Optimal Decision Strategies In Byzantine Environments

Optimum decision fusion in the presence of malicious nodes—often referred to as Byzantines—is hindered by the necessity of exactly knowing the statistical behavior of Byzantines. In this paper, we focus on a simple, yet widely adopted, setup in which a fusion center (FC) is asked to make a binary decision about a sequence of system states by relying on the possibly corrupted decisions provided by local nodes. We propose a game-theoretic framework, which permits to exploit the superior performance provided by optimum decision fusion, while limiting the amount of a priori knowledge required. We use numerical simulations to derive the optimum behavior of the FC and the Byzantines in a game-theoretic sense, and to evaluate the achievable performance at the equilibrium point of the game. We analyze several different setups, showing that in all cases, the proposed solution permits to improve the accuracy of data fusion. We also show that, in some cases, it is preferable for the Byzantines to minimize the mutual information between the status of the observed system and the reports submitted to the FC, rather than always flipping the decision made by the local nodes.

Cornell University - arXiv, 2022

Since the mid-1980s it has been known that Byzantine Agreement can be solved with probability 1 asynchronously, even against an omniscient, computationally unbounded adversary that can adaptively corrupt up to f < n/3 parties. Moreover, the problem is insoluble with f ≥ n/3 corruptions. However, Bracha's [Bra87] 1984 protocol (see also Ben-Or [Ben83]) achieved f < n/3 resilience at the cost of exponential expected latency 2 Θ(n) , a bound that has never been improved in this model with f = ⌊(n − 1)/3⌋ corruptions. In this paper we prove that Byzantine Agreement in the asynchronous, full information model can be solved with probability 1 against an adaptive adversary that can corrupt f < n/3 parties, while incurring only polynomial latency with high probability. Our protocol follows earlier polynomial latency protocols of King and Saia [KS16, KS18] and Huang, Pettie, and Zhu [HPZ22], which had suboptimal resilience, namely f ≈ n/10 9 [KS16, KS18] and f < n/4 [HPZ22], respectively. Resilience f = (n−1)/3 is uniquely difficult as this is the point at which the influence of the Byzantine and honest players are of roughly equal strength. The core technical problem we solve is to design a collective coin-flipping protocol that eventually lets us flip a coin with an unambiguous outcome. In the beginning the influence of the Byzantine players is too powerful to overcome and they can essentially fix the coin's behavior at will. We guarantee that after just a polynomial number of executions of the coin-flipping protocol, either (a) the Byzantine players fail to fix the behavior of the coin (thereby ending the game) or (b) we can "blacklist" players such that the blacklisting rate for Byzantine players is at least as large as the blacklisting rate for good players. The blacklisting criterion is based on a simple statistical test of fraud detection. if any corrupt player initiates a broadcast, then either all good players accept the same value v, and only v, or all good players accept nothing. See [Bra87] for details of this primitive. Validation. The Reliable-Broadcast primitive allows us to assume that all relevant communication is public, via broadcasts. Fix any protocol P based on broadcasts. Informally, a player p validates a message m originating from q if p has already accepted and validated a set of broadcasts that, were they to be received by q, would have caused q to make a suitable state transition according to P and broadcast m. See [Bra87] for details of validation. The reliable broadcast primitive prevents the adversary from sending conflicting messages to different players, or convincing one player to accept a broadcast and another not to. The validation mechanism prevents it from making state transitions logically inconsistent with the protocol P. Note, however, that in general P is probabilistic and validation permits a series of transitions that are logically possible but statistically unlikely. In summary, the adversary is characterized by the following powers. Full Information & Scheduling. The adversary knows the internal state of all players and controls the order in which messages are delivered. It may delay messages arbitrarily. Corruption & Coin Flipping. The adversary may adaptively corrupt up to f players as the execution of the protocol progresses. Once corrupted, a player continues to follow protocol , except the adversary now chooses the outcomes of all of its coin flips. Algorithm 1 Bracha-Agreement() from the perspective of player p Require: v p ∈ {−1, 1}. 1: loop 2: Reliable-Broadcast v p and wait until n − f messages are validated from some set of players S p. set v p ← sgn(q∈Sp v q). ⊲ sgn(x) = 1 if x ≥ 0 and −1 otherwise. 3: Reliable-Broadcast v p and wait until n − f messages are validated. if more than n/2 messages have some value v * then set v p ← v * , otherwise set v p ← ⊥.

Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing

It has been known since the early 1980s that Byzantine Agreement in the full information, asynchronous model is impossible to solve deterministically against even one crash fault [FLP 1985], but that it can be solved with probability 1 [Ben-Or 1983], even against an adversary that controls the scheduling of all messages and corrupts up to < /3 players [Bracha 1987]. The main downside of [Ben-Or 1983, Bracha 1987] is that they terminate with 2 Θ() latency in expectation whenever = Θ(). King and Saia [KS 2016, KS 2018] developed a polynomial protocol (polynomial latency, polynomial local computation) that is resilient to < (1.14 × 10 −9) Byzantine faults. The new idea in their protocol is to detect-and blacklist-coalitions of likely-bad players by analyzing the deviations of random variables generated by those players over many rounds. In this work we design a simple collective coin-flipping protocol such that if any coalition of faulty players repeatedly does not follow protocol, then they will eventually be detected by one of two simple statistical tests. Using this coin-flipping protocol, we solve Byzantine Agreement in polynomial latency, even in the presence of up to < /4 Byzantine faults. This comes close to the < /3 upper bound on the maximum number of faults [LSP 1982, BT 1985, FLM 1986].

Lecture Notes in Computer Science, 2011

The standard Byzantine Agreement (BA) problem requires non-faulty processes to agree on a common value. In many real-world applications, it is important that the processes agree on the correct value rather than any value. In this paper, we present a problem called Accurate Byzantine Agreement (ABA) in which all processes get a common feedback (or payoff) from the environment indicating if the value they agreed upon was correct or not. The solution to this problem, referred to as the ABA algorithm, requires the non-faulty processes to incorporate the feedback so that their chance of choosing the correct value improves over subsequent iterations of the algorithm. We present an algorithm that solves the ABA problem based on two key ingredients: a standard solution to the BA problem and a multiplicative method to maintain and update process weights indicative of how often they are correct. We give guarantees on the accuracy of the algorithm based on assumptions on the accuracy of the processes and the proportion of faulty and non-faulty processes in the system. For each iteration, if the weight of accurate processes is at least 3/4 th the weight of the non-faulty processes, the algorithm always decides on the correct value. When the non-faulty processes are accurate with probability greater than 1/2, the algorithm decides on the correct value with very high probability after some initial number of mistakes. In fact, among n processes, if there exists even one process which is accurate for all iterations, the algorithm is wrong only O(log n) times for any large number of iterations of the algorithm.

2013 IEEE International Conference on Acoustics, Speech and Signal Processing, 2013

This paper considers the problem of optimal distributed detection with independent identical sensors in the presence of Byzantine attacks. By considering the attacker to be strategic in nature, we address the issue of designing the optimal fusion rule and the local sensor thresholds that minimize the probability of error at the fusion center (FC). We first consider the problem of finding the optimal fusion rule under the constraint of fixed local sensor thresholds and fixed Byzantine strategy. Next, we consider the problem of joint optimization of the fusion rule and local sensor thresholds for a fixed Byzantine strategy. Then we extend these results to the scenario where both the FC and the Byzantine attacker act in a strategic manner to optimize their own utilities. We model the strategic behavior of the FC and the attacker using game theory and show the existence of Nash Equilibrium. We also provide numerical results to gain insights into the solution.

Networks, 1993

We consider the problem of efficient information exchange in a communication network whose nodes and/or links are subject to Byzantine faults that are randomly and independently distributed through the network. The goal is almost safe communication, i.e., getting to every fault-free node information about every other fault-free node, with probability converging to one as the number of nodes grows. We present nonadaptive almost-safe communication schemes working for various networks in asymptotically optimal time and using an asymptotically optimal number of message bits.

Proceedings of the 2012 ACM symposium on Principles of distributed computing, 2012

In the Byzantine agreement problem, a set of n processors, any f of whom may be arbitrarily faulty, must reach agreement on a value proposed by one of the correct processors. It is a celebrated result that unless n > 3 f , Byzantine agreement is impossible in a variety of computation and communication models. This is due to the fact that faulty processors can equivocate, that is, say different things to different processors. If this ability is mitigated, for example by assuming a global broadcast channel, then n > 2 f is sufficient. With very few exceptions, the literature on Byzantine agreement has been confined to the n > 2 f and n > 3 f paradigms. We bridge the gap between these two paradigms by assuming partial broadcast channels among sets of three processors, observing that equivocation is fundamentally an act involving three parties: a faulty processor that lies (inconsistently) to two correct processors. We characterize the conditions under which Byzantine agreement is possible for all n = 2 f + h, h an integer in [1.. f ], by giving asymptotically tight bounds on the number of necessary and sufficient partial broadcast channels. We prove these bounds by a reduction to a problem in extremal combinatorics, which itself is a natural generalization of a well-studied hypergraph coloring problem. Algorithmically, we show that deciding whether a given set of broadcast channels enables Byzantine agreement is co-NPcomplete. Although partial broadcast channels have been studied in prior work, the bounds obtained on the number of required channels were sub-optimal by up to a factor of Θ(n 2 ). Moreover, this work has been confined to the synchronous model. In contrast, we apply our results to several distinct models and provide stronger motivation for using partial broadcast channels in practice, drawing from recent work in the systems community.

Proceedings of the 17th International Conference on Distributed Computing and Networking, 2016

k-Set agreement is a central problem of fault-tolerant distibuted computing. Considering a set of n processes, where up to t may commit failures, let us assume that each process proposes a value. The problem consists in defining an algorithm such that each non-faulty process decides a value, at most k dfferent values are decided, and the decided values satisfy some context-depending validity condition. Synchronous message-passing algorithms solving k-set agreement have been proposed for different failure models (mainly process crashes, and process Byzantine failures). Differently, k-set agreement cannot be solved in failure-prone asynchronous message-passing systems when t ≥ k. To circumvent this impossibility an asynchronous system must be enriched with additional computational power. Assuming t ≥ k, this paper presents a distributed algorithm that solves k-set agreement in an asynchronous message-passing system wher up to t processes may commit Byzantine failures. To that end, each process is enriched with randomization power. While randomized k-set agreement algorithms exist for the asynchronous process crash failure model where t ≥ k, to our knowledge the proposed algorithm is the first that solves k-set agreement in the presence of up to t ≥ k Byzantine processes. Interestingly, this algorithm is signature-free, and ensures that no value proposed only by Byzantine processes can be decided by a non-faulty process. Its design is based on a modular construction which rests on a "no-duplicity" one-to-all broadcast abstraction, and two all-to-all communication abstractions.

Distributed Computing, 2013

We present an efficient, optimally-resilient Asynchronous Byzantine Agreement (ABA) protocol involving n = 3t + 1 parties over a completely asynchronous network, tolerating a computationally unbounded Byzantine adversary, capable of corrupting at most t out of the n parties. In comparison with the best known optimally-resilient ABA protocols of Canetti and Rabin (STOC 1993) and Abraham, Dolev and Halpern (PODC 2008), our protocol is significantly more efficient in terms of the communication complexity. Our ABA protocol is built on a new statistical asynchronous verifiable secret sharing (AVSS) protocol with optimal resilience. Our AVSS protocol significantly improves the communication complexity of the only known statistical and optimally-resilient AVSS protocol of Canetti et al. Our AVSS protocol is further built on an asynchronous primitive called asynchronous weak commitment (AWC), while the AVSS of Canetti et al. is built on the primitive called asynchronous weak secret sharing (AWSS). We observe that AWC has weaker requirements than AWSS and hence it can be designed more efficiently than AWSS.

2018

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 ...