Decision Fusion with Unbalanced Priors under Synchronized Byzantine Attacks: a Message-Passing Approach Andrea Abrardo, Mauro Barni, Kassem Kallas, and Benedetta Tondi University of Siena, Siena, Italy abrardo@diism.unisi.it, barni@dii.unisi.it, k kallas@hotmail.com, benedettatondi@gmail.com Abstract—We consider a variant of the decision fusion problem in the presence of Byzantines where the two states of the system under observation are not equiprobable. In this setup, the Byzantines can not adopt a simple corruption strategy consisting in flipping the local decisions regardless of the estimated state of the system. Doing so, in fact, they would reveal their presence to the fusion center, since their reports would not follow the expected statistics. On its side, the fusion center can exploit the knowledge of the a-priori probabilities to improve its decision. In view of the above observations, we first introduce a new corruption strategy for the Byzantines, which permits them to make the statistics of their reports indistinguishable from those of the honest nodes. Then, we adopt the perspective of the fusion center and we propose a nearly-optimum, efficient, fusion strategy based on message passing, to face with the new attack. We do so in the most challenging scenario wherein the Byzantines are synchronised, i.e. they share a common source of randomness allowing them to submit wrong reports in a simultaneous way. We prove the validity of the proposed approach under several working conditions with regard to the percentage of byzantine nodes, the length of the observation window and the a priori- probabilities of the system states. I. I NTRODUCTION Decision fusion in the presence of Byzantines [1] has received an increasing attention in the last years due to its relevance in several scenarios including: wireless sensor networks [2], cognitive radio [3], distributed detection [4] and many others [5], [6], [7], [8]. In the most studied version of the problem, a Fusion Center (FC) has to make a binary decision about the status of an observed system, by collecting the decisions made locally by the nodes. In doing so, the FC must take into account the possible presence of Byzantines, that is nodes submitting a wrong decision in the attempt to induce a decision error. Most of the works published so far, assume that the corruption strategy adopted by the Byzantines consists in flipping the local decision made by the node with a certain probability P mal , often assumed equal to 1. In the non-adversarial version of the problem, the Bayesian optimal fusion rule (known as Chair-Varshney rule) has been derived in [9] . Extending the Chair-Varshney rule to consider the presence of the Byzantines requires that the FC knows the positions of the Byzantines as well as the flipping probability P mal . This information is rarely available hence calling for the adoption of suboptimal fusion rules. In [4], for instance, by adopting a Neyman-Pearson setup and assuming that the Byzantines know the true system state, the asymptotic per- formance achievable by the FC when the size of the network (number of nodes) increases is analysed as a function of the percentage of Byzantines in the network. In order to improve the estimation of the system states, the FC can make its decision by relying on a sequence of reports sent over an observation window of length m, referring to m subsequent states of the observed system. In this way, it is possible for the FC to isolate the Byzantines and consequently ignore their reports. In this vein, the analysis of [4] is extended in [10] to a situation in which the Byzantines are unaware of the true system state. Byzantines isolation is achieved by counting the mismatches between the reports received from each node and the global decision made by the FC. To over- come the lack of knowledge about the exact strategy adopted by the Byzantines, the authors adopt a game-theoretic setup in which each party makes its best choice without knowing the strategy of the other party. A soft isolation scheme is proposed in [11], where the reports from suspect nodes are given a lower reputation rather than being completely discarded. Even in [11], the lack of knowledge at the FC about the strategy adopted by the attacker (and viceversa) is tackled by adopting a game-theoretic formulation. A rather different approach is adopted in [12], where a tolerant scheme that mitigates the impact of Byzantines on the global decision is used rather that ignoring the reports submitted by suspect nodes. When the value of P mal and the probability that a node is a Byzantine are known, the optimum fusion rule under multiple observations can be derived [13]. Since P mal is usually unknown to the FC, in [13] the value of P mal used within the optimum fusion rule and the value actually used by the Byzantines are strategically chosen in a game-theoretic setting. One of the main results in [13] is that the best option for the Byzantines is not to always flip the local decision (corresponding to P mal =1), since, once the malicious nodes are identified, a flipped report still brings useful information about the sate of the system. On the contrary, for certain combinations of the distribution of Byzantines within the network and the length of the observation window, it is better for the Byzantines to minimize the mutual information between the reports submitted to the FC and the system states (P mal =0.5). One of the main drawbacks of the optimum fusion rule proposed in [13] is that the computational cost grows exponentially with the size of the 1160 Proceedings, APSIPA Annual Summit and Conference 2018 12-15 November 2018, Hawaii 978-988-14768-5-2 ©2018 APSIPA APSIPA-ASC 2018