Byzantines and Mitigation Techniques

Transcription

1 Distributed Detection in Tree Networs: 1 Byzantines and Mitigation Techniques Bhavya Kailhura, Student Member, IEEE, Swasti Brahma, Member, IEEE, Beran Dule, Member, IEEE, Yunghsiang S Han, Fellow, IEEE, arxiv: v1 cs.cr] 22 Oct 2014 Pramod K. Varshney, Fellow, IEEE Abstract In this paper, the problem of distributed detection in tree networs in the presence of Byzantines is considered. Closed form expressions for optimal attacing strategies that minimize the miss detection error exponent at the fusion center (FC) are obtained. We also loo at the problem from the networ designer s (FC s) perspective. We study the problem of designing optimal distributed detection parameters in a tree networ in the presence of Byzantines. Next, we model the strategic interaction between the FC and the attacer as a Leader-Follower (Stacelberg) game. This formulation provides a methodology for predicting attacer and defender (FC) equilibrium strategies, which can be used to implement the optimal detector. Finally, a reputation based scheme to identify Byzantines is proposed and its performance is analytically evaluated. We also provide some numerical examples to gain insights into the solution. Index Terms Distributed detection, data falsification, Byzantines, tree networs, error exponent, leader-follower game, reputation based mitigation scheme This wor was supported by the Center for Advanced Systems and Engineering at Syracuse University. The authors would lie to than Aditya Vempaty for his valuable comments and suggestions to improve the quality of the paper. B. Kailhura, S. Brahma and P. K. Varshney are with Department of EECS, Syracuse University, Syracuse, NY ( bailhu@syr.edu; sbrahma@syr.edu; varshney@syr.edu) B. Dule is with Department of Electrical and Electronics Engineering, Hacettepe University, Beytepe Campus, Anara, Turey. ( beran@ee.hacettepe.edu.tr) Y. S. Han is with EE Department, National Taiwan University of Science and Technology, Taiwan, R. O. C. ( yshan@mail.ntust.edu.tw)

2 2 I. INTRODUCTION Distributed detection deals with the problem of maing a global decision regarding a phenomenon based on local decisions collected from several remotely located sensing nodes. Distributed detection research has traditionally focused on the parallel networ topology, in which nodes directly transmit their observations or decisions to the Fusion Center (FC) 1] 2] 3]. Despite its theoretical importance and analytical tractability, parallel topology may not always reflect the practical scenario. In certain cases, it may be required to place the nodes outside their communication range with the FC. Then, the coverage area can be increased by forming a multi-hop networ, where nodes are organized hierarchically into multiple levels (tree networs). Some examples of tree networs include wireless sensor and military communication networs. For instance, the IEEE (Zigbee) specifications 4] and IEEE b 5] support tree networs. Typically, a networ embodies a large number of inexpensive sensors, which are deployed in an open environment to collect the observations regarding a certain phenomenon and, therefore, are susceptible to many inds of attacs. A typical example is a Byzantine attac. While Byzantine attacs (originally proposed in 6]) may, in general, refer to many types of malicious behavior, our focus in this paper is on data-falsification attacs 7] 16], where an attacer sends false (erroneous) data to the FC to degrade detection performance. In this paper, we refer to such data falsification attacers as Byzantines, and the data thus fabricated as Byzantine data. A. Related Wor Recently, distributed detection in the presence of Byzantine attacs has been explored in 10], 11], where the problem of determining the most effective attacing strategy for the Byzantines was investigated. However, both wors focused only on parallel topology. The problem considered in this paper is most related to our earlier papers 12], 16]. In 12], 16], we studied the problem of distributed detection in perfect tree networs (all intermediate nodes in the tree have the same number of children) with Byzantines under the assumption that the FC does not now which decision bit is sent from which node and assumes each received bit to originate from nodes at depth with a certain probability. Under this assumption, the attacer s aim was to maximize the false alarm probability for a fixed detection probability. When the number of nodes is large, by Stein s lemma 17], we now that the error exponent of the false alarm probability

3 3 can be used as a surrogate for the false alarm probability. Thus, the optimal attacing strategy was obtained by maing the error exponent of the false alarm probability at the FC equal to zero, which maes the decision fusion scheme to become completely incapable (blind). Some counter-measures were also proposed to protect the networ from such Byzantines. There are several notable differences between this paper and our earlier papers 12], 16]. First, in contrast to 12], 16], in this paper, the problem of distributed detection in regular tree networs 1 with Byzantines is addressed in a practical setup where the FC has the nowledge of which bit is transmitted from which node. Note that, in practice, the FC nows which bit is transmitted from which node, e.g., using MAC schemes 2, and can utilize this information to improve system performance. Next, for the analysis of the optimal attac, we consider nodes residing at different levels of the tree to have different detection performance. We also allow Byzantines residing at different levels of the tree to have different attacing strategies and, therefore, provide a more general and comprehensive analysis of the problem as compared to 12], 16]. We also study the problem from the networ designer s perspective. Based on the information regarding which bit is transmitted from which node, we propose schemes to mitigate the effect of the Byzantines. B. Main Contributions In this paper, it is assumed that the FC nows which bit is transmitted from which node. Under this assumption, the problem of distributed detection in tree networs in the presence of Byzantines is considered. The main contributions of this paper are summarized below: Detection performance in tree networs with Byzantines is characterized in terms of the error exponent and a closed form expression for the optimal error exponent is derived. The minimum attacing power required by the Byzantines to blind the FC in a tree networ is obtained. It is shown that when more than a certain fraction of individual node decisions are falsified, the decision fusion scheme is completely jeopardized. The problem is also investigated from the networ designer s perspective by focusing on the design of optimal distributed detection parameters in a tree networ. 1 For a regular tree, intermediate nodes at different levels are allowed to have different degrees, i.e., number of children. 2 In practice, one possible way to achieve this is by using the buffer-less TDMA MAC protocol, in which, distinct nonoverlapping time slots are assigned (scheduled) to the nodes for communication. One practical example of such a scheme is given in 18].

4 4 Level 3 Level 2 Level 1 Fig. 1. Fusion Center (FC) A distributed detection system organized as a regular tree (a 1 = 2, a 2 = 3, a 3 = 2) is shown as an example. We model the strategic interaction between the FC and the attacer as a Leader-Follower (Stacelberg) game and identify attacer and defender (FC) equilibrium strategies. The nowledge of these equilibrium strategies can later be used to implement the optimal detector at the FC. We propose a simple yet efficient reputation based scheme, which wors even if the FC is blinded, to identify Byzantines in tree networs and analytically evaluate its performance. The rest of the paper is organized as follows. Section II introduces the system model. In Section III, we study the problem from Byzantine s perspective and provide closed form expressions for optimal attacing strategies. In Section IV, we investigate the problem of designing optimal distributed detection parameters in the presence of Byzantines. In Section V, we model the strategic interaction between the FC and the attacer as a Leader-Follower (Stacelberg) game and find equilibrium strategies. In Section VII, we introduce an efficient Byzantine identification scheme and analyze its performance. Finally, Section VII concludes the paper. II. SYSTEM MODEL We consider a distributed detection system organized as a regular tree networ rooted at the FC (See Figure 1). For a regular tree, all the leaf nodes are at the same level (or depth) and all the intermediate nodes at level have degree a. The regular tree is assumed to have a set N = {N } K =1 of transceiver nodes, where N = N is the total number of nodes at level. We assume that the depth of the tree is K > 1 and a 2. The total number of nodes in the networ is denoted as N = K =1 N and B = {B } K =1 denotes the set of Byzantine nodes

5 5 with B = B, where B is the set of Byzantines at level. The set containing the number of Byzantines residing at each level, 1 K, is referred to as an attac configuration, i.e., {B } K =1 = { B } K =1. Next, we define the modus operandi of the nodes. A. Modus Operandi of the Nodes We consider a binary hypothesis testing problem with two hypotheses H 0 (signal is absent) and H 1 (signal is present). Under each hypothesis, it is assumed that the observations Y,i at each node i at level are conditionally independent. Each node i at level acts as a source in the sense that it maes a one-bit (binary) local decision v,i {0,1} regarding the absence or presence of the signal using the lielihood ratio test (LRT) 3 p (1) Y,i (y,i ) p (0) Y,i (y,i ) v,i =1 v,i =0 λ, (1) where λ is the threshold used at level (it is assumed that all the nodes at level use the same threshold λ ) and p (j) Y,i (y,i ) is the conditional probability density function (PDF) of observation y,i under hypothesis H j for j {0,1}. We denote the probabilities of detection and false alarm of a node at level by P d = P(v,i = 1 H 1 ) and P fa = P(v,i = 1 H 0 ), respectively, which are functions of λ and hold for both Byzantines and honest nodes. After maing its one-bit local decision v,i {0,1}, node i at level sends u,i to its parent node at level 1, where u,i = v,i if i is an honest node, but for a Byzantine node i, u,i need not be equal to v,i. Node i at level also receives the decisions u,j of all successors j at levels +1,K], which are forwarded to node i by its immediate children, and forwards 4 them to its parent node at level 1. We assume error-free communication between children and the parent nodes. Next, we present a mathematical model for the Byzantine attac. B. Byzantine Attac Model We define the following strategies P H j,1(), P H j,0() and P B j,1(), P B j,0() (j {0,1} and = 1,,K) for the honest and Byzantine nodes at level, respectively: 3 Notice that, under the conditional independence assumption, the optimal decision rule at the local sensor is a lielihood-ratio test 19]. 4 For example, IEEE j mandates tree forwarding and IEEE s standardizes a tree-based routing protocol.

6 6 Honest nodes: P H 1,1 () = 1 PH 0,1 () = PH (x = 1 y = 1) = 1 (2) P H 1,0 () = 1 PH 0,0 () = PH (x = 1 y = 0) = 0 (3) Byzantine nodes: P B 1,1() = 1 P B 0,1() = P B (x = 1 y = 1) (4) P B 1,0 () = 1 PB 0,0 () = PB (x = 1 y = 0) (5) where P (x = a y = b) is the conditional probability that a node at level sends a to its parent when it receives b from its child or its actual decision is b. For notational convenience, we use (P1,0,P 0,1 ) to denote the flipping probability of the Byzantine node at level. Furthermore, we assume that if a node (at any level) is a Byzantine, then none of its ancestors and successors are Byzantine (non-overlapping attac configuration); 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 decisions of its successors). This means that every path from a leaf node to the FC will have at most one Byzantine. Notice that, for the attac configuration {B } K =1, the total number of corrupted paths (i.e., paths containing a Byzantine node) from level to the FC are i=1 B i N N N i, where B i N i is the total number of nodes covered 5 at level by the presence of B i Byzantines at level i. If we denote α = B N, then, i=1 B i N N i N = i=1 α i is the fraction of decisions coming from level that encounter a Byzantine along the way to the FC. We also approximate the probability that the FC receives the flipped decision x of a given node at level when its actual decision is x as β x,x = j=1 α jp j x,x, x {0,1}. C. Binary Hypothesis Testing at the Fusion Center We consider the distributed detection problem under the Neyman-Pearson (NP) criterion. The FC receives decision vectors, z 1,,z K ], where z for {1,,K} is a decision vector with its elements being z 1,,z N, from the nodes at different levels of the tree. Then the FC maes the global decision about the phenomenon by employing the LRT. Due to system vulnerabilities, some of the nodes may be captured by the attacer and reprogrammed 5 Node i at level covers all its children at levels +1 to K and itself.

7 7 to transmit false information to the FC to degrade detection performance. We assume that the only information available at the FC is the probability β x,x, which is the probability with which the data coming from level has been falsified. Using this information, the FC calculates the probabilities π j,0 = P(z i = j H 0,) and π j,1 = P(z i = j H 1,), which are the distributions of received decisions z i originating from level and arriving to the FC under hypotheses H 0 and H 1. The FC maes its decision regarding the absence or presence of the signal using the following lielihood ratio test ( ) s ( ) N s K π 1,1 1 π H 1,1 1 η (6) H 0 =1 π 1,0 1 π 1,0 where s is the number of decisions that are equal to one and originated from level, and the threshold η is chosen in order to minimize the missed detection probability (P M ) while eeping the false alarm probability (P F ) below a fixed value δ. 6 Next, we derive a closed form expression for the optimal missed detection error exponent for tree networs in the presence of Byzantines, which will later be used as a surrogate for the probability of missed detection. Proposition 1: For ak level tree networ employing the detection scheme as given in (6), the asymptotic detection performance can be characterized using the missed detection error exponent given below D = K =1 N j {0,1} πj,0log π j,0. (7) Proof: Let Z = Z 1,,Z N1 ] denote the received decision vectors from the nodes at level 1, where Z i is the decision vector forwarded by the node i at level 1 to the FC. Observe that, Z i for i = 1 to N 1 are independent and identically distributed (i.i.d.). Therefore, using Stein s lemma 17], the optimal error exponent for the detection scheme as given in (6) is the Kullbac- Leibler divergence (KLD) 20] between the distributions P(Z H 0 ) and P(Z H 1 ). Summation term in (7) follows from the additive property of the KLD for independent distributions. Note that, (7) can be compactly written as K =1 N D (π j,1 π j,0) with D (π j,1 π j,0) being the π j,1 6 This type of problem setup is important, for instance, in Cognitive Radio Networs (CRN). In order to coexist with the primary user (PU), secondary users (SUs) must guarantee that their transmissions will not interfere with the transmission of the PU who have higher priority to access the spectrum.

8 8 KLD between the data coming from node i at level under H 0 and H 1. The FC wants to maximize the detection performance, while, the Byzantine attacer wants to degrade the detection performance as much as possible which can be achieved by maximizing and minimizing the KLD, respectively. Next, we explore the optimal attacing strategies for the Byzantines that degrade the detection performance most by minimizing the KLD. III. OPTIMAL BYZANTINE ATTACK As discussed earlier, the Byzantines attempt to mae the KL divergence as small as possible. Since the KLD is always non-negative, Byzantines attempt to choose P(z i = j H 0,) and P(z i = j H 1,) such that D = 0,. In this case, an adversary can mae the data that the FC receives from the nodes such that no information is conveyed from them. This is possible when P(z i = j H 0,) = P(z i = j H 1,) j {0,1},. (8) Notice that, π j,0 = P(z i = j H 0,) and π j,1 = P(z i = j H 1,) can be expressed as π 1,0 = β 1,0 (1 P fa )+(1 β 0,1 )P fa (9) π 1,1 = β 1,0 (1 P d )+(1 β 0,1 )P d. (10) with β1,0 = j=1 α jp j 1,0 and β 0,1 = j=1 α jp j 0,1. Substituting (9) and (10) in (8) and after simplification, the condition to mae the D = 0 for a K-level networ becomes j=1 α j(p j 1,0+ P j 0,1) = 1,. Notice that, when j=1 α j < 0.5, there does not exist any attacing probability distribution (P j 0,1,Pj 1,0 ) that can mae D = 0, and, therefore, the KLD cannot be made zero. In the case of j=1 α j = 0.5, there exists a unique solution (P j 0,0,P j 1,0) = (1,1), j that can mae D = 0,. For the j=1 α j > 0.5 case, there exist infinitely many attacing probability distributions (P j 0,1,Pj 1,0 ) which can mae D = 0,. Lemma 1: In a tree networ with K levels, the minimum number of Byzantines needed to blind the FC (or to mae D = 0, ) is given by B 1 = N 1 2. Proof: The proof follows from the fact that the condition j=1 α j = 0.5,, is equivalent to α 1 = 0.5, α = 0, = 2,,K. Next, we explore the optimal attacing probability distribution (P 0,1,P 1,0 ) that minimizes D when j=1 α j < 0.5, i.e., in the case where the attacer cannot mae D = 0. To analyze

9 9 the problem, first we investigate the properties of D with respect to (P 0,1,P 1,0) assuming (P j 0,1,P j 1,0), 1 j 1 to be fixed. We show that attacing with symmetric flipping probabilities is the optimal strategy in the region where the attacer cannot mae D = 0. In other words, attacing with P 1,0 = P 0,1 is the optimal strategy for the Byzantines. Lemma 2: In the region where the attacer cannot mae D = 0, i.e., for j=1 α j < 0.5, the optimal attacing strategy comprises of symmetric flipping probabilities (P 0,1 = P 1,0 = p). In other words, any non zero deviation ǫ i (0,p] in flipping probabilities (P 0,1,P 1,0) = (p ǫ 1,p ǫ 2 ), where ǫ 1 ǫ 2, will result in an increase in D. Proof: Please see Appendix A. In the next theorem, we present the solution for the optimal attacing probability distribution (P j,1,p j,0 ) that minimizes D in the region where the attacer cannot mae D = 0. Theorem 1: In the region where the attacer cannot mae D = 0, i.e., for j=1 α j < 0.5, the optimal attacing strategy is given by (P 0,1,P 1,0) = (1,1). Proof: Observe that, in the region where the attacer cannot mae D = 0, the optimal strategy comprises of symmetric flipping probabilities (P0,1 = P 1,0 = p). The proof is complete if we show that D is a monotonically decreasing function of the flipping probability p. After plugging in (P0,1,P 1,0 ) = (p,p) in (9) and (10), we get π 1,1 = β 1 1,0 (1 P d )+(1 β 1 0,1 )P d ]+α (p P d (2p))+P d ] (11) π1,0 = β1,0 1 (1 P fa )+(1 β 1 0,1 )P fa ]+α (p Pfa (2p))+P fa ]. (12) Now we show that D is a monotonically decreasing function of the parameter p or in other words, dd dp < 0. After plugging in π 1,1 = α (1 2Pd ) and π 1,0 = α (1 2Pfa ) in the expression of dd dp and rearranging the terms, the condition dd dp < 0 becomes (1 2P d) ( 1 π 1,0 1 π 1,1 π 1,0 π 1,1 ) +(1 2P fa)log ( 1 π 1,1 π 1,0 1 π1,0 π1,1 ) < 0 (13)

10 KLD P 1, P 0,1 0.1 Fig. 2. KLD D vs. flipping probabilities when Pd = 0.8, Pfa = 0.2, and the probability that the bit coming from level encounters a Byzantine is j=1αj = 0.4. Since P d > P fa and β x,x < 0.5, we have π 1,1 > π 1,0. Now, using the fact that 1 P d 1 P fa > 1 2P d 1 2P fa and (33), we have 1 2Pd 1 π 1,0 1 2Pfa 1 π1,1 1 2P d 1 π 1,0 1 2Pfa 1 π1,1 π 1,0 π1,1 π 1,0 π 1,1 ] < (π 1,1 π 1,0) ] ] π 1,0 + 1 π 1,1 1 π 1, π 1,0 ] (14) < 1 1 π 1,1 1 π1,0. (15) Applying the logarithm inequality (x 1) logx x 1, for x > 0 to (15), one can prove x that (13) is true. Next, to gain insights into the solution, we present some numerical results in Figure 2. We plot D as a function of the flipping probabilities (P1,0, P 0,1 ). We assume that the probability of detection is Pd = 0.8, the probability of false alarm is P fa = 0.2, and the probability that the bit coming from level encounters a Byzantine is j=1 α j = 0.4. We also assume that P 0,1 = P 0,1 and P 1,0 = P 1,0,. It can be seen that the optimal attacing strategy comprises of symmetric flipping probabilities and is given by (P 0,1,P 1,0) = (1,1), which corroborates our theoretical result presented in Lemma 2 and Theorem 1. We have shown that, for all, D (P 0,1,P 1,0) D (1,1). (16)

11 11 Now, by multiplying both sides of (16) by N and summing it over all K we can show that the KLD, D, is minimized by (P0,1,P 1,0 ) = (1,1), for all, in the region K α < 0.5. Now, we explore some properties of D with respect to j=1 α j in the region where the attacer cannot mae D = 0, i.e., for j=1 α j < 0.5. This analysis will later be used in exploring the problem from the networ designer s perspective. Lemma 3: D = min D (π (Pj,1,P j,0 ) j,1 π j,0 ) is a continuous, decreasing and convex function of j=1 α j for j=1 α j < 0.5. Proof: The continuity of D (π j,1 π j,0) with respect to the involved distributions implies the continuity of D. To show that D is a decreasing function of t = j=1 α j, we use the fact that arg min (P 0,1,P 1,0 ) D (π j,1 π j,0) is equal to (1,1) for j=1 α j < 0.5 (as shown in Theorem 1). After plugging (P0,1,P 1,0) = (1,1),, in the KLD expression, it can be shown that dd < 0. Hence, dt D is a monotonically decreasing function of j=1 α j for j=1 α j < 0.5. The convexity of D follows from the fact thatd (π j,1 π j,0 ) is convex inπ j,1 andπ j,0, which are affine transformations of j=1 α j (Note that, convexity holds under affine transformation). It is worth noting that Lemma 3 suggests that minimization/maximization of j=1 α j is equivalent to minimization/maximization of D. Using this fact, one can consider the probability that the bit coming from level encounters a Byzantine (i.e., t = j=1 α j) in lieu of D for optimizing the system performance. Observe that, the expression t = j=1 α j is much more tractable than the expression for D. Next, to gain insights into the solution, we present some numerical results in Figure 3. We plot min D as a function of the probability that the bit coming from level encounters a (Pj,1,P j,0 ) Byzantine, i.e., t. We assume that the probabilities of detection and false alarm are Pd = 0.8 and P fa = 0.2, respectively. Notice that, when t = 0.5, D between the two probability distributions becomes zero. It is seen that D is a continuous, decreasing and convex function of the fraction of covered nodes, t, for t < 0.5, which corroborates our theoretical result presented in Lemma 3. Until now, we have explored the problem from the attacer s perspective. In the rest of the paper, we loo into the problem from a networ designer s perspective and propose techniques to mitigate the effect of Byzantines. First, we study the problem of designing optimal distributed detection parameters in a tree networ in the presence of Byzantines. =1

12 minkld t Fig. 3. min (P j,1,p j,0 ) D vs probability that the bit coming from level encounters a Byzantine for P d = 0.8 and P fa = 0.2. IV. SYSTEM DESIGN IN THE PRESENCE OF BYZANTINES For a fixed attac configuration {B } K =1, the detection performance at the FC is a function of the local detectors used at the nodes in the tree networ and the global detector used at the FC. This motivates us to study the problem of designing detectors, both at the nodes at different levels in a tree and at the FC, such that the detection performance is maximized. More specifically, we are interested in answering the question: How does the nowledge of the attac configuration {B } K =1 affect the design of optimal distributed detection parameters? By Stein s lemma 17], we now that in the NP setup for a fixed false alarm probability, the missed detection probability of the optimal detector can be minimized by maximizing the KLD. For an optimal detector at the FC, the problem of designing the local detectors can be formalized as follows: max {P d,p fa }K =1 K N =1 j {0,1} P(z i = j H 0,)log P(z i = j H 0,) P(z i = j H 1,). (17) The local detector design problem as given in (17) is a non-linear optimization problem. Furthermore, it is difficult to obtain a closed form solution for this problem. Also, observe that the solution space is not constrained to the lielihood ratio based tests. To solve the problem, we need to find the pairs {Pd,P fa }K =1 which maximize the objective function as given in (17). However, P d and P fa are coupled and, therefore, cannot be optimized independently. Thus, we first analyze the problem of maximizing the KLD for a fixed P fa. We assume that P fa = y

13 13 and P d = y +x. Next, we analyze the properties of KLD with respect to x, i.e., (P d P fa ) in the region where attacer cannot blind the FC, i.e., for j=1 α j < 0.5, in order to study the local detector design problem. Notice that, in the region j=1 α j 0.5, D = 0 and optimizing over local detectors does not improve the performance. Lemma 4: For a fixed P fa = y, when j=1 α j < 0.5, the KLD, D, as given in (7) is a monotonically increasing function of x = (P d P fa ). Proof: To prove this, we calculate the partial derivative of D with respect to x. By substituting P fa = y and P d = y + x into (7), the partial derivative of D with respect to x can be calculated as D = N x x D x = N π 1,1 π1,0 log π 1,0 +(1 π1,0 )log 1 π 1,0 ( 1 π 1,0 1 π 1,1 π 1,1 π 1,0 π 1,1 ), 1 π 1,1 where π 1,0 and π 1,1 are as given in (9) and (10), respectively and π 1,1 = (1 β 0,1 β 1,0). Notice that, ( 1 π 1,0 1 π 1,1 π 1,1 > π 1,0. Thus, the condition to mae D x > 0 simplifies to π 1,0 π 1,1 ) > 0 π 1,1 > 0 1 > (β 0,1 +β 1,0 ) (18) Substituting the values of β1,0 and β 1,1, the above condition can be written as: ] α j P j 1,0 + α j P j 0,1 < 1 (19) j=1 j=1 α j (P j 1,0 +Pj 0,1 ) < 1 (20) j=1 The above condition is true for any 0 P j 0,1,P j 1,0 1 when j=1 α j < 0.5. This completes the

14 14 proof. Lemma 4 suggests that one possible solution to maximize D is to choose the largest possible x constrained to 0 x 1 y. The upper bound results from the fact that {P d,p fa }K =1 are probabilities and, thus, must be between zero and one. In other words, the solution is to maximize the probability of detection for a fixed value of probability of false alarm. In detection theory, it is well nown that the lielihood ratio based test is optimum for this criterion. Thus, under the conditional independence assumption, lielihood ratio based test as given in (6) is optimal for local nodes, even in the presence of Byzantines, and the optimal operating points {P d,p fa }K =1 are independent of the Byzantines parameters {α } K =1. To summarize, optimal local detectors for distributed detection in tree networs are lielihood ratio based detectors and are independent of the Byzantines parameter {α } K =1. We further explore the problem from the networ designer s (FC) perspective. In our previous analysis, we have assumed that the attac configuration {B } K =1 is nown and shown that the optimal local detector is independent of {α } K =1. However, notice that the KLD is the exponential decay rate of the error probability of the optimal detector. In other words, while optimizing over KLD, we implicitly assumed that the optimal detector, which is a lielihood ratio based detector, is used at the FC. Taing logarithm on both sides of (6), the optimal decision rule simplifies to K a 1 s +a 0 (N s )] =1 H 1 logη (21) H 0 where the optimal weights are given by a 1 = log π 1,1 and a π1,0 0 = log 1 π 1,1. To implement the 1 π1,0 optimal detector, the FC needs to now the optimal weights a j, which are functions of {α } K =1. In the next section, we are interested in answering the question: Is it possible for the FC to predict the attac configuration {B } K =1 in the tree? The nowledge of this attac configuration can be used for determining the optimal detector at the FC to improve the system performance. Notice that, learning/estimation based techniques can be used on data to determine the attac configuration. However, the FC has to acquire a large amount of data coming from the nodes over a long period of time to accurately estimate {B } K =1. In the next section, we propose a novel technique to predict the attac configuration by considering the following scenario: The FC, acting first, commits to a defensive strategy by deploying the defensive resources to protect the tree networ, while the attacer chooses its best

15 15 response or attac configuration after surveillance of this defensive strategy. Both, the FC and the Byzantines have to incur a cost to deploy the defensive resources and attac the nodes in the tree networ, respectively. We consider both the FC and the attacer to be strategic in nature and model the strategic interaction between them as a Leader-Follower (Stacelberg) game. This formulation provides a framewor for identifying attacer and defender (FC) equilibrium strategies, which can be used to implement the optimal detector. The main advantage of this technique is that the equilibrium strategies can be determined a priori and, therefore, there is no need to observe a large amount of data coming from the nodes over a long period of time to accurately estimate {B } K =1. V. STACKELBERG GAME FOR ATTACK CONFIGURATION PREDICTION PROBLEMS We model the strategic interaction between the FC and the attacer as a Leader-Follower (Stacelberg) game. We assume that the FC has to incur a cost for deploying the networ and the Byzantine has to incur a cost 7 for attacing the networ. It is assumed that the networ designer or the FC has a cost budget C networ budget and the attacer has a cost budget Cbudget attacer 8. More specifically, the FC wants to allocate the best subset of defensive resources (denoted as { c } K =1 )9 from a set of available defensive resources C = (c 1,,c n ) (arranged in a descending order, i.e., c 1 c 2 c n ), where n K, complying with its budget constraint C networ budget different levels of the tree networ. After the FC allocates the defensive resources or budget to different levels of the tree networ, an attacer chooses an attac configuration, {B } K =1 complying with his budget constraint C attacer budget networ. to maximally degrade the performance of the Next, we formalize the Stacelberg game as a bi-level optimization problem. For our problem, the upper level problem (ULP) corresponds to the FC who is the leader of the game, while the to 7 Due to variations in hardware complexity and the level of tamper-resistance present in nodes residing at different levels of the tree, the resources required to capture and tamper nodes at different levels may be different and, therefore, nodes have varying costs of being attaced. 8 In this paper, we assume that the attacer budget C attacer budget is such that K α < 0.5, i.e., the attacer cannot mae D = 0,. Notice that, if the attacer can mae D = 0 for some = l, then, it can also mae D = 0, l. Also, D = 0 implies that π 1,1 = π 1,0 and, therefore, the weights (a 1,a 0) in (21) are zero. In other words, the best the FC can do in the case when D = 0, l is to ignore or discard the decisions of the nodes residing at level l. This scenario is equivalent to using the tree networ with (l 1) levels for distributed detection. 9 Let c denote the resources deployed or budget allocated by the FC to protect or deploy a node at level. =1

16 16 lower level problem (LLP) belongs to the attacer who is the follower. maximize { c } K =1 C D({ c } K =1 ) subject to K =1 c N C networ budget minimize B Z + D({B } K =1 ) subject to K =1 c B C attacer budget 0 B N, = 1,2,...,K (22) where Z + is the set of non-negative integers. Notice that, the bi-level optimization problem, in general, is an NP-hard problem. In fact, the LLP is a variant of the pacing formulation of the bounded napsac problem with a non-linear objective function. This is, in general, NP-hard. Using existing algorithms, cost set { c } K =1 and attac configuration {B } K =1 can be determined at the cost of computational efficiency. In this paper, we identify a special case of the above problem which can be solved in polynomial time to determine the equilibrium strategies. Next, we discuss the relationships that enable our problem to have a polynomial time solution. We define profit P(S) of an attac configuration S = {B } K =1 as follows10 P(S) = D(φ) D(S) = D(φ) D({B } K =1 ), whered(φ) is the KLD when there are no Byzantines in the networ and D(S) = D({B } K =1 ) is the KLD with{b } K =1 Byzantines in the tree networ. Next, we define the concept of dominance which will be used later to explore some useful properties of the optimal attac configuration {B } K =1. Definition 1: We say that a set S 1 dominates another 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 set S i, respectively. If 10 In this section, we assume that the optimal operating point, i.e., (P d,p fa), is the same for all the nodes in the tree networ. It has been shown that the use of identical thresholds is asymptotically optimal for parallel networs 21]. We conjecture that this result is valid for tree networs as well and employ identical thresholds.

17 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 wealy dominates S 2. To solve the bi-level optimization problem, we first solve the LLP assuming the solution of the ULP to be some fixed( c 1,, c K ). This approach will give us a structure of the optimal{b } K =1 for any arbitrary { c } K =1, which can later be utilized to solve the bi-level optimization problem. We refer to LLP as a maximum damage Byzantine attac problem. Observe that, nowing that the FC chooses ( c 1,, c K ), the LLP can be reformulated as follows: minimize {B } K =1 subject to K N D ({B i } i=1 ) =1 K =1 c B C attacer budget 0 B N, and integer Next, we discuss the relationships that enable our maximum damage Byzantine attac problem to admit a polynomial time solution. A. Analysis of the Optimal Attac Configuration In this section, we identify a special case of the bounded napsac problem (LLP) which can be solved in polynomial time. More specifically, ( we show that if the) set of defensive resources C = N +1 (c 1,,c n ) satisfy the cost structure c max min c 11 min or c 1 mina c n, {1,,K 1} N then, the optimal solution {B } K =1 exhibits the properties given in the lemma below. ( Lemma 5: Given ak level tree networ with cost structure satisfyingc max c min, the best response of an attacer with cost budget C attacer budget B 1 = C attacer budget c 1 is {B } K =1 with and the remaining elements of B for 2 K can be calculated recursively. Proof: To prove Lemma 5, it is sufficient to show that: 1) KLD is a monotonically decreasing function of B, and, 2) Attacing parent nodes is a strictly dominant strategy. min {1,,K 1} ) N +1 N 11 Notice that, in the case of the perfect M-ary tree networs, the proposed cost structure simplifies to c max M c min.

18 18 Lemma 3 suggests that the KLD is a monotonically decreasing function of B in the region where attacer cannot mae D = 0 and, therefore, (1) is proved. Next, we show that attacing parent nodes is a strictly dominant strategy. In other words, given a cost budget C attacer budget, it is more profitable for an attacer to attac the parent nodes. Observe that the KLD at level is a function of Byzantines parameter (B 1,,B ). Thus, we denote it as D (B 1,,B ). In order to prove that attacing parent nodes is a strictly dominant strategy, it is sufficient to show that the attac configurations 1 = (B 1,,B j,b j+1,,b K ) strictly dominates the attac 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 that P(S 1 ) > P(S 2 ) and C(S 1 ) C(S 2 ). From the cost inequality it follows thatc(s 1 ) C(S 2 ) becausec max (min N +1 /N ) c min c j (N j+1 /N j ) c j+1. Also, note that if the attac configuration S 1 strictly dominates the attac configuration S 2, then, it will also strictly dominate any attac configuration S 2 with S 2 = (B 1,,B j δ,b j+1 +δγ,,b K ), where γ N j+1 N j. Next, we show that 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 j 1 D (B 1,,B ) < D (B 1,,B )+D j (B 1,,B j 1,B j δ) =1 =1 K j 1 D (B 1,,B ) < D (B 1,,B )+D j (B 1,,B j 1,B j δ) =1 =1 + K =j+1 D (B 1,,B j δ,b j+1 +δ N j+1 N j,b j+2,,b ), where the last inequality follows from the fact that B j N j + B j+1 N j+1 = B j δ N j + B N j+1 j+1+ δ N j N j+1 D (B 1,,B j,b j+1,,b ) = D (B 1,,B j δ,b j+1 + N j+1 N j δ,,b ). and, therefore, This implies that the set S 1 strictly dominates the set S 2. From the results in Lemma 3, it is seen that the profit is an increasing function of the attac nodes. Lemma 3 in conjunction with the fact that attacing parent nodes is a strictly dominant strategy implies Lemma 5. It can also be shown that the solution {B } K =1 condition that the attacer cannot mae D = 0,. will be non-overlapping and unique under the

19 19 B. Bi-Level Optimization Algorithm Based on Lemma 5, in this section we will present a polynomial time algorithm to solve the bi-level ( optimization ) problem, i.e., to find { c } K =1 and {B } K =1. Using the cost structure N +1 c max min c min, the attac configuration {B } K =1 as given in Lemma 5 can be N determined in a computationally efficient manner. Due to structure of the optimal {B } K =1, the bi-level optimization problem simplifies to finding the solution { c } K =1 of the ULP. To solve this problem, we use an iterative elimination approach. We start by listing all ( ) n K combinations from the set C, denoted as, S = {s i } K) (n i=1. Without loss of generality, we assume that the elements of s i = {c i 1,,c i K } are arranged in descending order, i.e., ci ci +1,. Notice that, all these ( ) n K combinations will satisfy c i N +1 N c i +1, because c i c max min j N j+1 N j c min min j N j+1 N j c i +1 N +1 N c i +1. Next, we discard all those subsets s i from S which violate the networ designer s cost budget constraint. If the set S is empty, then there does not exist any solution for the ULP. Otherwise, the problem reduces to finding the subset s i which maximizes the KLD. To find the subset s i which maximizes the KLD, using the dominance relationship we start with assigning the cost C attacer c 1 = min s c budget 1, where s has the elements which are solutions of argmin. Next, we i c i 1 discard all those subsets s i from S which do not have c 1 as their first element and solve the problem recursively. Pseudo code of the polynomial time algorithm to find { c } K =1 and {B } K =1 Algorithm 1. is presented as C. An Illustrative Example Let us consider a two-level networ withn 1 = 6 andn 2 = 12. We assume thatc = {4, 3, 2}, C networ budget = 60 and C attacer budget = 11. Next, we solve the bi-level optimization problem. Observe that, costs satisfy c 1 2 c 3. So the algorithm chooses the solution of the ULP as ( c 1 = 4, c 2 = 3) and the solution of the LLP as (B 1 = 11 4 = 2, B2 = = 1). To corroborate these result, in Figure 4, we plot the min KLD for all combinations of the parameters B 1 P 1,0,P 0,1 and B 2 in the tree. We vary the parameter B 1 from 0 to 6 and B 2 from 0 to 12. All the feasible solutions are plotted in red and unfeasible solutions are plotted in blue. Figure 4 corroborates

20 20 Algorithm 1 Bi-Level Optimization Algorithm ( ) Require: C = {c } n =1 with c N j+1 max min c min j N j 1: S All K out of n combinations {s i } K) (n i=1 with elements of s i arranged in decreasing order 2: for i = 1 to ( n K) do K 3: if c i N > Cbudget networ =1 4: S S/s i then 5: end if 6: end for 7: if S is an empty set then 8: return (φ, φ) 9: else 10: for = 1 to K do 11: c = min 12: B j s cj C attacer budget c 13: C attacer budget (C attacer budget c B ) 14: end for 15: return ({ c } K =1, {B } K =1 ) 16: end if where s has elements which are solutions of argmin i C attacer budget c i the results of our algorithm. Notice that, the attac configuration {B } K =1 is the set containing the number of Byzantines residing at different levels of the tree. However, the FC cannot identify the Byzantines in the networ. Also, notice that when the adversary attacs more than 50% of nodes at level 1, the decision fusion scheme becomes completely incapable. In these scenarios, where the FC is blind, the nowledge of attac configuration will not incur any performance benefit. Next, we present a reputation-based Byzantine identification/mitigation scheme, which wors even when

21 21 Fig. 4. min KLD vs. attac configuration (B 1,B 2) for P d = 0.9, P fa = 0.1. the networ is blind, in order to improve the detection performance of the networ. We propose a simple yet efficient Byzantine identification scheme and analyze its performance. VI. AN EFFICIENT BYZANTINE IDENTIFICATION SCHEME In this section, we propose and analyze a Byzantine identification scheme to be implemented at the FC. A. Byzantine Identification Scheme We assume that the FC has the nowledge of the attac model and utilizes this nowledge to identify the Byzantines. The FC observes the local decisions of each node over a time window T, which can be denoted by (,i) = u 1 (,i),...,u T (,i)] for 1 i N at level 1 K. We also assume that there is one honest anchor node with probability of detection Pd A and probability of false alarm Pfa A present and nown to the FC. We employ the anchor node to provide the gold standard which is used to detect whether or not other nodes are Byzantines. The FC can also serve as an anchor node when it can directly observe the phenomenon and mae a decision. We denote the Hamming distance between reports of the anchor node and an honest node i at level over the time window T by d A H (,i) = UA U H (,i), that is the number of elements that are different between U A and U H (,i). Similarly, the Hamming distance between reports of the anchor node and a Byzantine node i at level over the time window T is denoted by d A B (,i) = UA U B (,i). Since the FC is aware of the fact that Byzantines might be

22 22 present in the networ, it compares the Hamming distance of a node i at level to a threshold η, i, (a procedure to calculate η is discussed later in the paper), to mae a decision to identify the Byzantines. In tree networs, a Byzantine node alters its decision as well as received decisions from its children prior to transmission in order to undermine the networ performance. Therefore, solely based on the observed data of a node i at level, the FC cannot determine whether the data has been flipped by the node i itself or by one of its Byzantine parent node. In our scheme, the FC maes the inference about a node being Byzantine by analyzing the data from the node i as well as its predecessor nodes data. FC starts from the nodes at level 1 and computes the Hamming distance between reports of the anchor node and the nodes at level 1. FC declares node i at level 1 to be a Byzantine if and only if the Hamming distance of node i is greater than a fixed threshold η 1. Children of identified Byzantine nodes C(B 1 ) are not tested further because of the non-overlapping condition. However, if a level 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 level, for 2 K, to be a Byzantine if and only if the Hamming distance of node i is greater than a fixed threshold η and Hamming distances of all predecessors of node i is less than equal to their respective thresholds η j. In this way, it is possible to counter the data falsification attac by isolating Byzantine nodes from the information fusion process. The probability that a Byzantine node i at level is isolated at the end of the time window T, is denoted as P iso B (,i). B. Performance Analysis As aforementioned, local decisions of the nodes are compared to the decisions of the anchor node over a time window of length T. The probability that an honest node i at level maes a decision that is different from the anchor node is given by P AH diff(,i) = P(u A i = 1,u H,i = 0,H 0)+P(u A i = 0,u H,i = 1,H 0) +P(u A i = 1,u H,i = 0,H 1)+P(u A i = 0,u H,i = 1,H 1) = P 0 (P fa +P A fa) 2P fap A fa]+p 1 (P d +P A d ) 2P dp A d ]. = P 0 P AH diff(,i,0)]+p 1 P AH diff(,i,1)].

23 23 where the prior probabilities of the two hypotheses H 0 and H 1 are denoted by P 0 and P 1, respectively. The probability that a Byzantine node i at level sends a decision different from that of the anchor node is given by P AB diff (,i) = P(u A i = 1,u B,i = 0,H 0)+P(u A i = 0,u B,i = 1,H 0) +P(u A i = 1,u B,i = 0,H 1 )+P(u A i = 0,u B,i = 1,H 1 ) = P 0 P A fap fa +(1 P A fa)(1 P fa)]+p 1 P A d P d +(1 P A d )(1 P d)]. = P 0 P AB diff (,i,0)]+p 1P AB diff (,i,1)]. The difference between the reports of a node and the anchor node under hypothesis l {0,1} (i.e., d A I (,i,l), I {H,B}) is a Bernoulli random variable with mean PAH diff (,i,l) for honest nodes and Pdiff AB (,i,l) for Byzantines. FC declares node i at level to be a Byzantine if and only if the Hamming distance of node i is greater than a fixed threshold η and Hamming distances of all predecessors of node i are less than equal to their respective thresholds η j. The probability that a Byzantine node i at level is isolated at the end of the time window T can be expressed as = = P iso B (,i) = P(dA B (,i) > η ),(d A H ( 1,i) η 1),,(d A H (1,i) η 1)] ] 1 P l Pd A B(,i,l) > η ] Pd A H(m,i,l) η m ] l {0,1} l {0,1} P l T j=η +1 m=1 ( ) 1 T (Pdiff AB j (,i,l))j (1 Pdiff AB (,i,l))t j m=1 η m j=0 ( ) T (Pdiff AH j (m,i,l))j (1 Pdiff AH (m,i,l))t j. For large T, by using the normal approximation, we get P iso B (,i) = l {0,1} P l Q η TP AB diff (,i,l) (TP AB diff (,i,l)(1 PAB diff (,i,l))) This can be written recursively as follows PB iso ( +1,i) = P l (1 b(,l)) l {0,1} 1 m=1 Q ( a( +1,l) a(, l) TP AH diff (m,i,l) η m (TP AH diff (m,i,l)(1 PAH diff (m,i,l))) ) P iso B (,i,l) ], (24).

24 24 with P iso B (,i). = l {0,1} P l P iso B (,i,l)], and a(,l) = Q b(,l) = Q η TP AB diff (,i,l) (TP AB diff (,i,l)(1 PAB diff (,i,l))) η TP AH diff (,i,l) (TP AH diff (,i,l)(1 PAH diff (,i,l))) One can choose η such that the isolation probability of honest nodes at level based solely on its data under the hypothesis H l (i.e., b(,l)) is constrained to some value δ << 0.5. In other words, we choose η such that max b(,l) = δ, i.e., l {0,1} η = Q 1 (δ ) TPdiff AH (,i,l )(1 Pdiff AH (,i,l ))+TPdiff(,i,l AH ) (25) where l = argmax l b(,l). Now, the expression for a(,l) can be written as a(,l) = Q Q 1 (δ ) Pdiff AH (,i,l )(1 Pdiff AH (,i,l ))+ T(Pdiff AH (,i,l ) Pdiff AB (,i,l)) Pdiff AB (,i,l)(1 PAB diff (,i,l)) Now using the fact that max l P AH diff P AB diff (,i,l) < minpdiff AB (,i,l), it can be shown that(pah diff l (,i,l ) a(,l) = 1. (,i,l)) < 0, i and, therefore, lim T Lemma 6: For a K level tree networ, for our proposed Byzantine identification scheme, the asymptotic (i.e., T ) probability that a Byzantine node i at level +1, for 1 K 1, is isolated is lower-bounded by, (1 δ j ). j=2 Proof: Notice that, lim a(,l) = 1. The asymptotic performance of the proposed scheme T can be analyzed as follows: lim T Piso B ( +1,i) = l {0,1} (1 δ ) = P l lim (1 b(,l)) T l {0,1} (1 δ j ). j=2 ( a( +1,l) a(, l) P l lim P iso B (,i,l)] T,. ) P iso B (,i,l) ]

25 25 P B iso () iso P (=1) B iso P B (=2) iso P (=3) B iso P B (=4) iso P B (=5) Time window T Fig. 5. Isolation probability P iso B (,i) vs. time window T. Notice that, the parallel networ topology is a special case of the tree networ topology with K = 1. For K = 1, our scheme can identify all the Byzantines with probability one because lim (1,i) = lim P l a(1,l)] = 1. When K > 1, we can choose η appropriately such T Piso B T l {0,1} that Byzantines can be identified with a high probability. Next, to gain insights into the solution, we present some numerical results in Figure 5 that corroborate our theoretical results. We consider a tree networ with K = 5 and plot P iso B (,i), 1 5, as a function of the time window T. We assume that the operating points (P d,p fa ), 1 5, for the nodes at different levels are given 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 (Pd A,PA fa ) = (0.9,0.1). We also assume that the hypotheses are equiprobable, i.e., P 0 = P 1 = 0.5, and the maximum isolation probability of honest nodes at level based solely on its data is constrained by δ = 0.01,. It can be seen from Figure 5 that in a span of only T = 25 time windows, our proposed scheme isolates/identifies almost all the Byzantines in the tree networ. VII. CONCLUSION In this paper, we considered the problem of optimal Byzantine attacs on distributed detection mechanism in tree networs. We analyzed the performance limit of detection performance with Byzantines and obtained the optimal attacing strategies that minimize the detection error exponent. The problem was also studied from the networ designer s perspective. It was shown

26 26 that the optimal local detector is independent of the Byzantine s parameter. Next, we modeled the strategic interaction between the FC and the attacer as a Leader-Follower (Stacelberg) game and attacer and defender (FC) equilibrium strategies were identified. We also proposed a simple yet efficient scheme to identify Byzantines and analytically evaluated its performance. There are still many interesting questions that remain to be explored in the future wor such as analysis of the problem for arbitrary networ topologies. The case where Byzantines collude in several groups (collaborate) to degrade the detection performance can also be investigated. APPENDIX A PROOF OF LEMMA 2 To prove the lemma, we first show that any positive deviationǫ (0,p] in flipping probabilities (P 1,0,P 0,1 ) = (p,p ǫ) will result in an increase in D. After plugging in (P 1,0,P 0,1 ) = (p,p ǫ) in (9) and (10), we get π 1,0 = β 1 1,0 (1 P fa)+(1 β 1 0,1 )P fa]+α (p P fa(2p ǫ))+p fa] (26) π1,1 = β1,0 1 (1 P d )+(1 β 1 0,1 )P d ]+α (p Pd (2p ǫ))+p d ]. (27) Now we show that D is a monotonically increasing function of the parameter ǫ or in other words, dd dǫ > 0. dd dǫ = π 1,0 ( π 1,0 π 1,0 + (1 π 1,0) π 1,1 π1,1 ) ( π 1,1 1 π 1,1 +π 1,0 log π 1,0 π 1,0 1 π1,0 π 1,1 ) π 1,0log 1 π 1,0 1 π 1,1 (28) = π1,1 = α Pd and dπ 1,0 dǫ where dπ 1,1 dǫ equation, the condition dd dǫ > 0 becomes = π1,0 = α Pfa. After rearranging the terms in the above 1 π 1,0 1 π 1,1 + P fa P d log π 1,0 π 1,1 > π 1,0 + P fa π1,1 Pd log 1 π 1,0. (29) 1 π1,1 Since P d > P fa and β x,x < 0.5, π 1,1 > π 1,0. It can also be proved that P d have π 1,0 Pfa π1,1 > 1. Hence, we