central noe in the chain, noe i+1. Here ADV,i 1 ( k) ADV,i( k) i,i 1( k) ( k) i,i 1. Especially for noe m, it oes not nee to calculate ADV for the next central noe. ADV i 5) Also, the eavesropper E woul overhear all the public iscussion through all the public iscussion. The public iscussion overhear by eavesropper E E ( k) can be presente as the combination of all ADV information incluing star an chain: pub E ( k) [ ( k),, ( k), ( k),, ( k),, ( k),, ( k)], pub 1 m1 1 m2 1 m ADV,1 ADV,1 ADV,2 ADV,2 ADV,m ADV,m m [ ( k), ( k),, ( k)] ADV,2 ADV,3 ADV,m [( ( k) ( k)),,( ( k) ( k)),( ( k) ( k)),, 0,1 1 0,1 m1 0,1 1 1,1 1,1 2,2 ( ( k) ( k)),,( ( k) ( k)),,( ( k) ( k))], (4-4) 0,1 m2 0,1 1 2,2 m,m 0,1 m m,m m [( ( k) ( k)),( ( k) ( k)),,( ( k) ( k))] 0,1 1,2 0,1 2,3 0,1 m1,m Phase 3. Secret Bit Extraction In this phase, an r bit secret key is extracte base on the RD receive by the legitimate noes in each time slot through three ifferent secret bit extraction steps. Here the bit number r can be selecte to give an aequate length to the secret key. For a single time slot r bits secret key is generate an there are totally n time slots, therefore the final generate secret key woul have a length of nr bits. Let the RD calculate by noe i be RD, 1,2@ ( ), i (1,2,, m ), (1, 2,, ) i k i k k n, where ( ) 1,2@ i k means the 1,2 value calculate by the noe i at time slot k, an for every noe 1 25

39 to noe m, n time slot RD measurements are mae as RD. Three types of bit extraction ik, steps are utilize for each RD measurement collaboratively to achieve an r bit secret key Sik, ( ), (1,2,,r) for each time slot k. Let the mean value of, RD be ik m i k R Di, k n an F( RD ik, ) be the cumulative istribution function of RD ik,. The following secret bit etermination algorithm presents the propose secret bit extraction protocol for S ik, (). For a set of RD measurements RD, i(1,2,, m ), k (1, 2,, n) ik, 0, if RD i, k i i) Sik, (1) 1, if RD m 0 (4-5) ik, m 0 i 0, if RDi, k RDi, k1 0 ii) Sik, (2) 1, if RDi, k RDi, k1 0 (4-6) iii) S, ( ), (3,4,, r), here 2 r-2 quantization levels are use where ik q min( RD ), q max( RD ) (4-7) 0 i, k r2 2 i, k 1 u r 2 an q u F ( ),u 1, 2,, 2 1 (4-8) r2 2 Here u is to make sure that qu is selecte base on the same 2 1 q u F ( ) r2 istribution function of RD. The u th quantization bin is efine as the interval ik, [q,q ], an Gray coing is employe for bit extraction. u 1 u 1) For a single observe RDik, 1,2, the value is mainly ecie by the istance between noe 1 an noe 2. Let the threshol base on RD minus the mean istance m ik, i 26

40 coul reuce the influence of noe istance an amplify the ranomness of the RD in step i. 2) For step ii, the evaluation of RDi, k RDi, k implies the moving tren of noe 1 an 1 noe 2. 3) Step iii, secret key bit number r is selecte by the user in orer to give an aequate length to the generate secret key S ik, (). By increasing the secret key bit number r, the secrecy privilege of the legitimate noes over the eavesropper is increase since it is harer for the eavesropper to ecipher the same secret key. However, it will also make the secret bit extraction algorithm more vulnerable to the noise an thus increase the bit mismatch rate of the legitimate noes. In this thesis, r is selecte as 4 consiering this traeoff an the simplicity for simulation. 4) Finally, by combining all the S ik, () extracte from n time slots, noe i is able to generate the secret key as S i (), (1,2,, n*r). Since RD is ientical for each noe, all the m noes are then able to agree on a synchronize secret key S i (). Example: Let the RD of {98m, 99m, 100m, 101m, 102m, 103m} is receive in n = 6 time slots at noe i. RD follows a uniform istribution that the probability for each RD value is ientical. Base on each receive RD value, an r = 4 bit secret key nees to be extracte. Here the mean value of RD is Since RD is uniformly istribute, the 2 r- 2 =4 quantization bins are [98, 99.25], [99.25, ], [100.50, ], [101.75, 103]. The corresponing 2 bit Gray coe is [00, 01, 11, 10]. 27

41 1) For time slot k = 1, RDi,1 98 is less than mean value mi = , therefore S (1) 0 i,1. Since RD is the first RD value, i,1 S (2) 0 i,1. RD falls in the first quantization bin [98, i, ] so that Si,1 (3,4) 00. Thus, the generate secret key S () for time slot 1 is 0000.,1 i 2) For time slot k = 2, RDi,2 99 is less than mean value mi = , therefore S (1) 0 i,2. Since RDi,2 RD, i,1 S (2) 1 i,2. RD i,2 falls in the first quantization bin [98, 99.25] an Si,2 (3,4) 00. Thus, the generate secret key S (),2 for time slot 2 is i 3) For time slot k = 3, RDi,3 100 is less than mean value mi = , therefore S (1) 0 i,3. Since RDi,3 RD, S (2) 1 i,2 i,3. RD i,3 falls in the secon quantization bin [99.25, ] an Si,3 (3,4) 01. Thus, the generate secret key S (),3 for time slot 3 is i 4) Repeat the previous proceure for each time slot an combine all the S () together i,k an then noe i coul generate the secret key as In summary, the propose secret key generation algorithm is presente in this chapter, incluing m noe wireless network topology esign an secret bit extraction metho. Later, the maximum secret key generation rate for the propose algorithm will be erive in Chapter 5. 28

42 Chapter 5 Theoretical Analysis for MSR A theoretical analysis for the Maximum achievable Secret key generation Rate (MSR) is conucte in this chapter. Base on ifferent network topology, the MSR is ifferent from each other. Therefore, the theoretical analysis is conucte base on star, chain an hybri topology respectively. 5.1 Star Topology Base MSR Following is a brief review of the star topology: Step 1: Noe 1 an noe 2 is ranomly selecte to get Reference Distance (RD) as RD ( k). Noe 1 an noe 2 estimate RD as ( k ) an ( k ) respectively. 1,2 1,2 2,1 Step 2: Noe 1 publishes the ADV to noe i, i (3,4,,m) as ADV,i( k) 1,2 ( k) 1,i ( k). Noe i calculates istance from noe 1 as ( ) i,1 k. Step 3: Noe i estimates RD as 1,2 ( k) ADV, i( k) i,1( k). Step 4: Secret bit extraction base on RD is processe to generate final secret key. Accoring to the theoretical analysis propose by Maurer an other researchers in [6-8], secret key generation rate is the mutual information between two noes. Since they only 29

43 propose the MSR base on 3-noe network, the mutual information between central noe 1 an all other noe i incluing eavesropper, Noe 2 Noe 1 Noe E i(2,3,,m,e) shoul be examine for the star topology wireless network. For simplicity, let s assume that all Noe 3 Noe 4 Noe m Figure 5-1: Brief on star topology information sequences are inepenent ientically istribute (i.i.). Let any ranom sequence z(k) through time slot 1, 2,, n be Z [z(1), z(2),, z(n)]. Then the localization information sequence acquire for noe 1 shoul be D1 [ D1,2, D1,3,, D1,m ] [ 1,2 (1),, 1,2 ( n),, (1),, (n)]. For noe 2, RD 1,2 2,1 is given. All other noe i, i (3,4,,m) have 1,m 1,m to calculate RD through ADV public iscussion an may suffer from noise or other istractions. Therefore, secret key generation rate for noe 2 is higher than any other noes an the MSR is base on the mutual information between noe 1 an noe i, i (3,4,,m). Let s consier noe i, i (3,4,,m). Noe 1 broacasts the ADV to all noes with ADV sequence in the public iscussion. Thus noe i overhear all ADV public iscussion as Epub ( k) [ ADV,3( k), ADV,4( k),, ADV,m( k) ] [ 1,2 ( k) 1,3 ( k), 1,2 ( k) 1,4 ( k),, ( k) ( k)] an E [ E (1), E (2),, E ( n)]. Noe i then estimate the RD 1,2 1,m pub pub pub pub with its own localization sequence D an the ADV sequence E i,1 pub. Therefore the information for noe i is the joint information of [ Di,1, E pub ]. The ADV sequence E pub is also overhear by eavesropper noe e. Here, noe e is consiere to overhear the 30

44 localization information as well. Let the localization information acquire by noe e be D [ D, D,, D ]. Next, the MSR is erive as T. A same approach is mae as time e e,1 e,2 e,m slot n, which yiels MSR, uner large enough time scale. MSR (bits/sample) for noe i can be presente as: R i 1 lim I( D1; Di,1, E pub) (5-1) n n MSR for the m noe wireless network is limite by the worst noe (eg. the noe has higher noise interference than others), which can be presente as: R noe min R (5-2) 3 i m i Consiering the presence of eavesropper, the final MSR can be presente as the information obtaine by noes minus the information obtaine by eavesroppers: R 1 R lim I( D ; D, E ) (5-3) n final noe 1 e pub n Since the i.i. assumption has been mae for all information sequence, R i for any noe i (3,4,,m) is ientical. Therefore, an arbitrary noe 3 is consiere for MSR. Further, base on the i.i. assumption an the simplicity of erivation, a single time slot is consiere. Base on former assumptions, the final MSR can be expane as: R I([,,, ];, E ) I([,,, ];[,,, ], E ) final 1,2 1,3 1,m 3,1 pub 1,2 1,3 1,m e,1 e,2 e,m pub I([,,, ]; E ) I([,,, ]; E ) 1,2 1,3 1,m pub 1,2 1,3 1,m 3,1 I([ 1,2, 1,3,, 1,m ]; Epub 1,2 1,3 1,m e,1 e,2 e,m pub I([,,, ]; E ) 1,2 1,3 1,m 3,1 = I( ; E ) I([,, ];, E ]) 1,2 3,1 pub 1,3 1,m 3,1 1,2 ) I([,,, ];[,,, ] E ) pub 31 pub..(5-4)...(5-5)....(5-6) pub..(5-7)

45 = I( ; E ) 1,2 3,1 pub = h( E ) h(, E ) 1,2 pub 1,2 3,1...(5-8)..(5-9) = h( E ) h( E ) h( ) (h(, E ) h(, E ) h( )).(5-10) pub 1,2 pub 1,2 3,1 pub 1,2 3,1 pub 1,2 = h([,, ] 1,2 1,3 1,2 1,4, 1,2 1,m 1,2 1,2 1,3 1,2 1,4 1,2 1,m ) h([,,, ]).(5-11) h(,,,, ) h(,,,, )... (5-12) Here, Eq.(5-5) follows Eq.(3-9) such that I(X; Y, Z) I(X; Z) I(X; Y Z). Eq.(5-6) is ue to the i.i. assumption, i, j is inepenent from e,i. Eq.(5-7) follows the chain rule. Eq.(5-8) is because, given that 1,2 an E, pub 1,3,, 1,m are etermine, the entropy is 0. Eq.(5-9) follows Eq.(3-8) since I(X;Y Z) H(X Z) H(X Y, Z). Eq.(5-10) follows Eq.(3-10) since H(Y X ) H(X Y) H(X) H(Y). Eq.(5-11) is the expansion of Eq.(5-10). Eq.(5-12) is because, for given 1,2, the ranomness in [ 1,2 1,3, 1,2 1,4,, 1,2 1,m ] can be simply presente as [ 1,3, 1,4,, 1,m ]. pub h(,[,,, ] ) h(,[,,, ]) 3,1 1,2 1,3 1,2 1,4 1,2 1,m 1,2 3,1 1,2 1,3 1,2 1,4 1,2 1,m = h([,, ]) h([, 1,3 1,4, 1,m 1,2 1,3 1,2 1,4, 1,2 1,m, ]) 3,1 1,3 1,4 1,m 3,1 1,2 1,3 1,2 1,4 1,2 1,m Next, the MSR is further estimate through Gaussian istribution assumption that all observations of istance terms are consiere as i.i. Gaussian processes. In real wireless communication environment, the istance information can be presente as the sum of noe istance an noise: (k) (k) w (k), where w (k) i, j is an aitive Gaussian noise. ` i,j i,j i,j ` The istance measure at both en i an j shoul be ientical, so (k) i,j ` j,i (k). However the noise at ifferent noe is uncorrelate, hence w (k) an i, j w (k) j,i are inepenent. Since the entropy of Gaussian istribution `i, (k) is only a function of its variance 2 j as is efine in Eq.(3-11) that ` h ( i, j(k)) 1 2 ln(2 e ), the mean value oes not affect the MSR 2 32