Distributed Detection in Tree Networks: Byzantines and Mitigation Techniques

1 Distrib uted Detecti on in T r ee Networks: Byzantines and Mit igation T echniques Bhavya Kailkhura, Student Member , IEEE , S w astik Brahma, Member , IEEE , Berkan Dulek, Member , IEEE , Y unghsiang S Han, F ellow , IEEE , Pramod K. V arshney , F ellow , I EEE Abstract In this paper , the problem of distrib uted detection in tree networks in the presence of Byzantines is considered . Closed form expressions for optima l attacking strategies that min imize the miss d etection error exponent at the f usion cen ter (FC) are obtained. W e also lo ok at the problem from the network de- signer’ s (FC’ s) pe rspecti ve. W e study th e prob lem of designing o ptimal distributed d etection parame ters in a tree network in the presence of Byzan tines. Next, we model the stra te gic in teraction between th e FC and the attacker as a L eader -Follower (Stackelberg) gam e. This for mulation pr ovides a metho dology for predicting attacker and defender (FC) equilibrium strategies, which can be used to implement the optimal detector . Finally , a r eputation based schem e to iden tify Byzantines is p roposed and its perfor mance is analytically evaluated. W e also provid e some num erical examples to gain insigh ts into the solu tion. Index T erms Distributed detection , data falsification, Byzantines, tree networks, erro r exp onent, leade r -f ollo wer game, reputatio n based mitiga tion scheme This work was supported by the Center for Advan ced Systems and Engineering at Syracuse Univ ersity . The authors would like to t han k Aditya V empaty for his valuable comments and suggestions to i mpro ve the quality of the paper . B. Ka ilkhura, S. Brahma and P . K. V a rshney are with Department of E ECS, Syracuse Unive rsity , S yrac use, NY 13244. (email: bkailkhu@syr .edu; skbrahma@syr .edu; varshne y@syr .edu) B. Dulek is with Department of Electrical and El ectron ics Engineering, Hacettepe Univ ersity , Beytepe C amp us, 06800 Ankara, T urke y . (email: berkan@ee.hace ttepe.edu.tr) Y . S . Han is with EE Department, National T aiw an Unive rsity of S cience and T echno logy , T aiwan , R. O. C. (email: yshan@mail.ntust.edu.tw) July 10, 2018 DRAFT 2 I . I N T R O D U C T I O N Distributed detecti on d eals with the problem of m aking a global decision regarding a phe- nomenon based on local decis ions col lected from sev eral remotely located sensin g nodes. Dis- tributed detection research has tradit ionally focused on the parallel network topology , i n which nodes directly transmit their observa tions or decisions to the Fusion C enter (FC) [1] [2] [3]. Despite its th eoretical impo rtance and analytical tractabil ity , parallel topology may not alw ays reflect the practical scenario. In certain cases, it m ay be required to place the nodes outs ide their communication range with t he FC. Then, the cover age area can be increased by forming a multi-hop n etw ork, where nodes are organized hi era rchically in to mul tiple le vels (tree networks). Some examples of t ree networks inclu de wireless senso r and military communication networks. For instance, the IEEE 802.15.4 (Zigbee) specifications [4] and IEEE 802.22b [5] support t ree networks. T ypicall y , a netw o rk emb odies a large number of inexpensive sensors, which are deployed in an open en vironment to coll ect the obs erv ations regarding a certain p henomenon and, t herefore , are susceptible to many kin ds o f attacks. A typical example i s a Byzantine attack. Whil e Byzantine attacks (originally p roposed in [6]) may , i n general, refer to many types of malicious beha vior , our focus in this paper i s on data-f alsi fica tion attacks [7]–[16], where an attacker sends false (erroneous) dat a to the FC to degrade detection p erformance. In this paper , we refer to su ch data falsification attackers as Byzantines , and the data thus fabricated as Byzantin e data . A. Related W ork Recently , di strib uted detection in the presence of Byzantine attacks h as been explored i n [10], [11], where the pro blem of determi ning the most ef fectiv e attacking st rate g y for the Byzanti nes was in vestigated. Howe ver , bo th works focused only on parallel topology . The p roblem consid- ered in this paper is most related to our earlier papers [12 ], [16]. In [12], [16], we studied the problem of di strib uted detection i n perfect tree networks (all intermediate nodes in the tree have the same n umber o f children) with Byzantines un der the assumption that the FC d oes not know which decisio n bit is sent from which node and ass umes each receive d bit t o originate from nodes at depth k with a certain probability . Under this assumpt ion, the attacker’ s ai m was to maximize the false alarm probabili ty for a fixed det ection probability . When t he number of nodes is lar ge, by Stein ’ s lemma [17], w e know that the error exponent of the false alarm probabil ity July 10, 2018 DRAFT 3 can be used as a su rrogate for the false alarm probability . Thus, t he optimal attacking st rate g y was obtained by making the error exponent of t he false alarm probability at the FC equal to zero, whi ch makes th e decis ion fusi on scheme to become completely incapable (bl ind). Some counter-measures were also proposed to protect the network from such Byzantines. There are se veral notable dif ferences between this p aper and our earlier papers [12], [16]. First, i n cont rast to [12], [16], in this paper , the problem of distri b uted detection in regular tree networks 1 with Byzantines i s addressed in a practical setup where the FC has the kn o wledge of which bit is transm itted from which node. Note that, i n p rac tice, the FC knows which bit is transmitted from which node, e.g., using M A C schemes 2 , and can utilize this informatio n to improve system performance. Next, for the analysis of the optimal attack, we consider nodes residing at different leve ls of the tree to hav e diff erent detection performance. W e also allow Byzantines r esiding at diffe rent levels of the tree to h a ve different attacki ng st rate g ies and, therefore, provide a mo re g enera l and comprehensiv e analysis of the problem as comp ared to [12], [16]. W e als o study the problem from the network designer’ s perspectiv e. Based on t he information regarding w hich bit is transmitt ed from wh ich node, we propose s chemes to mitig ate the ef fect o f the Byzantines. B. Main Contributions In t his paper , it is assumed that t he FC kno ws which bit is transmitted from which nod e. Under th is assumpt ion, the problem of distributed detection i n tree networks in the presence of Byzantines is con sidered. The main contributions of this paper are summ arized b elo w: • Detection performance in tree networks wi th Byzanti nes is characterized in terms of the error exponent and a closed form expression for the opti mal error exponent is derived. • The minimu m attacking power requi red by t he Byzantines to bli nd t he FC in a tree network is obtained. It is shown that when more than a certain fraction of ind i vid ual n ode decisi ons are falsified, the decision fusion scheme is completely jeopardized. • The problem is also in vestigated from the network designer’ s perspecti ve by focusin g on the des ign of optimal distributed detection parameters in a tree network. 1 For a regular tree, intermediate nodes at differen t lev els are allowed t o hav e different degrees, i.e., number of children. 2 In practice, one possible way to achie ve this is by using the buf fer-less TDMA MA C protocol, in which, distinct non- ov erl app ing time slots are assigned (scheduled) to the nodes for communication. One practical example of such a scheme is gi ven in [18 ]. July 10, 2018 DRAFT 4 Fusion Center (F C) Level 1 Level 2 Level 3 Fig. 1. A distributed detection system organized as a regular tree ( a 1 = 2 , a 2 = 3 , a 3 = 2) is shown as an example. • W e model the strategic interaction between the FC and t he attacker as a Leader -Follower (Stackelber g) game and identify attacker and defender (FC) equilibri um strategies. The knowledge of these equilibrium st rate g ies can later be used to implement the optimal detec tor at th e FC. • W e propose a simple yet ef ficient reputation b ased scheme, which works ev en if the FC i s blinded, t o identi fy Byzantines in tree networks and analytically ev aluate its performance. The rest o f the paper i s organized as foll o ws. Section II introduces the s ystem mo del. In Sec- tion III, we st udy the probl em from Byzantine’ s perspectiv e and provide clo sed form expressions for optimal attackin g st rate g ies. In Section IV, we in vestigate the problem of design ing optim al distributed detection parameters in the presence of Byzantines. In Section V, we m odel the strategic interaction between the FC and the attacker as a Leader -Follo wer (Stacke lberg) game and find equilibrium strategies. In Section VII, we introduce an effic ient Byzantin e ident ifica tion scheme and analyze i ts performance. Finally , Section VII concludes the paper . I I . S Y S T E M M O D E L W e consi der a dist rib ut ed detection system organized as a regular tree network rooted at the FC (See Fig ure 1). For a regular tree, all the leaf nodes are at the same le vel (or d epth) and all the in termediate no des at lev el k hav e degree a k . The regular tree i s assumed to have a s et N = { N k } K k =1 of transceiv er nodes, where | N k | = N k is the total number of no des at level k . W e assume that the depth o f the tree is K > 1 and a k ≥ 2 . The total num ber of nodes in th e network is denoted as N = P K k =1 N k and B = { B k } K k =1 denotes the set of Byzantine nodes July 10, 2018 DRAFT 5 with | B k | = B k , where B k is the s et of Byzantines at lev el k . The set containing the number o f Byzantines residing at each le vel k , 1 ≤ k ≤ K , is referred to as an attack configuration, i.e., { B k } K k =1 = {| B k |} K k =1 . Next, we define the modus operandi of t he nodes. A. Modus Operandi of the Nodes W e consider a bin ary hypothesi s testing problem with two h ypotheses H 0 (signal is absent) and H 1 (signal is present). Under each hypothesis , it is a ssumed that the observations Y k ,i at each node i at l e vel k are conditionall y independent. Each n ode i at level k acts as a source in the sens e that it m ak es a one-bit (binary) local decision v k ,i ∈ { 0 , 1 } regarding t he absence or presence of th e signal using the likelihood ratio test (LR T) 3 p (1) Y k,i ( y k ,i ) p (0) Y k,i ( y k ,i ) v k,i =1 ≷ v k,i =0 λ k , (1) where λ k is th e threshold used at leve l k (it is assumed that all th e nodes at leve l k use the same threshold λ k ) and p ( j ) Y k,i ( y k ,i ) i s t he condi tional prob ability density functi on (PDF) of observation y k ,i under hypothesis H j for j ∈ { 0 , 1 } . W e denot e the probabiliti es of detecti on and false alarm of a node at lev el k by P k d = P ( v k ,i = 1 | H 1 ) and P k f a = P ( v k ,i = 1 | H 0 ) , respectively , which are functions of λ k and hold for both Byzantines and honest n odes. After m aking its o ne-bit local d ecision v k ,i ∈ { 0 , 1 } , n ode i at level k sends u k ,i to its parent node at leve l k − 1 , where u k ,i = v k ,i if i is an h onest node, but for a Byzantine nod e i , u k ,i need n ot be equal to v k ,i . Node i at level k also receive s the decisions u k ′ ,j of all successors j at leve ls k ′ ∈ [ k + 1 , K ] , which are forwarded to node i b y its imm ediate chi ldren, and forwards 4 them to its parent nod e at lev el k − 1 . W e assume error -free com munication between chi ldren and the parent nodes. Next, we present a m athematical model for t he Byzantine attack. B. Byzantine At tac k Model W e d efine the fol lo wi ng strategies P H j, 1 ( k ) , P H j, 0 ( k ) and P B j, 1 ( k ) , P B j, 0 ( k ) ( j ∈ { 0 , 1 } and k = 1 , · · · , K ) for the h onest and Byzantine nodes at l e vel k , respectively: 3 Notice t hat, under the conditional independ ence assumption, the optimal decision rule at the local sensor is a likelihood -ratio test [19]. 4 For example, IEEE 802.16j mandates tree forwarding and IE EE 802.11s standardizes a tree-based routing protocol. July 10, 2018 DRAFT 6 Honest nodes: P H 1 , 1 ( k ) = 1 − P H 0 , 1 ( k ) = P H k ( x = 1 | y = 1) = 1 (2) P H 1 , 0 ( k ) = 1 − P H 0 , 0 ( k ) = P H k ( x = 1 | y = 0) = 0 (3) Byzantine nodes: P B 1 , 1 ( k ) = 1 − P B 0 , 1 ( k ) = P B k ( x = 1 | y = 1) (4) P B 1 , 0 ( k ) = 1 − P B 0 , 0 ( k ) = P B k ( x = 1 | y = 0) (5) where P k ( x = a | y = b ) is the conditional probability t hat a node at leve l k sends a to its parent when it receives b from its child or its actual decisi on is b . For notati onal con venience, we us e ( P k 1 , 0 , P k 0 , 1 ) to denote the flipping probability of the Byzantine node at le vel k . Furthermore, we assume t hat if a node (at any le vel) i s a Byzantine, then none of its ancestors and s uccessors are Byzantine (non-overlapping attack configuratio n); otherwise, the effect of a Byzantine due to other Byzantines on the same path may be nullified (e.g., Byzantine ancestor re-flipping the already flipped decision s of i ts successors). This means that ev ery path from a leaf node to t he FC w ill hav e at most one Byzantin e. Notice that, for the attack configuration { B k } K k =1 , t he total number of corrupt ed paths (i.e., paths contain ing a Byzantine node) from level k to t he FC are P k i =1 B i N k N i , where B i N k N i is the total number of nodes cov ered 5 at lev el k by the presence o f B i Byzantines at le vel i . If we deno te α k = B k N k , then, P k i =1 B i N k N i N k = P k i =1 α i is the fraction of decisions coming from lev el k that encounter a Byzantine along the way to the FC. W e also approximate th e probabili ty t hat t he FC recei ves t he flipped d ecision ¯ x of a given node at lev el k when its actual decision is x as β k ¯ x,x = P k j =1 α j P j ¯ x,x , x ∈ { 0 , 1 } . C. Binary Hypot hesis T esting at the Fusion Center W e consider the distributed detection problem under the Neyman-Pearson (NP) criterion. The FC recei ves d ecision vec tors, [ z 1 , · · · , z K ] , where z k for k ∈ { 1 , · · · , K } is a decisio n vector wi th its elements b eing z 1 , · · · , z N k , from the nodes at different levels of t he tree. Then the FC mak es the global decision abo ut t he phenomenon by employing the LR T . Due to system vul nerabilities, s ome of t he nodes may be captured b y th e attacker and reprogrammed 5 Node i at lev el k ′ cov ers all its children at lev els k ′ + 1 to K and itself. July 10, 2018 DRAFT 7 to transmit false information to the FC to degrade detectio n performance. W e assume that the only in formation av ailable at the FC is t he probability β k ¯ x,x , wh ich is the probabilit y with which the data coming from l e vel k has been falsified. Using this information, the FC calculates t he probabilities π k j, 0 = P ( z i = j | H 0 , k ) and π k j, 1 = P ( z i = j | H 1 , k ) , which are the distributions of recei ved decision s z i originating from level k and arriving to the FC under h ypotheses H 0 and H 1 . The FC makes its decis ion regarding the absence or p resence of the signal us ing t he following likelihood ratio test K Y k =1 π k 1 , 1 π k 1 , 0 ! s k 1 − π k 1 , 1 1 − π k 1 , 0 ! N k − s k H 1 ≷ H 0 η (6) where s k is the number of decisions that are equal to one and originated from le vel k , and the threshold η is chos en in order to minim ize the m issed detection p robability ( P M ) while keeping the false alarm probability ( P F ) below a fixe d value δ . 6 Next, we deri ve a closed form expression for the opt imal missed detection error e x ponent for tree networks in the presence of Byzantines, which wi ll l ater be used as a surrogate for the probability of m issed detection. Pr oposition 1: For a K lev el tree n etw ork employing the detection scheme as giv en i n (6), the asymptotic det ection performance can be characterized using t he missed det ection error exponent giv en below D = K X k =1 N k   X j ∈{ 0 , 1 } π k j, 0 log π k j, 0 π k j, 1   . (7) Pr oof: Let Z = [ Z 1 , · · · , Z N 1 ] denot e the recei ved decision vectors from the nodes at le vel 1 , wh ere Z i is the decision vector forwarded by the node i at lev el 1 to the FC. Observe t hat, Z i for i = 1 t o N 1 are i ndependent and identi cally distributed (i.i.d.). Therefore, using Stein’ s lemma [17], the optim al error exponent for the d etection scheme as given in (6) is t he Kullback- Leibler di vergence (KLD) [20] between the distributions P ( Z | H 0 ) and P ( Z | H 1 ) . S ummation term i n (7) follows from the additiv e property of t he KLD for independent distributions. Note that, (7) can be compactly written as P K k =1 N k D k ( π k j, 1 || π k j, 0 ) with D k ( π k j, 1 || π k j, 0 ) being the 6 This type of problem setup is important, for instance, in Cognitiv e Radio Networks (CR N). I n order to coexist with the primary user (PU), secondary users (SUs) must guarantee t hat their t ransmission s will not interfere wi th the transmission of the PU who hav e higher priority to access the spectrum. July 10, 2018 DRAFT 8 KLD between the data comin g from node i at le vel k under H 0 and H 1 . The FC wants to maximize the detection performance, while, the B yzantine attacker wants to degrade the detection performance as much as po ssible which can be achiev ed by maxi mizing and minimizing the KLD, respectiv ely . Next, we e xplore the optim al attacking strategies for the Byzantines t hat degrade the detection performance mos t by minimizing th e KLD. I I I . O P T I M A L B Y Z A N T I N E A T T AC K As discussed earlier , th e Byzantines attempt to make the KL diver gence as small as p ossible. Since the KLD is always non-negativ e, Byzantines at tempt to choose P ( z i = j | H 0 , k ) and P ( z i = j | H 1 , k ) such that D k = 0 , ∀ k . In thi s case, an adversary can make the data t hat t he FC recei ves from t he nodes such that no information is con veyed from them. Thi s i s po ssible when P ( z i = j | H 0 , k ) = P ( z i = j | H 1 , k ) ∀ j ∈ { 0 , 1 } , ∀ k. (8) Notice t hat, π k j, 0 = P ( z i = j | H 0 , k ) and π k j, 1 = P ( z i = j | H 1 , k ) can be expressed as π k 1 , 0 = β k 1 , 0 (1 − P k f a ) + (1 − β k 0 , 1 ) P k f a (9) π k 1 , 1 = β k 1 , 0 (1 − P k d ) + (1 − β k 0 , 1 ) P k d . (10) with β k 1 , 0 = P k j =1 α j P j 1 , 0 and β k 0 , 1 = P k j =1 α j P j 0 , 1 . Substituting (9) and (10) in (8) and after simplification, the conditi on to make the D = 0 for a K -leve l network becomes P k j =1 α j ( P j 1 , 0 + P j 0 , 1 ) = 1 , ∀ k . Notice that, when P k j =1 α j < 0 . 5 , there does not exist any attacking probabi lity distribution ( P j 0 , 1 , P j 1 , 0 ) that can make D k = 0 , and, therefore, t he KLD cannot be made zero. In the case of P k j =1 α j = 0 . 5 , there exists a unique solution ( P j 0 , 0 , P j 1 , 0 ) = (1 , 1) , ∀ j that can make D k = 0 , ∀ k . For the P k j =1 α j > 0 . 5 case, th ere exist infinitely many attacking probability distributions ( P j 0 , 1 , P j 1 , 0 ) which can make D k = 0 , ∀ k . Lemma 1: In a tree network with K lev el s, the mini mum number of Byzantin es needed to blind the FC (or to make D k = 0 , ∀ k ) is given by B 1 =  N 1 2  . Pr oof: The proof follows from the fact th at the condi tion P k j =1 α j = 0 . 5 , ∀ k , is equiv alent to α 1 = 0 . 5 , α k = 0 , ∀ k = 2 , · · · , K . Next, we explore the optimal attacking probabil ity distribution ( P k 0 , 1 , P k 1 , 0 ) that mi nimizes D k when P k j =1 α j < 0 . 5 , i.e., in t he case where the attacker cannot make D = 0 . T o analyze July 10, 2018 DRAFT 9 the problem, first we in vestigate the properties of D k with respect to ( P k 0 , 1 , P k 1 , 0 ) assuming ( P j 0 , 1 , P j 1 , 0 ) , 1 ≤ j ≤ k − 1 t o be fixe d. W e show that attacking with s ymmetric flipping probabilities i s the opti mal st rate g y in the region where t he att ack er cannot make D k = 0 . In other words, attacking with P k 1 , 0 = P k 0 , 1 is the o ptimal st rate gy for the Byzantines. Lemma 2: In the region where the attacker cannot make D k = 0 , i.e., for P k j =1 α j < 0 . 5 , the op timal attacking st rate g y comprises of symm etric flipping probabil ities ( P k 0 , 1 = P k 1 , 0 = p ) . In other words, any non zero deviation ǫ i ∈ (0 , p ] i n flipping probabilities ( P k 0 , 1 , P k 1 , 0 ) = ( p − ǫ 1 , p − ǫ 2 ) , where ǫ 1 6 = ǫ 2 , will result in an increase in D k . Pr oof: Please see Append ix A. In t he next theorem, we present the solut ion for the optimal attackin g probabilit y di strib ution ( P k j, 1 , P k j, 0 ) that m inimizes D k in the region where the at tack er canno t make D k = 0 . Theor em 1: In the region where the attacker cannot make D k = 0 , i.e., for P k j =1 α j < 0 . 5 , the optimal att acking strategy i s given by ( P k 0 , 1 , P k 1 , 0 ) = (1 , 1) . Pr oof: Observe that, i n t he re gion where the attacker cannot make D k = 0 , the optimal strategy comprises of symmetric flipping probabili ties ( P k 0 , 1 = P k 1 , 0 = p ) . The p roof is complete if we sho w that D k is a monotonically decreasing function of the flip ping probabili ty p . After plugging in ( P k 0 , 1 , P k 1 , 0 ) = ( p, p ) in (9) and (10 ), we get π k 1 , 1 = [ β k − 1 1 , 0 (1 − P k d ) + (1 − β k − 1 0 , 1 ) P k d ] + [ α k ( p − P k d (2 p )) + P k d ] (11) π k 1 , 0 = [ β k − 1 1 , 0 (1 − P k f a ) + (1 − β k − 1 0 , 1 ) P k f a ] + [ α k ( p − P k f a (2 p )) + P k f a ] . (12) Now we show that D k is a monotonically decreasing functio n of the parameter p or in other words, dD k dp < 0 . After pl ugging in π k ′ 1 , 1 = α k (1 − 2 P k d ) and π k ′ 1 , 0 = α k (1 − 2 P k f a ) in the expression of dD k dp and rearranging the terms, the condition dD k dp < 0 becomes (1 − 2 P k d ) 1 − π k 1 , 0 1 − π k 1 , 1 − π k 1 , 0 π k 1 , 1 ! + (1 − 2 P k f a ) log 1 − π k 1 , 1 1 − π k 1 , 0 π k 1 , 0 π k 1 , 1 ! < 0 (13) July 10, 2018 DRAFT 10 0 0.5 1 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 P 0,1 P 1,0 KLD 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Fig. 2. KLD D k vs. fl ipping probabilities when P k d = 0 . 8 , P k f a = 0 . 2 , and the probability that the bit coming f rom lev el k encounters a Byzantine is P k j =1 α j = 0 . 4 . Since P k d > P k f a and β k ¯ x,x < 0 . 5 , we ha ve π k 1 , 1 > π k 1 , 0 . No w , using t he f act th at 1 − P k d 1 − P k f a > 1 − 2 P k d 1 − 2 P k f a and (33), we have 1 − 2 P k d 1 − 2 P k f a " 1 − π k 1 , 0 1 − π k 1 , 1 − π k 1 , 0 π k 1 , 1 # < ( π k 1 , 1 − π k 1 , 0 ) " 1 π k 1 , 1 + 1 1 − π k 1 , 0 # (14) ⇔ 1 − 2 P k d 1 − 2 P k f a " 1 − π k 1 , 0 1 − π k 1 , 1 − π k 1 , 0 π k 1 , 1 # + " π k 1 , 0 π k 1 , 1 − 1 # < 1 − 1 − π k 1 , 1 1 − π k 1 , 0 . (15) Applying the l ogarithm inequ ality ( x − 1) ≥ log x ≥ x − 1 x , for x > 0 to (15), one can prove that (13) i s true. Next, to gain in sights into the solution , we present some num erical results in Figure 2. W e plot D k as a functio n of the flipping probabilit ies ( P k 1 , 0 , P k 0 , 1 ) . W e assume that the probability of detection i s P k d = 0 . 8 , t he prob ability o f false alarm is P k f a = 0 . 2 , and t he prob ability t hat the bit coming from level k encounters a Byzantine is P k j =1 α j = 0 . 4 . W e also assume t hat P k 0 , 1 = P 0 , 1 and P k 1 , 0 = P 1 , 0 , ∀ k . It can be seen that the optimal attacking strategy comprises of sy mmetric flipping prob abilities and is give n by ( P k 0 , 1 , P k 1 , 0 ) = (1 , 1) , which corroborates our theoretical result presented i n Lemma 2 and Theorem 1. W e hav e shown t hat, for all k , D k ( P k 0 , 1 , P k 1 , 0 ) ≥ D k (1 , 1) . (16) July 10, 2018 DRAFT 11 Now , by multipl ying both sides of (16) b y N k and s umming it over all K we can show t hat t he KLD, D , is minimi zed by ( P k 0 , 1 , P k 1 , 0 ) = (1 , 1) , for all k , in the region K P k =1 α k < 0 . 5 . Now , we e xp lore some properties of D k with respect to P k j =1 α j in the re g ion where the attacker cannot make D k = 0 , i.e., for P k j =1 α j < 0 . 5 . Thi s analysis will later b e used in exploring the problem from t he network designer’ s perspective . Lemma 3: D ∗ k = min ( P k j, 1 ,P k j, 0 ) D k ( π k j, 1 || π k j, 0 ) i s a continuous , decre asing and con vex function of P k j =1 α j for P k j =1 α j < 0 . 5 . Pr oof: The continuit y o f D k ( π k j, 1 || π k j, 0 ) with respect t o t he inv olved di strib utions implies the continuity of D ∗ k . T o show that D ∗ k is a decreasing function of t = P k j =1 α j , we use t he fact that ar g min ( P k 0 , 1 ,P k 1 , 0 ) D k ( π k j, 1 || π k j, 0 ) is equal to (1 , 1) for P k j =1 α j < 0 . 5 (as shown in Theorem 1). After plugging ( P k 0 , 1 , P k 1 , 0 ) = (1 , 1) , ∀ k , in the K LD expression, i t can be sh o wn t hat dD k dt < 0 . Hence, D ∗ k is a monot onically decreasing function of P k j =1 α j for P k j =1 α j < 0 . 5 . The con vexity of D ∗ k follows from the f act that D ∗ k ( π k j, 1 || π k j, 0 ) is con vex in π k j, 1 and π k j, 0 , which are affine transform ations of P k j =1 α j (Note t hat, con ve xity ho lds under af fine transformati on). It i s worth no ting that Lemm a 3 suggests that minim ization/maximization o f P k j =1 α j is equiv alent t o m inimization/maxim ization of D k . Using this fact, one can consider the pro bability that the bit com ing from le vel k encounters a Byzantine (i.e., t = P k j =1 α j ) in li eu of D k for optimizing the syst em performance. Observe that, t he expression t = P k j =1 α j is much more tractable than th e expression for D k . Next, t o gain insights into t he s olution, we present some nu merical results in Figure 3. W e plot min ( P k j, 1 ,P k j, 0 ) D k as a function of the probabil ity that the bit coming from level k encounters a Byzantine, i.e., t . W e assume that th e probabi lities of det ection and false alarm are P k d = 0 . 8 and P k f a = 0 . 2 , respectively . Not ice that, when t = 0 . 5 , D k between t he two p robability distributions becomes zero. It is seen that D ∗ k is a cont inuous, decreasing and con vex functi on o f the fraction of covered no des, t , for t < 0 . 5 , wh ich corroborates o ur th eoretical result presented in Lemma 3. Until now , we ha ve explored the probl em from the attacker’ s perspective. In t he rest of the paper , we look i nto the problem from a network designer’ s perspective and propose techniques to mi tigate the effe ct of Byzantines. First, we st udy the problem of designing optim al di strib uted detection parameters i n a tree network in the presence of Byzantines. July 10, 2018 DRAFT 12 0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 t minKLD Fig. 3. min ( P k j, 1 ,P k j, 0 ) D k vs probability that the bit coming from lev el k encounters a Byzantine for P k d = 0 . 8 and P k f a = 0 . 2 . I V . S Y S T E M D E S I G N I N T H E P R E S E N C E O F B Y Z A N T I N E S For a fixed attack configuration { B k } K k =1 , the d etection performance at the FC is a function of the local detectors used at the nodes in the tree network and the glo bal detector used at the FC. This motivates us t o stu dy the problem of desi gning detectors, both at the no des at diffe rent levels in a tree and at the FC, such that the detection performance is maximized. More specifically , we are int erested in answering the question: How does the knowledge of the attack configuration { B k } K k =1 af fect the d esign of optimal distributed detection parameters? By Stein’ s l emma [17], we know that in the NP setup for a fixed false alarm probability , t he missed detection prob ability of the opt imal detector can be m inimized by maximizing the KL D. For an optim al detector at t he FC, the probl em of designi ng the local detectors can b e formalized as follows: max { P k d ,P k f a } K k =1 K X k =1 N k X j ∈{ 0 , 1 } P ( z i = j | H 0 , k ) log P ( z i = j | H 0 , k ) P ( z i = j | H 1 , k ) . (17) The l ocal detector design problem as giv en in (17 ) is a non-linear optimi zation problem. Furthermore, it is diffi cult to obtain a closed form sol ution for t his p roblem. Also, observe that the solution space is n ot const rained to t he likelihood ratio based test s. T o solve the problem , we need to find the pairs { P k d , P k f a } K k =1 which maximize the objective function as given in (17). Howe ver , P k d and P k f a are coupled and, therefore, cannot be optimized independentl y . Thu s, we first analyze the prob lem of maxim izing t he KLD for a fixed P k f a . W e assume that P k f a = y k July 10, 2018 DRAFT 13 and P k d = y k + x k . Next, we analyze the properties of KLD with respect to x k , i. e., ( P k d − P k f a ) in the region where attacker cannot blind the FC, i .e., for P k j =1 α j < 0 . 5 , in order to study t he local detector design pro blem. Notice that, in the region P k j =1 α j ≥ 0 . 5 , D k = 0 and optimizi ng over l ocal detectors does n ot im prov e the performance. Lemma 4: For a fixed P k f a = y k , when P k j =1 α j < 0 . 5 , t he KLD, D , as given in (7) is a monotonicall y increasing function of x k = ( P k d − P k f a ) . Pr oof: T o prove this, we calcul ate the partial deri va tive of D with respect to x k . By substitut ing P k f a = y k and P k d = y k + x k into (7) , the partial deriv ativ e of D with respect to x k can be calculated as ∂ D ∂ x k = N k ∂ ∂ x k " π k 1 , 0 log π k 1 , 0 π k 1 , 1 + (1 − π k 1 , 0 ) log 1 − π k 1 , 0 1 − π k 1 , 1 # ⇔ ∂ D ∂ x k = N k π k ′ 1 , 1 1 − π k 1 , 0 1 − π k 1 , 1 − π k 1 , 0 π k 1 , 1 ! , where π k 1 , 0 and π k 1 , 1 are as giv en in (9) and (10), respectively and π k ′ 1 , 1 = (1 − β k 0 , 1 − β k 1 , 0 ) . Notice that, 1 − π k 1 , 0 1 − π k 1 , 1 − π k 1 , 0 π k 1 , 1 ! > 0 ⇔ π k 1 , 1 > π k 1 , 0 . Thus, the condition to make ∂ D ∂ x k > 0 sim plifies t o π k ′ 1 , 1 > 0 ⇔ 1 > ( β k 0 , 1 + β k 1 , 0 ) (18) Substitutin g the values of β k 1 , 0 and β k 1 , 1 , the abov e condi tion can be w ritten as: k X j =1 α j P j 1 , 0 + k X j =1 α j P j 0 , 1 < 1 (19) ⇔ k X j =1 α j ( P j 1 , 0 + P j 0 , 1 ) < 1 (20) The above conditi on i s t rue for any 0 ≤ P j 0 , 1 , P j 1 , 0 ≤ 1 when P k j =1 α j < 0 . 5 . Thi s com pletes the July 10, 2018 DRAFT 14 proof. Lemma 4 sugg ests that one possi ble s olution to maxi mize D is to choose the largest p ossible x k constrained t o 0 ≤ x k ≤ 1 − y k . The upper bound results from the fact that { P k d , P k f a } K k =1 are probabil ities and, thus, must be between zer o and one. In other w ords, the solution is to maximize the probability o f detectio n for a fixed value of probability of false alarm. In detection theory , it is well known t hat the likelihood ratio based t est is op timum for thi s criterion. Thus , under t he condi tional independence assumption, likelihood ratio based test as gi ven in (6) is optimal for local no des, e ven in the presence of Byzantines, and t he o ptimal operating po ints { P k ∗ d , P k ∗ f a } K k =1 are i ndependent of the Byzantines’ parameters { α k } K k =1 . T o summarize, optimal l ocal detectors for dis trib u ted detection in tree networks are likelihood ratio based detectors and are i ndependent of the Byzantines’ parameter { α k } K k =1 . W e further explore the problem from the network designer’ s (FC) perspectiv e. In our previous analysis, we hav e assumed th at the attack con figuration { B k } K k =1 is known and shown that the opti mal local detector i s i ndependent of { α k } K k =1 . Howev er , notice that the KLD is the exponential decay rate of t he error probability o f the optimal detector. In other words, while opt imizing over KLD, we implicitl y assu med that the optimal detector , which is a likelihood ratio based detector , is used at the FC. T aking logarithm o n both sides of (6) , the optimal decision rule si mplifies to K X k =1 [ a k 1 s k + a k 0 ( N k − s k )] H 1 ≷ H 0 log η (21) where the optimal weights are gi ven by a k 1 = log π k 1 , 1 π k 1 , 0 and a k 0 = log 1 − π k 1 , 1 1 − π k 1 , 0 . T o implement the optimal detector , the FC needs to know the op timal weights a k j , which are functi ons of { α k } K k =1 . In t he next section, we are int ere sted in answering the question: Is i t possible for the FC t o predict the attack configuration { B k } K k =1 in the tree? The knowledge of t his attack configuration can be used for determin ing the optim al d etector at the FC to i mprov e the system performance. Notice that, learning/est imation based techniques can be us ed on data t o determine the attack configuration. Howe ver , the FC h as to acquire a l ar ge am ount o f data coming from the no des over a long period of time to accurately estim ate { B k } K k =1 . In th e next section, we propose a novel t echnique to predict the attack configuration by considering the following scenario: Th e F C, acting first, comm its to a defensive strategy by deploying the defensive resou rces to protect the tree network, while the attacker chooses it s best July 10, 2018 DRAFT 15 response or attack configuration after surveillance of thi s defensiv e strategy . Both, the FC and the Byzantines ha ve to incur a cost to deploy th e defensive resources and attack the nodes in the tree net w ork, respectively . W e cons ider bo th the FC and the attacker to be strategic in nature and model the st rate gi c interaction between t hem as a Leader -Follo wer (Stackelber g) gam e. This formulation provides a framework for ident ifying attacker and defender (FC) equilibrium strategies, which can be used to implement the op timal detector . The main adv antage of thi s technique is that t he equ ilibrium strategies can b e d etermined a pri ori and, th ere fore, t here is no n eed to observe a l ar ge amount of data coming from the nodes over a l ong period of time to accurately estimate { B k } K k =1 . V . S T AC K E L B E R G G A M E F O R A T TAC K C O N FI G U R A T I O N P R E D I C T I O N P RO B L E M S W e model th e strate gic interaction between the FC and the attacker as a Leader -Follower (Stackelber g) game. W e assum e t hat the FC has to incur a cost for deploying the n etw ork and the B yzantine has to incur a cost 7 for attacki ng the network. It is assu med that the network designer or the FC has a cost budget C netw ork budg et and t he attacker h as a cost budget C attacker budg et 8 . More specifically , the FC wants to allocate the best subset of defensive resources (denoted as { ˜ c k } K k =1 ) 9 from a set of av ailable defensive resources C = ( c 1 , · · · , c n ) (arranged in a d escending order , i.e., c 1 ≥ c 2 · · · ≥ c n ), where n ≥ K , complying with i ts budget constraint C netw ork budg et to diffe rent levels of the tree netw ork. After the FC allocates the defensive resources or budget to differe nt le vels of the tree network, an attacker chooses an attack configuration, { B k } K k =1 complying with his budget cons traint C attacker budg et to maximally d e grade the performance of the network. Next, we formali ze the Stackelber g game as a bi -le vel opt imization problem . For our prob lem, the up per lev el problem (ULP) corresponds to the FC who is the leader of the game, whi le th e 7 Due to variations in hardware complex ity and the lev el of tamper-resistance present in node s residing at different l e vels of t he tree, the resources required to capture and tamper nodes at different lev els may be different and, t herefo re, nodes have v arying costs of being att ack ed. 8 In this paper, we assume that the attack er budget C attack er budget is such that K P k =1 α k < 0 . 5 , i.e., the attacker canno t make D k = 0 , ∀ k . Notice that, if the attacker can make D k = 0 for some k = l , then, i t can also make D k = 0 , ∀ k ≥ l . Also, D k = 0 implies that π k 1 , 1 = π k 1 , 0 and, therefore, the weights ( a k 1 , a k 0 ) in (21) are zero. In other words, the best the FC can do in t he case when D k = 0 , ∀ k ≥ l is to ignore or discard the decisions of t he nodes residing at lev el k ≥ l . This scenario is equi v alent to using t he tree network with ( l − 1) leve ls for distributed detection. 9 Let ˜ c k denote the resources deployed or budget al located by the FC to protect or deploy a node at level k . July 10, 2018 DRAFT 16 lower l e vel problem (LLP) belongs to the attacker who is the foll o wer . maximize { ˜ c k } K k =1 ∈ C D ( { ˜ c k } K k =1 ) subject to K X k =1 ˜ c k N k ≤ C netw ork budg et minimize B k ∈ Z + D ( { B k } K k =1 ) subject to K X k =1 ˜ c k B k ≤ C attacker budg et 0 ≤ B k ≤ N k , ∀ k = 1 , 2 , . . . , K (22) where Z + is the set of non-negati ve in te gers. Not ice that, the bi-level optimi zation pro blem, in general, is an NP-hard problem. In fac t, the L LP is a variant of t he packing formulation of t he bounded k napsack problem with a non-linear ob jecti ve function. This is, in general, NP-hard. Using existi ng al gorithms, cost set { ˜ c k } K k =1 and attack configuration { B k } K k =1 can be determined at the cost of computation al ef ficiency . In this paper , we ident ify a sp ecial case of the above problem whi ch can be solved in polynom ial time to determin e the equi librium strategies. Next, we discuss the relationships that enable our problem to have a polynom ial time solution. W e define profit P ( S ) of an attack con figuration S = { B k } K k =1 as follows 10 P ( S ) = D ( φ ) − D ( S ) = D ( φ ) − D ( { B k } K k =1 ) , where D ( φ ) is the KLD when t here ar e no Byzantines in the network and D ( S ) = D ( { B k } K k =1 ) is the KLD with { B k } K k =1 Byzantines in the tree network. Next, we define the concept of dominance which will be used later to explore some us eful properties of the optim al attack configuratio n { B k } K k =1 . Definition 1: W e say t hat a set S 1 dominates ano ther set S 2 if P ( S 1 ) ≥ P ( S 2 ) and C ( S 1 ) ≤ C ( S 2 ) , (23) where P ( S i ) and C ( S i ) denote the profit and cost incurred by using s et S i , respectively . If 10 In this section, we assume that the optimal operating point, i.e., ( P k ∗ d , P k ∗ f a ) , is the same for all the nodes in the tree network. It has been shown that the use of identical thresholds is asymptotically optimal for parallel networks [21 ]. W e conjecture that this result is valid for tree networks as well and employ identical thresholds. July 10, 2018 DRAFT 17 in (23), P ( S 1 ) > P ( S 2 ) , S 1 strictly dominates S 2 and if P ( S 1 ) = P ( S 2 ) , S 1 weakly dominates S 2 . T o so lve the bi-level optimization problem, we first sol v e the LLP assum ing the soluti on of the ULP to be some fixed (˜ c 1 , · · · , ˜ c K ) . This approach will gi ve us a st ructure of the optimal { B k } K k =1 for any arbitrary { ˜ c k } K k =1 , which can later be utilized to solve t he bi -le vel opt imization problem. W e refer to LLP as a maximum damage Byzantin e attack problem. Observe t hat, knowing th at the FC choos es ( ˜ c 1 , · · · , ˜ c K ) , the LLP can be reformulated as follows: minimize { B k } K k =1 K X k =1 N k D k ( { B i } k i =1 ) subject to K X k =1 ˜ c k B k ≤ C attacker budg et 0 ≤ B k ≤ N k , and integer ∀ k Next, we dis cuss the relations hips that enable our m aximum damage Byzantine attack probl em to admit a polyn omial time solution. A. Analysis o f the Optimal Attack Configuration In this section, we identify a special case of the bounded kn apsack problem (LLP) which can be solved i n p olynomial ti me. More specifically , we show t hat i f the set o f defensive resources C = ( c 1 , · · · , c n ) satisfy th e cost structure c max ≤  min k ∈{ 1 , ··· ,K − 1 } N k +1 N k  × c min 11 or c 1 ≤ min k a k × c n , then, the o ptimal solutio n { B k } K k =1 exhibits the properties giv en in the lemma b elo w . Lemma 5: Given a K level t ree network wi th cost structure satisfying c max ≤  min k ∈{ 1 , ··· ,K − 1 } N k +1 N k  × c min , the best respons e of an attacker wit h cost b udget C attacker budg et is { B k } K k =1 with B 1 = j C attack er budget ˜ c 1 k and the remaini ng elements of B k for 2 ≤ k ≤ K ca n be calculated recursi vely . Pr oof: T o prove Lemma 5, it is sufficient to show that: 1) KLD is a monoto nically decreasing function of B k , and, 2) Attacking p are nt nodes is a s trictly dominant strategy . 11 Notice that, in the case of the perfect M -ary tree networks, the proposed cost structure simplifies to c max ≤ M × c min . July 10, 2018 DRAFT 18 Lemma 3 suggests that the KLD is a monotonically decreasing function of B k in the region where attacker cannot make D k = 0 and, t herefore , (1) is proved. Next, we sho w that attacking parent nod es is a st rictly do minant strategy . In other words, giv en a cost budget C attacker budg et , it is more p rofitable for an attacker to attack the parent nod es. Observe that the KLD at level k is a function of Byzantines’ parameter ( B 1 , · · · , B k ) . Thus, we denot e it as D k ( B 1 , · · · , B k ) . In order to prov e that attacking parent nodes is a strictl y dom inant strategy , it is sufficient t o show that the att ack configuration S 1 = ( B 1 , · · · , B j , B j +1 , · · · , B K ) strictly dominates the attack configuration S 2 = ( B 1 , · · · , B j − δ, B j +1 + δ N j +1 N j , · · · , B K ) for δ ∈ { 1 , · · · , B j } . In other words, we want to show t hat P ( S 1 ) > P ( S 2 ) and C ( S 1 ) ≤ C ( S 2 ) . From the cost inequal ity it follows that C ( S 1 ) ≤ C ( S 2 ) because c max ≤ (min k N k +1 /N k ) × c min ⇒ ˜ c j ≤ ( N j +1 /N j ) × ˜ c j +1 . Also, note that if t he attack configuration S 1 strictly dominates t he attack configuration S 2 , then, it will also strictly domin ate any attack configuration ˜ S 2 with ˜ S 2 = ( B 1 , · · · , B j − δ, B j +1 + δ γ , · · · , B K ) , where γ ≤ N j +1 N j . Next, we sho w t hat P ( S 1 ) > P ( S 2 ) . Since D j ( B 1 , · · · , B j − 1 , B j ) < D j ( B 1 , · · · , B j − 1 , B j − δ ) , for δ ∈ { 1 , · · · , B j } , ∀ j , it follows that D j ( B 1 , · · · , B j − 1 , B j ) < D j ( B 1 , · · · , B j − 1 , B j − δ ) ⇔ j X k =1 D k ( B 1 , · · · , B k ) < j − 1 X k =1 D k ( B 1 , · · · , B k ) + D j ( B 1 , · · · , B j − 1 , B j − δ ) ⇔ K X k =1 D k ( B 1 , · · · , B k ) < j − 1 X k =1 D k ( B 1 , · · · , B k ) + D j ( B 1 , · · · , B j − 1 , B j − δ ) + K X k = j +1 D k ( B 1 , · · · , B j − δ, B j +1 + δ N j +1 N j , B j +2 , · · · , B k ) , where the last inequality follows from t he f act that B j N j + B j +1 N j +1 = B j − δ N j + B j +1 + N j +1 N j δ N j +1 and, therefore, D k ( B 1 , · · · , B j , B j +1 , · · · , B k ) = D k ( B 1 , · · · , B j − δ, B j +1 + N j +1 N j δ , · · · , B k ) . This implies th at the set S 1 strictly dominates t he set S 2 . From the results in Lemma 3, it is seen that th e profit is an increasing function o f the attack nodes. Lem ma 3 in conjunction wit h the fact that attacking parent nodes is a strictly dominant strategy implies Lemma 5. It can also be shown that the s olution { B k } K k =1 will be non-overlapping and unique under the condition that the attacker cannot make D k = 0 , ∀ k . July 10, 2018 DRAFT 19 B. Bi-Level Optimizati on A lgorithm Based on Lemma 5, in this section we will present a polynomial time algorithm to sol v e the bi-lev el opti mization problem, i.e., to find { ˜ c k } K k =1 and { B k } K k =1 . Using the cost structure c max ≤  min k N k +1 N k  × c min , the at tack configuration { B k } K k =1 as given in Lem ma 5 can be determined in a com putationally ef ficient m anner . Due to s tructure of t he opti mal { B k } K k =1 , th e bi-lev el optim ization problem simplifies to finding the solution { ˜ c k } K k =1 of the ULP . T o sol v e th is problem, we use an iterative elimination approach. W e start by li sting all  n K  combinations from the set C , denoted as, S = { s i } ( n K ) i =1 . W ithout loss of generality , we assume that the element s of s i = { c i 1 , · · · , c i K } are arr anged in descending order , i.e., c i k ≥ c i k +1 , ∀ k . Notice t hat, all these  n K  combinations will satisfy c i k ≤ N k +1 N k c i k +1 , because c i k ≤ c max ≤ min j N j +1 N j c min ≤ min j N j +1 N j c i k +1 ≤ N k +1 N k c i k +1 . Next, we dis car d all those su bsets s i from S which violate the network designer’ s cost budget constraint. If the s et S is empty , then t here do es not exist any so lution for the ULP . Otherwise, the problem reduces to finding the subset s i which maximizes the KLD. T o find the s ubset s i which maximi zes the KLD, using the domi nance relationshi p we start with assigning the cost ˜ c 1 = min k ∈ s c k 1 , where s has th e elements which are solutions of arg min i $ C attacker budg et c i 1 % . Next, we discard all those subsets s i from S which do no t have ˜ c 1 as their first element and sol v e the problem recursi vely . Pseudo code of t he polynomial ti me algorithm to find { ˜ c k } K k =1 and { B k } K k =1 is presented as Algorithm 1. C. An Illustrative Examp le Let us consider a tw o-lev el network with N 1 = 6 and N 2 = 12 . W e assume that C = { 4 , 3 , 2 } , C netw ork budg et = 60 and C attacker budg et = 11 . Next, we solve the bi-level opti mization problem. Observe that, cost s satisfy c 1 ≤ 2 × c 3 . So the algorithm chooses the solution of the ULP as ( ˜ c 1 = 4 , ˜ c 2 = 3 ) and the sol ution of the LLP as ( B 1 =  11 4  = 2 , B 2 =  11 − 2 × 4 3  = 1 ). T o corroborate these result, in Figure 4 , we plo t the min P 1 , 0 ,P 0 , 1 KLD for all combinations of the parameters B 1 and B 2 in the tree. W e var y the parameter B 1 from 0 to 6 and B 2 from 0 to 12 . All t he feasible solutions are pl otted i n red and unfeasible soluti ons are plot ted in b lue. Figure 4 corroborates July 10, 2018 DRAFT 20 Algorithm 1 Bi-Level Optimization Algorithm Requir e: C = { c k } n k =1 with c max ≤  min j N j +1 N j  × c min 1: S ← All K out of n combinations { s i } ( n K ) i =1 with elements o f s i arranged in decreasing order 2: f or i = 1 to  n K  do 3: if K P k =1 c i k × N k > C netw ork budg et then 4: S ← S/s i 5: end if 6: end f or 7: if S is an empty set then 8: r eturn ( φ, φ ) 9: else 10: f or k = 1 to K do 11: ˜ c k = min j ∈ s c j k where s has element s which are solutions of arg min i $ C attacker budg et c i k % 12: B k ← j C attack er budget ˜ c k k 13: C attacker budg et ← ( C attacker budg et − ˜ c k B k ) 14: end f or 15: r etur n ( { ˜ c k } K k =1 , { B k } K k =1 ) 16: end i f the results of our algorithm . Notice t hat, t he attack configuration { B k } K k =1 is the set containing t he number of Byzantines residing at differ ent le vels of the t ree . Ho we ver , the FC cannot id entify the Byzantines in the network. Also, notice that when the adversary attacks more than 50% of nodes at le vel 1 , the decision fusi on s cheme becomes completely in capable. In these scenarios, where the FC is blind, the knowledge of attack configuratio n will not incur any performance benefit. Next, we present a reputation-based Byzantine i dentification/mitigation schem e, wh ich works e ven when July 10, 2018 DRAFT 21 Fig. 4. min KLD vs. attack configuration ( B 1 , B 2 ) for P d = 0 . 9 , P f a = 0 . 1 . the network is blind, i n order to im pro ve the detection performance of t he network. W e propose a simple yet efficient Byzantine identification scheme and analyze its performance. V I . A N E FFI C I E N T B Y Z A N T I N E I D E N T I FI C A T I O N S C H E M E In t his section, we propose and analyze a Byzantine ident ification s cheme to be imp lemented at the FC. A. Byzantine Identi fication Scheme W e assume that the FC h as the knowledge of t he attack model and ut ilizes this knowledge to identify th e Byzantines. The FC observes the local decisions of each node ov er a time window T , which can be denoted by ( k , i ) = [ u 1 ( k , i ) , . . . , u T ( k , i )] for 1 ≤ i ≤ N k at l e vel 1 ≤ k ≤ K . W e als o assume that th ere is one honest anchor no de with probability of det ection P A d and probability of f alse alarm P A f a present and kno wn to th e FC. W e em ploy the anchor node to provide the g old standard wh ich is used to detect whether or not other nodes are Byzantines. The FC can also s erv e as an anchor no de when i t can di rectly observe the pheno menon and m ak e a decisi on. W e denot e the Hammi ng dist ance b etween reports of the anchor node and an honest node i at level k over the tim e window T by d A H ( k , i ) = || U A − U H ( k , i ) | | , that is t he number of elements th at are di f ferent between U A and U H ( k , i ) . Simi larly , the H amming d istance b etween reports of the anchor node and a Byzantine node i at level k ove r t he time window T is d enoted by d A B ( k , i ) = || U A − U B ( k , i ) | | . Since the FC is aware of the fact that Byzantines migh t be July 10, 2018 DRAFT 22 present i n the network, it compares the Hamm ing distance of a node i at le vel k to a thresh old η k , ∀ i, ∀ k (a procedure to calculate η k is discussed later in the p aper) , to make a decision to identify the Byzantines. In tree networks, a Byzantine node alt ers its decision as well as received decisions from it s chi ldren p rior to transmi ssion in order to undermine the network performance. Therefore, sol ely based on t he o bserv ed data of a n ode i at le vel k , the FC cannot determine whether th e data has been flipped by t he node i itself or by on e of its Byzantine parent n ode. In ou r scheme, the FC makes the inference about a node being Byzantine by analyzing the data from the no de i as well as its p redec essor nodes’ data. FC starts from th e nodes at le vel 1 and computes th e Hamming d istance between report s of the anchor node and the nod es at l e vel 1 . FC declares n ode i at le vel 1 t o be a Byzantine i f and only if t he Hamming distance of node i is greater than a fixed thresho ld η 1 . Chil dren of identified Byzantine nodes C ( B 1 ) are not tested further because of the non-ov erlapping condi tion. Howe ver , if a le vel 1 node is determined not to be a Byzantine, then, the FC tests its children nodes at level 2 . The FC declares node i at lev el k , for 2 ≤ k ≤ K , to be a Byzantin e if and only if the Ham ming distance of node i is greater than a fixed t hreshold η k and Hammi ng di stances of all predecessors of no de i is less than equ al to their respectiv e thresho lds η j . In t his way , it is poss ible to counter t he data falsification attack by isolat ing Byzantine nodes from the i nformation fusion p rocess. The p robability that a Byzantine node i at lev el k is is olated at the end of the time window T , is denoted as P iso B ( k , i ) . B. P erfor mance Analysis As aforementioned, local decision s of the nodes are compared t o th e decisions of the anchor node over a tim e window of length T . The probabili ty t hat an honest no de i at level k makes a decision that is di f ferent from the anchor node is giv en by P AH dif f ( k , i ) = P ( u A i = 1 , u H k ,i = 0 , H 0 ) + P ( u A i = 0 , u H k ,i = 1 , H 0 ) + P ( u A i = 1 , u H k ,i = 0 , H 1 ) + P ( u A i = 0 , u H k ,i = 1 , H 1 ) = P 0 [( P k f a + P A f a ) − 2 P k f a P A f a ] + P 1 [( P k d + P A d ) − 2 P k d P A d ] . = P 0 [ P AH dif f ( k , i, 0)] + P 1 [ P AH dif f ( k , i, 1)] . July 10, 2018 DRAFT 23 where the prior probabi lities of th e two hy potheses H 0 and H 1 are denoted by P 0 and P 1 , respectiv ely . The probability that a Byzantine node i at level k sends a decision different from that of th e anchor node is giv en by P AB dif f ( k , i ) = P ( u A i = 1 , u B k ,i = 0 , H 0 ) + P ( u A i = 0 , u B k ,i = 1 , H 0 ) + P ( u A i = 1 , u B k ,i = 0 , H 1 ) + P ( u A i = 0 , u B k ,i = 1 , H 1 ) = P 0 [ P A f a P k f a + (1 − P A f a )(1 − P k f a )] + P 1 [ P A d P k d + (1 − P A d )(1 − P k d )] . = P 0 [ P AB dif f ( k , i, 0)] + P 1 [ P AB dif f ( k , i, 1)] . The difference between the reports of a node and th e anchor node under hyp othesis l ∈ { 0 , 1 } (i.e., d A I ( k , i, l ) , I ∈ { H , B } ) is a Bernoulli random variable with m ean P AH dif f ( k , i, l ) for honest nodes and P AB dif f ( k , i, l ) for Byzantines. FC declares node i at lev el k to be a Byzantine if and only i f the Hamming di stance of nod e i is greater than a fixed threshold η k and Hammi ng distances of all predecessors of node i are less than equal to thei r respectiv e thresho lds η j . The probability that a Byzantine node i at level k i s isolated at th e end of the ti me window T can be expressed as P iso B ( k , i ) = P [( d A B ( k , i ) > η k ) , ( d A H ( k − 1 , i ) ≤ η k − 1 ) , · · · , ( d A H (1 , i ) ≤ η 1 )] = X l ∈{ 0 , 1 } P l " P [ d A B ( k , i, l ) > η k ] k − 1 Y m =1 P [ d A H ( m, i, l ) ≤ η m ] # = X l ∈{ 0 , 1 } P l   T X j = η k +1  T j  ( P AB dif f ( k , i, l )) j (1 − P AB dif f ( k , i, l )) T − j k − 1 Y m =1   η m X j =0  T j  ( P AH dif f ( m, i, l )) j (1 − P AH dif f ( m, i, l )) T − j     . For large T , by using the normal approximation , we get P iso B ( k , i ) = X l ∈{ 0 , 1 } P l   Q   η k − T P AB dif f ( k , i, l ) q ( T P AB dif f ( k , i, l )(1 − P AB dif f ( k , i, l )))   k − 1 Y m =1 Q   T P AH dif f ( m, i, l ) − η m q ( T P AH dif f ( m, i, l )(1 − P AH dif f ( m, i, l )))     . This can be writt en recursiv ely as follows P iso B ( k + 1 , i ) = X l ∈{ 0 , 1 } P l  (1 − b ( k , l ))  a ( k + 1 , l ) a ( k , l )  P iso B ( k , i, l )  , (24) July 10, 2018 DRAFT 24 with P iso B ( k , i ) . = X l ∈{ 0 , 1 } P l [ P iso B ( k , i, l )] , and a ( k , l ) = Q   η k − T P AB dif f ( k , i, l ) q ( T P AB dif f ( k , i, l )(1 − P AB dif f ( k , i, l )))   , b ( k , l ) = Q   η k − T P AH dif f ( k , i, l ) q ( T P AH dif f ( k , i, l )(1 − P AH dif f ( k , i, l )))   . One can choose η k such that the isolation p robability of ho nest nodes at lev el k based sol ely on its data under th e hypoth esis H l (i.e., b ( k , l ) ) is constrained to some value δ k << 0 . 5 . In other words, we choose η k such that max l ∈{ 0 , 1 } b ( k , l ) = δ k , i.e., η k = Q − 1 ( δ k ) q T P AH dif f ( k , i, l ∗ )(1 − P AH dif f ( k , i, l ∗ )) + T P AH dif f ( k , i, l ∗ ) (25) where l ∗ = arg max l b ( k , l ) . No w , the expression for a ( k , l ) can be written as a ( k , l ) = Q   Q − 1 ( δ k ) q P AH dif f ( k , i, l ∗ )(1 − P AH dif f ( k , i, l ∗ )) + √ T ( P AH dif f ( k , i, l ∗ ) − P AB dif f ( k , i, l )) q P AB dif f ( k , i, l )(1 − P AB dif f ( k , i, l ))   Now using th e fac t that max l P AH dif f ( k , i, l ) < min l P AB dif f ( k , i, l ) , it can be sho wn th at ( P AH dif f ( k , i, l ∗ ) − P AB dif f ( k , i, l )) < 0 , ∀ i and, t herefore , lim T → ∞ a ( k , l ) = 1 . Lemma 6: For a K level tree network, for our proposed Byzantine identification scheme, the asymptotic (i.e., T → ∞ ) probabi lity that a Byzantine node i at level k + 1 , for 1 ≤ k ≤ K − 1 , is isolated i s lower -bounded by , k Y j =2 (1 − δ j ) . Pr oof: Notice that, lim T → ∞ a ( k , l ) = 1 . The asymp totic p erf ormance of the proposed scheme can be analy zed as follows: lim T →∞ P iso B ( k + 1 , i ) = X l ∈{ 0 , 1 } P l lim T →∞  (1 − b ( k , l ))  a ( k + 1 , l ) a ( k , l )  P iso B ( k , i, l )  ≥ (1 − δ k ) X l ∈{ 0 , 1 } P l lim T →∞  P iso B ( k , i, l )  = k Y j =2 (1 − δ j ) . July 10, 2018 DRAFT 25 5 10 15 20 25 30 35 40 45 50 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time window T P B iso (k) P B iso (k=1) P B iso (k=2) P B iso (k=3) P B iso (k=4) P B iso (k=5) Fig. 5. Isolation probability P iso B ( k , i ) vs. t ime window T . Notice that, the parallel network topology is a special case of t he tree network topol ogy wi th K = 1 . F or K = 1 , our scheme can id entify all the Byzantines with prob ability one because lim T → ∞ P iso B (1 , i ) = lim T → ∞ X l ∈{ 0 , 1 } P l [ a (1 , l )] = 1 . When K > 1 , we can choose η k appropriately such that Byzantines can be identified with a hi gh probabilit y . Next, to gain insight s into the solutio n, we present some n umerical results i n Figure 5 that cor- roborate our theoretical results. W e consid er a tree network wit h K = 5 and plot P iso B ( k , i ) , 1 ≤ k ≤ 5 , as a function of the time window T . W e assume t hat t he operatin g poi nts ( P k d , P k f a ) , 1 ≤ k ≤ 5 , for the nodes at dif ferent le vels are g i ven by [(0 . 8 , 0 . 1) , ( 0 . 75 , 0 . 1) , (0 . 6 , 0 . 1) , (0 . 65 , 0 . 1) , (0 . 6 , 0 . 1)] and for anchor node ( P A d , P A f a ) = ( 0 . 9 , 0 . 1) . W e also assume t hat t he hypo theses are equi- probable, i .e., P 0 = P 1 = 0 . 5 , and the maximum isolation probability of honest nodes at le vel k based sol ely on its data i s constrained by δ k = 0 . 0 1 , ∀ k . It can be seen from Figure 5 that in a span of only T = 25 ti me win do ws, our proposed scheme isolates/i dentifies almost all the Byzantines in t he tree network. V I I . C O N C L U S I O N In this paper , we consi dered the probl em o f opt imal Byzantine attacks on dist rib ut ed detection mechanism in tree n etw orks. W e analyzed the performance limit of detection performance with Byzantines and o btained the optim al attacking s trate gies that mini mize t he detection error exponent. The problem was als o st udied from the net w ork designer’ s perspective. It was shown July 10, 2018 DRAFT 26 that the opt imal local det ector is i ndependent of the Byzantine’ s parameter . Next, we modeled the strategic int erac tion between the FC and the attacker as a Leader -Follower (Stackelberg) game and att ack er and defender (FC) equ ilibrium strategies were id entified. W e also p roposed a sim ple y et ef ficient scheme to ident ify Byzantin es and analytically ev aluated its performance. There are sti ll many int eresting questio ns that remain t o be explored in the future work such as analysis of the problem for arbitrary n etw ork topolo gies. The case where Byzantines col lude in sev eral groups (collaborate) to degrade the detection performance can also be i n vestigated. A P P E N D I X A P RO O F O F L E M M A 2 T o prove t he lemma, we first show that any positive deviation ǫ ∈ (0 , p ] i n flipping probabilities ( P k 1 , 0 , P k 0 , 1 ) = ( p, p − ǫ ) will result in an increase in D k . After p lugging in ( P k 1 , 0 , P k 0 , 1 ) = ( p, p − ǫ ) in (9) and (10), we get π k 1 , 0 = [ β k − 1 1 , 0 (1 − P k f a ) + (1 − β k − 1 0 , 1 ) P k f a ] + [ α k ( p − P k f a (2 p − ǫ )) + P k f a ] (26) π k 1 , 1 = [ β k − 1 1 , 0 (1 − P k d ) + (1 − β k − 1 0 , 1 ) P k d ] + [ α k ( p − P k d (2 p − ǫ )) + P k d ] . (27) Now we sho w that D k is a m onotonically increasing function of the parameter ǫ o r in other words, dD k dǫ > 0 . dD k dǫ = π k 1 , 0 π k ′ 1 , 0 π k 1 , 0 − π k ′ 1 , 1 π k 1 , 1 ! + π k ′ 1 , 0 log π k 1 , 0 π k 1 , 1 + (1 − π k 1 , 0 ) π k ′ 1 , 1 1 − π k 1 , 1 − π k ′ 1 , 0 1 − π k 1 , 0 ! − π k ′ 1 , 0 log 1 − π k 1 , 0 1 − π k 1 , 1 (28) where dπ k 1 , 1 dǫ = π k ′ 1 , 1 = α k P k d and dπ k 1 , 0 dǫ = π k ′ 1 , 0 = α k P k f a . After rearranging the terms i n the above equation, the condit ion dD k dǫ > 0 becomes 1 − π k 1 , 0 1 − π k 1 , 1 + P k f a P k d log π k 1 , 0 π k 1 , 1 > π k 1 , 0 π k 1 , 1 + P k f a P k d log 1 − π k 1 , 0 1 − π k 1 , 1 . (29) Since P k d > P k f a and β k ¯ x,x < 0 . 5 , π k 1 , 1 > π k 1 , 0 . It can als o be prov ed th at P k d P k f a π k 1 , 0 π k 1 , 1 > 1 . Hence, we hav e July 10, 2018 DRAFT 27 1 + ( π k 1 , 0 − π k 1 , 1 ) < P k d P k f a π k 1 , 0 π k 1 , 1 ⇔ ( π k 1 , 0 − π k 1 , 1 )[1 + ( π k 1 , 0 − π k 1 , 1 )] > P k d P k f a π k 1 , 0 π k 1 , 1 ( π k 1 , 0 − π k 1 , 1 ) ⇔ ( π k 1 , 0 − π k 1 , 1 ) " 1 + ( π k 1 , 0 − π k 1 , 1 ) π k 1 , 0 (1 − π k 1 , 1 ) # > P k d P k f a π k 1 , 0 π k 1 , 1 " π k 1 , 0 − π k 1 , 1 π k 1 , 0 (1 − π k 1 , 1 ) # ⇔ ( π k 1 , 0 − π k 1 , 1 )  1 1 − π k 1 , 1 + 1 π k 1 , 0  > P k d P k f a " π k 1 , 0 − π k 1 , 0 π k 1 , 1 + π k 1 , 0 π k 1 , 1 − π k 1 , 1 π k 1 , 1 (1 − π k 1 , 1 ) # ⇔ " 1 − π k 1 , 1 − (1 − π k 1 , 0 ) 1 − π k 1 , 1 + ( π k 1 , 0 − π k 1 , 1 ) π k 1 , 0 # > P k d P k f a " π k 1 , 0 π k 1 , 1 − 1 − π k 1 , 0 1 − π k 1 , 1 # ⇔ 1 − π k 1 , 0 1 − π k 1 , 1 + P k f a P k d 1 − π k 1 , 1 π k 1 , 0 ! > π k 1 , 0 π k 1 , 1 + P k f a P k d 1 − π k 1 , 0 1 − π k 1 , 1 − 1 ! . (30) T o prov e that (29) is true, we apply the l ogarithm inequal ity ( x − 1) ≥ log x ≥ x − 1 x , for x > 0 to (30). First, let us assume that x = π k 1 , 0 π k 1 , 1 . Now using the l ogarithm inequali ty we can show that log π k 1 , 0 π k 1 , 1 ≥ 1 − π k 1 , 1 π k 1 , 0 . Next, let us assu me that x = 1 − π k 1 , 0 1 − π k 1 , 1 . Now us ing the logarithm inequality it can be shown that " 1 − π k 1 , 0 1 − π k 1 , 1 − 1 # ≥ log 1 − π k 1 , 0 1 − π k 1 , 1 . Using these results and (30 ), one can prov e that condition (29) is true. Similarly , we can show that any non zero deviation ǫ ∈ (0 , p ] in flipping probabilities ( P k 1 , 0 , P k 0 , 1 ) = ( p − ǫ, p ) will result in an increase in D k , i.e., dD k dǫ > 0 , or π k 1 , 0 π k 1 , 1 + 1 − P k f a 1 − P k d log 1 − π k 1 , 0 1 − π k 1 , 1 > 1 − π k 1 , 0 1 − π k 1 , 1 + 1 − P k f a 1 − P k d log π k 1 , 0 π k 1 , 1 . (31) Since P k d > P k f a and β k ¯ x,x < 0 . 5 , π k 1 , 1 > π k 1 , 0 . It can also be proved that 1 − π k 1 , 0 1 − π k 1 , 1 < 1 − P k f a 1 − P k d . Hence, we have July 10, 2018 DRAFT 28 1 − π k 1 , 0 1 − π k 1 , 1 < 1 − P k f a 1 − P k d  1 − ( π k 1 , 0 − π k 1 , 1 )  (32) ⇔ 1 − π k 1 , 0 π k 1 , 1 (1 − π k 1 , 1 ) < 1 − P k f a 1 − P k d " 1 − ( π k 1 , 0 − π k 1 , 1 ) π k 1 , 1 # ⇔ 1 π k 1 , 1 (1 − π k 1 , 1 ) < 1 − P k f a 1 − P k d " 1 − ( π k 1 , 0 − π k 1 , 1 ) π k 1 , 1 (1 − π k 1 , 0 ) # ⇔ 1 π k 1 , 0 − π k 1 , 1 " π k 1 , 0 − π k 1 , 0 π k 1 , 1 + π k 1 , 0 π k 1 , 1 − π k 1 , 1 π k 1 , 1 (1 − π k 1 , 1 ) # < 1 − P k f a 1 − P k d " 1 − ( π k 1 , 0 − π k 1 , 1 ) π k 1 , 1 (1 − π k 1 , 0 ) # ⇔ 1 π k 1 , 0 − π k 1 , 1 " π k 1 , 0 π k 1 , 1 − 1 − π k 1 , 0 1 − π k 1 , 1 # < 1 − P k f a 1 − P k d " 1 π k 1 , 1 + 1 1 − π k 1 , 0 # (33) ⇔ π k 1 , 0 π k 1 , 1 − 1 − π k 1 , 0 1 − π k 1 , 1 > 1 − P k f a 1 − P k d " π k 1 , 0 − π k 1 , 1 π k 1 , 1 + π k 1 , 0 − π k 1 , 1 1 − π k 1 , 0 # (34) ⇔ π k 1 , 0 π k 1 , 1 − 1 − π k 1 , 0 1 − π k 1 , 1 > 1 − P k f a 1 − P k d " π k 1 , 0 − π k 1 , 1 π k 1 , 1 + 1 − π k 1 , 1 − (1 − π k 1 , 0 ) 1 − π k 1 , 0 # ⇔ π k 1 , 0 π k 1 , 1 + 1 − P k f a 1 − P k d " 1 − 1 − π k 1 , 1 1 − π k 1 , 0 # > 1 − π k 1 , 0 1 − π k 1 , 1 + 1 − P k f a 1 − P k d " π k 1 , 0 π k 1 , 1 − 1 # . 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