wChain: A Fast Fault-Tolerant Blockchain Protocol for Multihop Wireless Networks
Minghui Xu, Chunchi Liu, Yifei Zou, Feng Zhao, Jiguo Yu, Xiuzhen Cheng
JJOURNAL OF L A TEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 1 wChain: A Fast Fault-Tolerant BlockchainProtocol for Multihop Wireless Networks
Minghui Xu,
Student Member, IEEE,
Chunchi Liu, Yifei Zou, Feng Zhao,
Member, IEEE,
Jiguo Yu,
SeniorMember, IEEE,
Xiuzhen Cheng,
Fellow, IEEE, (cid:70)
Abstract —This paper presents wChain , a blockchain protocol specif-ically designed for multihop wireless networks that deeply integrateswireless communication properties and blockchain technologies underthe realistic SINR model. We adopt a hierarchical spanner as thecommunication backbone to address medium contention and achievefast data aggregation within O (log N log Γ) slots where N is the net-work size and Γ refers to the ratio of the maximum distance to theminimum distance between any two nodes. Besides, wChain employsdata aggregation and reaggregation, and node recovery mechanisms toensure efficiency, fault tolerance, persistence, and liveness. The worst-case runtime of wChain is upper bounded by O ( f log N log Γ) , where f = (cid:98) N (cid:99) is the upper bound of the number of faulty nodes. To validateour design, we conduct both theoretical analysis and simulation studies,and the results only demonstrate the nice properties of wChain , but alsopoint to a vast new space for the exploration of blockchain protocols inwireless networks. Index Terms —blockchain, fault-tolerance, multihop wireless networks,SINR model.
NTRODUCTION I N recent years, the popularity of 5G and IoT has arisenmore security and privacy issues relevant to identitymanagement, data sharing, and distributed computing inwireless networks. With the inception of Bitcoin, blockchainhas been envisioned as a promising technology that canbe utilized to support various applications such as onlinepayments and supply chain due to its salient propertiesof decentralization, immutability, and traceability. Corre-spondingly, effort has been put on protecting wireless ap-plications using the blockchain technology, e.g., mobile edgecomputing (MEC) [1], intelligent 5G [2], vehicular network-ing [3], and wireless sensor networking (WSN) [4].The com-mon idea of applying blockchain in wireless networks is M. Xu is with the Department of Computer Science, The George WashingtonUniversity, Washington, DC 20052 USA. E-mail: [email protected]. Liu, Y. Zou, X. Cheng are with the School of Computer Science andTechnology, Shandong University, Qingdao, 266510, P.R. China. E-mail: { [email protected]; [email protected]; [email protected] } .F. Zhao (Corresponding Author) is with the Guangxi Colleges andUniversities Key Laboratory of Complex System Optimization and BigData Processing, Yulin Normal University, Yulin, P.R. China. E-mail:[email protected]. Yu is with the Qilu University of Technology (Shandong Academy ofSciences), Jinan, Shandong, 250353, P.R. China; with Shandong ComputerScience Center (National Supercomputer Center in Jinan), Jinan, Shandong,250014, P.R. China; and with Shandong Laboratory of Computer Networks,Jinan, 250014, China. Email: [email protected]. to introduce trustlessness with blockchain so that functionssuch as identity management and data sharing become moreefficient and secure.However, previous studies on blockchain-enabled wire-less applications mostly focus on developing practical ap-plications or proposing architectures based on existingblockchain protocols, which were originally designed forwired network applications and thus are not suitable forwireless scenarios. To defend this point of view, let’s con-sider the state-of-the-art blockchain protocols. The conceptof proof of physical resources has been widely adopted,e.g., Proof-of-Work (PoW), Proof-of-Space, Proof-of-ElapsedTime, Proof-of-Space Time. The noteworthy drawback ofthese protocols is that they require high electricity, storage,or specific hardware (e.g., Intel SGX), which wireless de-vices cannot provide. On the other hand, protocols basedon virtual resources such as stake, reputation, or credibil-ity, i.e., Proof-of-Stake, Delegated Proof-of-Stake, Proof-of-Authority, Proof-of-Reputation, etc., always have a compli-cated design in order to avoid the centralization of wealthor power, which justifies why there is still no such a protocolparticularly designed for wireless networks.Another line of blockchain protocols, such as Ripple, Al-gorand, Tendermint, and Hotstuff, rely on message passing.They provide blockchain systems with safety and liveness inconfronting faulty nodes or even Byzantine failures. How-ever, when implemented in wireless networks, the followingtwo major problems need to be addressed: • Traditional protocols commonly used on the Internetare not efficient enough in wireless networks. For ex-ample, PBFT [5] and Tendermint [6] can reach a con-sensus within O ( N ) successful transmissions where N is the network size, while Hotstuff [7] reducescomplexity to O ( N ) but increases additional over-head due to the introduction of the cryptographictools. • Transforming fault-tolerant or Byzantine fault-tolerant protocols used in wired networks to wire-less environments remains a tricky problem. A re-cent work by Poirot et al. presented a fault-tolerantwireless Paxos for low-power wireless networks[8]. However, their results still indicate that tradi-tional consensus algorithms require many messageexchanges and high bandwidths, which may not be
This work has been submitted to the IEEE for possible publication. Copyright may be transferred without notice, after which this versionmay no longer be accessible. a r X i v : . [ c s . D C ] F e b OURNAL OF L A TEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 2 available in wireless networks.In this paper, we consider designing a message-passing-based blockchain protocol for wireless networks. To achievethis goal, we primarily need to overcome the challenge ofproperly implementing the medium access control (MAC)layer for blockchain. A promissing idea is to adopt theabstract MAC layer service (absMAC) proposed by Kuhn etal. [9], which provides low-level network functions to helpdesign distributed algorithms in wireless networks. Themost efficient implementation of absMAC was presentedin [10], which achieves the optimal bound for a successfultransmission in O (∆+log N ) slots via carrier sensing, where ∆ is the maximum number of neighbors a node may have.Obviously, with this efficient implementation and a fault-tolerant consensus algorithm that requires an optimal O ( N ) successful transmissions (e.g., Hotstuff), the best we canobtain is a protocol with a communication complexity ashigh as O ( N (∆ + log N )) slots.Nevertheless, this is far less than satisfactory. One needsa blockchain protocol that can take into consideration theunique features of wireless networking by deeply integrat-ing wireless communications with blockchain to achievethe necessary properties of efficiency, fault-tolerance, persis-tence, and liveness. Since the most basic primitives heavilyused in wireless consensus algorithms are broadcasts anddata aggregations, which are respectively responsible fordisseminating and collecting opinions or votes from peers,we use a spanner structure to accelerate the consensusprocess. A spanner is a hierarchical communication back-bone that can organize nodes carefully to speed up dataaggregation and dissemination processes. It introduces asparse topology in which only a small number of linksneed to be maintained such that efficiency and simplicitycan be well-balanced. Due to the need for decentralizationand practicality, our spanner is constructed in a distributedmanner and works under the realistic Interference-plus-Noise-Ratio (SINR) model. Facilitated with the spannerstructure, we develop the two primitives of data aggregationand reaggregation to handle interference with an adaptivepower scheme directly. These two primitives are adopted byour fault-tolerant wChain protocol to address faulty behav-iors caused by fail-stop errors and dynamic topologies suchthat the properties of resource conservation, fault-tolerance,efficiency, persistence, and liveness can be achieved.The main contributions of this paper are summarized asfollows.1) To the best of our knowledge, wChain is the firstblockchain protocol that is particularly designed formultihop wireless networks under a realistic SINRmodel to deeply integrate wireless communicationsand blockchain technologies.2) The wChain protocol ensures high performance byemploying a spanner as the communication back-bone. The runtime upper bound of the protocolis O (log N log Γ) when crash failures happen in alow frequency, and the worst-case upper bound is O ( f log N log Γ) .3) Our wChain protocol simultaneously achievesproperties of resource conservation, fault-tolerance,efficiency, persistence, and liveness, which are proved by theoretical analysis and verified by sim-ulation studies.The rest of the paper is organized as follows. Section 2introduces the most related work on the state-of-the-artblockchain protocols and consensus algorithms in wirelessnetworks. Section 3 presents our model and preliminaryknowledge. In Section 4, building blocks including utilitiesand data aggregation and reaggregation subroutines arefirst presented, then the three-phase wChain protocol isexplained in detail. Our wChain protocol is theoreticallyanalyzed in Section 5 in terms of efficiency, persistence, andliveness. We report the results of our simulation studies inSection 6 and conclude this paper in Section 7. ELATED W ORK
Blockchain protocols for wireless networks.
Blockchaintechnology has been studied for wireless applications suchas mobile edge computing (MEC) [1], intelligent 5G [2],vehicular networking [3], secure localization [4], and Wire-less D2D Transcoding [11]. Feng et al. [1] considered thejoint optimization of blockchain and MEC through a radioand computational resource allocation framework. Dai et al. [2] proposed a secure and intelligent architecture for next-generation wireless networks by integrating blockchain andAI technologies. In vehicular ad hoc networks, Malik etal. [3] utilized blockchain for secure key management. Ablockchain-based trust management model was proposedto ensure secure localization in wireless sensor networks[4]. A blockchain-enabled Device-to-Device (D2D) transcod-ing system was developed to provide trustworthy wirelesstranscoding services in [11]. Onireti et al. [12] provided ananalytic modeling framework to obtain a viable area forwireless PBFT-based blockchain networks. In [13], a trustlessmechanism with PoW-based blockchain was establishedto incentivize nodes to store data. Liu et al. [14] realizedcomputation offloading and content caching with mobileedge nodes in wireless blockchain networks. Sun et al. [15] proposed an analytic framework to explore how theperformance and security of wireless blockchain systems areaffected by wireless communication features such as SINR.Despite these extensive studies on applying blockchainto wireless networks, we are still in dire need of blockchainprotocols that are specifically designed for wireless net-works, which drives us to investigate and summarize ex-isting distributed leader election and consensus algorithmsfor wireless networks in the sequel.
Consensus and leader election algorithms for wirelessnetworks.
Consensus and leader election have also beenextensively explored in various wireless contexts, and thecorresponding solutions can guide us to design wirelessblockchains in a proper way. Most existing studies on con-sensus and leader election for wireless networks assume ahigh-level wireless network abstraction [16] [17] [18] [19][20] [21] or a realistic model grappling with issues in physi-cal and link layers [22] [23] [24] [25] [26].Moniz et al. [16] proposed a BFT consensus protocolwith runtime bounded by O ( N ) among k > (cid:98) N (cid:99) nodesin wireless ad hoc networks. They hid physical layer infor-mation but let nodes directly use high-level communicationprimitives. Leveraging the elegance of absMAC, Newport OURNAL OF L A TEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 3 provided upper and lower bounds for distributed consen-sus in wireless networks [17]. Subsequently, Newport andRobinson proposed a fault-tolerant consensus algorithmthat terminates within O ( N log N ) with unknown networksize [18]. A fully distributed leader election scheme wasproposed to make election value unforgeable and be re-sistant to jamming attacks in wireless sensor networks. Asecure extrema finding algorithm was proposed as a lock-step leader election algorithm that can be transformed toa secure preference-based leader election algorithm with autility function depicting nodes’ preference in wireless adhoc networks [20]. A top k -leader election algorithm withfaulty nodes was presented in [21].Chockler et al. [22] investigated the relationship betweencollision detection and fault-tolerant consensus under agraph-based model. Assuming a graph model with messagedelays, Scutari and Sergio [23] proposed a consensus algo-rithm in wireless sensor networks with multipath fading.Aysal et al. [24] studied the average consensus problem withprobabilistic broadcasts under a graph-based model. Richa et al. [25] focused on self-stabilization of leader election forsingle-hop wireless networks to mitigate jamming attacksby adaptively adjusting the transmission probability at theMAC layer. Gołe¸biewski and Klonowski [26] proposed a fairleader election scheme in ad-hoc single-hop radio sensornetworks, enabling resistance to adversaries who can trans-mit continuously to block channels or try to forge identities.In contrast, we observe that designing a blockchainprotocol is more challenging than consensus and leaderelection in wireless networks. On the one hand, blockchainintroduces new data structures such as a chain of blocks, andthus the corresponding computation, communication, andstorage overheads should be carefully addressed. On theother hand, it is harder to balance performance and securityin blockchain since consensus and leader elections offermore straightforward services without requiring a strongsecurity guarantee. More importantly, Blockchain requiresextra design elements (e.g., randomness, transaction andblock verification, blockchain update) to guarantee strictpersistence and liveness properties. ODELS AND P RELIMINARIES
Blockchain Basics.
Each node v maintains a blockchainlocally, denoted by BC v , which is a hash-chain of blocks. B iv refers to the i th block in BC v . We also denote BC i + v ( BC i − v ) as the partial blockchain of BC v before (after) B iv .Each block contains multiple transactions, and tx ji stands forthe j th transaction in B iv . Assume the latest block of BC v is B kv ; then v ’ view is defined as a tuple { seq, hash } v , where seq and hash are the sequence number and block hash of B kv , respectively. Besides, we adopt the UTXO model due toits remarkable properties such as large scalability and highlevel of security. We further assume nodes are supported bypublic key infrastructure, and the cryptographic primitivessuch as signature leveraged in our design are secure so thatno malicious entity can spoof the messages. Network Model.
We consider a multihop wireless adhoc network with a set V of N nodes deployed in a 2-dimensional geographic plane. Let d ( u, v ) denote the Eu-clidean distance between nodes u and v , D R ( v ) denote the disk centered at v with a radius R , and N R ( v ) denote theset of nodes excluding v within D R ( v ) . For simplicity, wenormalize the minimum distance between any two nodes tobe 1, and denote by Γ the ratio of the maximum distance tothe minimum distance between any two nodes. Each nodehas a unique id and knows no advanced information otherthan the network size N . The transmission power of eachnode can be controlled for interference mitigation. Assumenodes can crash at any time, which means that each node iseither functioning normally or completely stop working. Anode is regarded as faulty if it crashes in the current epochor does not have the latest view due to crash failures inprevious epochs. Without loss of generality, we assume N isodd. Our protocol can tolerate at most f faulty nodes where f = (cid:98) N (cid:99) . Interference and SINR Model.
We adopt the Signal-to-Interference-plus-Noise-Ratio (SINR) model, which capturesthe wireless network interference in a more realistic andprecise way than a graph-based one. A standard SINRmodel can be formulated as follows, which states that amessage sent by u is correctly received by v if and onlyif SIN R ( u, v ) = P u · d ( u, v ) − α N + (cid:80) w ∈ S \{ u } P · d ( w, v ) − α ≥ β (1)holds, where N is the ambient noise, α ∈ (2 , is the path-loss exponent, threshold β > is determined by hardware,and S ⊆ V denotes the set of nodes transmitting simulta-neously with u . Besides, we further assume that nodes canperform physical carrier sensing. Maximal Independent Set (MIS).
A set S ⊆ V is an independent set of V with respect to distance r if for any pairof nodes u and v in S , d ( u, v ) > r ; and S is referred toas a maximal independent set if for any node w / ∈ S , thereis a node x ∈ S such that d ( w, x ) ≤ r . MIS has beenwidely researched in recent years, and there exist a numberof methods computing an MIS in a distributed manner. Inthis paper, we adopt the approach presented in [27], whichcomputes a distributed MIS in optimal time O (log N ) if thenodes’ density is a constant. Spanner Construction.
A spanner is a network back-bone possessing the following properties: it only needsto maintain a small number of links, and can balancewell between efficiency and simplicity compared to othertopologies. Taking advantages of these features we employa spanner to facilitate the deployment of our data aggre-gation algorithm in wChain . More specifically, we adoptthe distributed spanner construction algorithm presentedin [27], which can construct a sparse spanner, denoted as H , with a bounded maximum degree, in O (log N log Γ) slots with a high probability. As illustrated in Fig. 1, theconstruction process of H contains log Γ rounds; and inthe i th round, where i = 1 , , · · · , log Γ , V i contains thenodes in a maximal independent set elected from V i − byrunning a distributed MIS algorithm with respect to r i = 2 i ;thereby V log Γ ⊆ · · · ⊆ V ⊆ V = V . One can see that theconstructed H holds the following properties: • nodes in V i constitute an MIS of V i − with respect to r i ; • each node v ∈ V i − \ V i has a parent node u ∈ V i and d ( v, u ) ≤ r i ; OURNAL OF L A TEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 4 (a) Construction of the first two layers of a spannerfollowed by an MIS; here hollow and solid circlesrepresent V and V , respectively. (b) A ( log Γ + 1 )-level full spanner where arrowsstand for the MIS relationship: each MIS node at acertain level covers a set of nodes in the next lowerlevel. Fig. 1. Spanner visualization. • V log Γ contains only one node, i.e., the root.For ease of explanation, nodes in V \ V log Γ and V log Γ are referred to as followers and collector , respectively. In thispaper, we say that an event E occurs with high probability(w.h.p.) if for any c ≥ , E occurs with probability at least − /N c . A summary of all critical notations and theirsemantic meanings is presented in Table 1. TABLE 1Summary of Notations
Symbol Description BC v the blockchain locally stored at node vB iv the i th block in BC v BC i + v the partial blockchain of BC v before B iv BC i − v the partial blockchain of BC v after B iv C v the set of child nodes of vd v v ’s data (to be aggregated) id v v ’s unique id id pv the unique id of v ’s parent M v the message queue maintained by vm v a single message sent by vp the uniform transmission probability P i the transmission power in the i th round ˆ P the maximum transmission power R i the i th round tx ji the j th transaction in the i th block V i the set of nodes as an MIS of V i − σ a sufficiently large constant to determine pλ (cid:48) an upper bound of the network density µ a constant to determine the round length HE P ROTOCOL
In this section, we first present an overview on wChain and demonstrate the involved utilities; then we detail thedata aggregation and reaggregation algorithms, the twosubroutines of our blockchain protocol; finally, we proposethe three-phase fast fault-tolerant wChain protocol.
The wChain protocol is executed in disjoint and consecutivetime intervals called epochs , and at each epoch, no more than one block can be generated. Non-faulty nodes should ap-pend the new block to their local blockchain so that they canjointly maintain a consistent global view. Within each epoch,a spanner is first established as a communication backbone,and the collector of the spanner is appointed as the leaderto take charge of wChain for the entire epoch. However, ifa crash failure occurs, a new spanner should be constructedin a reaggregation procedure for the same epoch. Under thiscircumstance, the collector might be changed, but the leaderholds the line. Data can be aggregated from followers tothe new collector who subsequently sends the aggregateddata to the leader to complete the data aggregation processof the current epoch. wChain proceeds by three phases,namely
PREPARE , COMMIT , and
DECIDE . In the
PREPARE phase, an incumbent leader aggregates view messages fromall followers to learn about the latest view. If more than f followers respond with the same view as that of the leader,the leader can aggregate transactions in the COMMIT phase.Otherwise, the leader sends a message to the followers toabandon the current epoch. The threshold ( f +1) (including f followers plus one leader) is specifically designated tosatisfy the quorum intersection property which implies thatamong the ( f + 1) nodes, at least one node is non-faultyand has the correct information. In the DECIDE phase, theleader verifies the collected transactions, organizes theminto a new block, and then sends the new block togetherwith the view update information to the entire network, bywhich the nodes can update their local blockchain to obtainthe latest view.
Before delving into details of the wChain protocol, we ex-plain its commonly used utilities. First of all,
MSG( data v ) isused the most often in wChain to generate a single message m embodying a variable data field, which might be a string,a transaction, or a block. For example, the parameter data v can be view v (view information), or tx v (transaction); it alsoindicates that the input data is signed by v . Other than thedata field, m also includes the timestamp , kindred , and role fields for the verification purpose. The kindred is atuple consisting of v ’s identity id v and the identity id pv of v ’s parent. The role field clarifies the role (i.e., follower, OURNAL OF L A TEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 5
Algorithm 1:
Utilities for node v Function
MSG( data v ) m.data ← data v m.timestamp ← time m.kindred ← { id v , id pv } m.role ← { role v , level v } return m Function add ( M v , M u ) return M v ← M v ∪ M u Function packup ( M v ) for m ∈ M v do if m.data is a valid transaction then organize m.data into B v return B v Function append ( BC v , B u ) if B u contains the hash of the B kv then return BC v ← BC v + B u Function extract ( BC v , M viewv ) i ← the ( f + s ) th highest seq searched in M viewv return BC i + v Function update ( BC v , BC i + u ) for j = 1 to | BC i + u | do if B ju / ∈ BC v then append ( BC v , B ju ) return BC v collector, or leader) and the specific level information suchas V \ V . A receiver recognizes m as valid if m includescorrect identity and role information. Moreover, M v is amessage queue to hold multiple messages. Duplicate mes-sages in M v , which can be identified based on the timestampinformation, should be discarded. A node v can performan add ( M v , M u ) operation to append M u to its local M v .For block formation, we provide packup ( M v ) to read allmessages from M v and organize valid transactions into ablock. Besides, append ( BC v , B u ) can append a block B u from node u to BC v only if B u contains the hash of the lastblock of BC v . In addition, extract ( BC v , M viewv ) intendsto extract a partial blockchain BC i + v from BC v where i isdetermined by the ( f + s ) th highest seq searched in M viewv and s is an adjustable constant to be determined later. Fornode recovery, the update ( BC v , BC i + u ) function is designedto help a node v update BC v by complementing missedblocks from the received BC i + u . In this subsection, we present two critical algorithms torealize data aggregation and reaggregation. The objective ofdata aggregation is to rapidly collect data from all followersto the leader in O (log n log Γ) w.h.p. However, messagesmight be lost due to crash failures. Therefore, we proposethe data reaggregation subroutine to remedy such a situ-ation and ensure that the leader can completely aggregatethe data from all non-faulty nodes within an epoch. When blockchains are to be implemented in a wirelessnetwork, unicast and multicast are generally not needed.Instead, we should exploit the broadcast nature of thewireless medium. Decomposing a typical consensus processone can see that there exist three major communicationpatterns, namely one-to-many, many-to-one, and many-to-many, that heavily employ broadcast and data aggregation,the two communication primitives. Broadcast can improveblockchain’s efficiency since a one-to-many communicationonly costs a one-time broadcast to disseminate a node’smessage to all its peers within the communication range. Formany-to-many, the communication complexity is O ( N ) fora network size of N if using unicast and multicast. Withbroadcast, the complexity can be reduced to O ( N ) .However, to reach a consensus, medium contention andpacket collision need to be carefully addressed in wirelessnetworks. For this purpose we restrict the transmissionprobability to be p and transmission power to be P i , whichare formally described in Section 4.2 (also see line 6 inAlgorithm. 2). Algorithm 2:
DataAggregation ( data v ) Subroutine Function
DataAggregation( data v ) Initially, m v ← MSG( data v ) , M v = { m v } (cid:46) In R i ( i = 1 , , · · · , log Γ) : if v ∈ V i − \ V i then for µ · log N slots do send M v with probability p = σλ (cid:48) andpower P i = 2 N βr αi else if v ∈ V i then for µ · log N slots do listen on the channel if receive a valid M u then M v ← add ( M v , M u ) Leveraging a spanner, we propose the
DataAggregation( data v ) subroutine to aggregate the datalevel by level. As shown in Algorithm 2, a node v executing DataAggregation( data v ) takes data v as input, generates amessage m v by MSG(data) , and initializes M v = { m v } .Then data aggregation proceeds by log Γ rounds. We denoteby R i the i th round. Recall that for i = 1 , , · · · , log Γ , anode v ∈ V i − \ V i has a parent node u ∈ V i and d ( v, u ) ≤ r i .As a consequence, R i is responsible for aggregating thedata from V i − \ V i to V i , and R log Γ is the final round whenthe collector receives the data from V log Γ − \ V log Γ .In a specific round R i , the nodes in V i − \ V i constantlysend M v for µ · log N slots with probability p = σλ (cid:48) , where λ (cid:48) = 25 , µ and σ are sufficiently large constants whoselower bounds are given in Section 5.1. Note that µ · log N is the optimal number of slots to ensure that without crashfailures, the data from all child nodes can be completelyaggregated to their parent w.h.p. The transmission power OURNAL OF L A TEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 6 is set to be P i = 2 N βr αi so that the transmission range ofeach node is /α r i , where r i = 2 i . This transmission rangeis slightly larger than r i such that for any child v , its parentis within the unit disk centered at v with a radius r i . Thispower control strategy improves child nodes’ ability to resistinterference outside the unit disk, thus contributing to thesuccess of the transmissions. Moreover, a node v ∈ V i asa parent in R i listens on the channel for µ · log N slots toreceive messages from its children. If v receives a M u froma child u , it appends M u to M v . The entire data aggregationprocess can be finished in µ log N log Γ slots, and the datafrom all non-faulty nodes can be aggregated w.h.p., whichis proved in Section 5.1. Algorithm 3:
Reaggregation ( data v ) Subroutine (cid:46) as a leader while true do (cid:46) slot one broadcast M data(cid:96) (cid:46) slot two listen on the channel (cid:46) slot three if sense noise > N in slot two then broadcast m (cid:96) ← MSG ( reaggregation (cid:96) ) else broadcast m (cid:96) ← MSG ( stop (cid:96) ) and break (cid:46) data reaggregation wait for aggregated data from a collector (cid:46) as a follower while true do (cid:46) slot one listen on the channel (cid:46) slot two if receive M data(cid:96) in slot one and data v / ∈ M data(cid:96) then broadcast m v ← MSG ( miss v ) (cid:46) slot three listen on the channel (cid:46) data reaggregation if receive reaggregation message in slot three then run SpannerConstruction run DataAggregation ( data v ) else breakThe reaggregation subroutine has two stages: a three-slot integrity check stage (lines 3-11, lines 16-22) and a datareaggregation stage (lines 12-13, lines 23-28). We define abroadcast operation (used in lines 4/9/11/20) as transmit-ting a message with ˆ P = 2 N βr α log Γ so that a node listen-ing on the channel can either receive a message from thesender or sense noise exceeding N . Concretely, the integritycheck stage intends to examine whether the leader losesany message from non-faulty nodes using physical carriersensing. In slot one, the leader l broadcasts its current M data(cid:96) to the entire network, where M data(cid:96) is the message queue embodying the messages whose type is data (e.g., M view(cid:96) isthe message queue of the view messages). Upon receiving M data(cid:96) , each node v examines if its data v is included in M data(cid:96) . If not, v broadcasts m v ← MSG ( miss v ) so thatin slot two the leader can get the notice saying that somemessages are missed by sensing noise greater than N , andbroadcast m (cid:96) ← MSG ( reaggregation (cid:96) ) in slot three to startthe second stage. In the data reaggregation stage, all nodesexcept the leader run SpannerConstrcution to reconstruct aspanner free from faulty nodes. The
SpannerConstrcution procedure is the same as the one we illustrate in Section 3.The new spanner does not include the leader and it elects anew collector who is responsible for sending the aggregateddata to the leader. Afterwards, the nodes whose messagesare missed in M data(cid:96) run DataAggregation ( data v ) . Onlywhen no messages from non-faulty nodes are missed canthe leader broadcast m (cid:96) ← MSG ( stop (cid:96) ) to end the reaggre-gation process. Algorithm 4:
Fast Fault-Tolerant Blockchain Proto-col (cid:46) PREPARE (cid:46) as a leader broadcast m (cid:96) ← MSG ( view (cid:96) ) listen on the channel for µ log N log Γ slots execute Reaggregation ( view (cid:96) ) (cid:46) as a follower if receive view u from a leader then run DataAggregation ( view v ) else abandon the current epoch execute Reaggregation ( view v ) (cid:46) COMMIT (cid:46) as a leader if |{ m ∈ M view(cid:96) | m.data = view (cid:96) }| ≥ f + 1 then broadcast m (cid:96) ← MSG ( correct (cid:96) ) listen on the channel for µ log N log Γ slots execute Reaggregation ( tx v ) (cid:46) as a follower if receive correct (cid:96) from a leader then run DataAggregation ( tx v ) else abandon the current epoch execute Reaggregation ( tx v ) (cid:46) DECIDE (cid:46) as a leader B (cid:96) ← packup ( M tx(cid:96) ) , and BC (cid:96) ← append ( BC (cid:96) , B (cid:96) ) broadcast BC i + (cid:96) ← extract ( BC (cid:96) , M view(cid:96) ) (cid:46) as a follower if receive BC i + (cid:96) from the leader then update( BC v , BC i + (cid:96) ) else abandon the current epoch wChain is a three-phase protocol that can achieve con-sensus on the sequence of blocks and handle failures caused OURNAL OF L A TEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 7 by wireless node crashes. Concretely, wChain leveragesbroadcast communications, data aggregation and reaggre-gation, which are all specifically designed for wireless net-works. At each epoch, wChain proceeds by three phases,namely
PREPARE , COMMIT , and
DECIDE . In the follow-ing, we depict each phase to demonstrate how fast fault-tolerance can be achieved in wChain . Prepare.
The
PREPARE phase intends to help a leaderobtain a global view. Recall that when a spanner is con-structed for the first time, wChain appoints the collector asthe leader to take charge of the current epoch. As a leader, (cid:96) broadcasts m (cid:96) ← MSG ( view (cid:96) ) to the entire network. Eachfollower v runs DataAggregation ( view v ) upon receivingthe view information from (cid:96) in the previous slot. Otherwise,the follower abandons the current epoch. Note that all nodesshould execute the reaggregation subroutine to ensure thatthe data from all non-faulty nodes are completely aggre-gated. Commit.
Denote by M view(cid:96) the message queue of (cid:96) embodying its view information. The requirement of |{ m ∈ M view(cid:96) | m.data = view (cid:96) }| ≥ f + 1 means that the leadershould successfully receive no less than f view messagesthat have identical views as itself. That is also to say, atleast f + 1 nodes ( f followers and one leader) have anidentical view. If such a requirement is satisfied, the leadercan broadcast a correct message and listen on the channelto receive transactions while the followers receiving a cor-rect signal start transaction aggregation. The reaggregationsubroutine is still executed to ensure the full aggregation oftransactions. Decide.
When the leader (cid:96) receives all transactions fromnon-faulty nodes, it packs up the transactions to form ablock B (cid:96) , and appends the B (cid:96) to its local BC (cid:96) . Thenthe leader (cid:96) executes extract ( BC (cid:96) , M view(cid:96) ) to formulate a BC i + (cid:96) , which is used to help recover at least s nodes thathave crashed in previous epochs and need to update theirblockchains to become non-faulty. Each non-faulty follower v updates BC v by running update ( BC v , BC i + (cid:96) ) . ROTOCOL A NALYSIS
In this section, we analyze the protocol in terms of efficiencyof data aggregation and reaggregation, persistence, andliveness.
Theorem 1.
The runtime of the data aggregation subrou-tine is upper bounded by O (log N log Γ) slots w.h.p., and theruntime of the reaggregation subroutine is upper bounded by O ( f log N log Γ) slots w.h.p.Proof. We first present Lemma 1, which focuses on one slotin a given round R i . Lemma 1.
For a given slot in round R i , if v ∈ V i is a parentnode of some node in V i − , for any node u ∈ V i − ∩ N r i ( v ) , if u transmits, v can receive the message with a constant probability.Proof. For a given slot in round R i , we denote the ag-gregated transmission probability of V i − ∩ N r i ( v ) by P i ( v ) = (cid:80) w ∈ V i − ∩ N ri ( v ) p w . Let’s first prove that P i ( v ) canbe bounded by σ , where σ is a sufficiently large constant. Due to the property of the maximum independent set, thedisks of radius r i / centered at any node w ∈ V i − ∩ N r i ( v ) are disjoint. We define density λ as the number of nodes in V i − ∩ N r i ( v ) , then the upper bound of density is λ ≤ π [ r + r / π ( r / = 25 , (2)where λ = 25 when i = 1 . Since for each w ∈ V i − ∩ N r i ( v ) , p w = σλ (cid:48) , we have P i ( v ) = λ · σλ (cid:48) ≤ σ , where λ (cid:48) = 25 .Then we partition the whole space outside D r i ( v ) intorings R j for j ≥ , where R j is the ring with a distancein the range [ jr i , ( j + 1) r i ] from v . Denote by S j the set ofnodes in V i − that also fall into R j . Considering the propertyof maximum independent set, one can see that the disks ofradius r i / centered at the nodes in R j are disjoint. Thenwe have | S j | ≤ π [( j + 1) r i + r i / − π [ jr i − r i / π ( r i / ≤ j. (3)Let I ( v, w ) be the interference at v caused by w . Denoteby I out the interference caused by the nodes outside D r i ( v ) .Then one can calculate I out as follows: I out = ∞ (cid:88) j =1 (cid:88) w ∈ S j I ( v, w ) = ∞ (cid:88) j =1 (cid:88) w ∈ S j P i d ( v, w ) α · p ≤ ∞ (cid:88) j =1 | S j | σ · N βr αi ( jr i ) α ≤ β ( α − σ ( α − · N ≤ N/ , (4)where the last inequality holds when σ > β ( α − α − . Consid-ering the case when u is the only node that transmits in thecurrent slot, since u ∈ V i − ∩ N r i ( v ) , d ( v, u ) ≤ r i , we have SIN R ( v, u ) = P i d ( v,u ) α N + I out ≥ N βr αi r αi N + N/ ≥ β, (5)which indicates that if u is the only node who transmits, v can receive the message. Then we bound the probabilitythat u is the only transmitting node. Since P i ( v ) ≤ σ , theprobability that only u transmits at each slot is p u (cid:89) w ∈ V i ∩ N ri ( v ) \ u (1 − p w ) ≥ p u (cid:89) w ∈ V i ∩ N ri ( v ) (1 − p w ) ≥ p u (cid:89) w ∈ V i ∩ N ri ( v ) e − pw − p = pe − Pi ( v )1 − p ≥ ( σλ (cid:48) ) − e − λ (cid:48) σλ (cid:48)− ∈ Ω(1) . (6)This implies that with a constant probability, v canreceive u ’s message.In Algorithm 2, each round consists of a fixed number of µ · log N slots. At each slot, a child u ∈ V i − \ V i transmitsconstantly with probability p = σλ (cid:48) . Lemma 1 indicates thatat each slot, u can succeed in sending a message to its parent v with a constant probability denoted by ˆ p . Thus, by apply-ing the Chernoff bound (see Lemma. 2 in Sec. 8.1), the proba-bility that u succeeds in sending a message to its parent after µ · log N slots is − (1 − ˆ p ) µ log N ≥ − e − ˆ pµ log N ≥ − N − if µ ≥ / ˆ p . Since the density of the active nodes is bounded OURNAL OF L A TEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 8 by λ (cid:48) = 25 , the probability that all children succeed is (1 − N ) λ (cid:48) .Next, assume that the nodes have synchronized clocks.Then at each round R i , the nodes in different independentsets can send messages to their parents at the same time.The probability that the data has been aggregated to allparent nodes in R i is at least (1 − N ) λ (cid:48) V i +1 ≥ (1 − N − ) since λ (cid:48) V i +1 < N . Thus, the one-round data aggregationsucceeds in O (log N ) w.h.p. Considering the (log Γ) -roundaggregation process, one can immediately derive that dataaggregation succeeds in O (log N log Γ) w.h.p.Unlike a normal data aggregation, Algorithm 3 termi-nates when the messages from all non-faulty nodes arereceived by the collector without any loss. A message canbe lost when crash failure happens. The number of faultynodes is bounded by f , thus during the data reaggregationprocess, there are at most f times of the execution of thespanner construction and data aggregation, which gives theupper bound of the runtime as O ( f log N log Γ) . In this subsection, we demonstrate how our protocol en-sures persistence and liveness properties whose definitionsare adapted from the rigorous ones proposed by Garay et al. [28].
Theorem 2.
Persistence.
If a non-faulty node v proclaims atransaction tx v in the position tx ji , other nodes, if queried, shouldreport the same result.Proof. To prove the persistence property, we need to showthat for any two blockchains BC v and BC u of nodes v and u , respectively, one cannot find two different transactions tx v ∈ BC v and tx u ∈ BC u that are in the same position tx ji . To prove by contradiction, we assume that such tx v and tx u exist, and there are two cases when the assumptioncan hold.C1: tx v and tx u are respectively appended toblockchains BC v and BC u at the same epoch. This indicatesthat a leader broadcasts two different blocks in the sameepoch, which is not permissible in wChain , thus contradict-ing our assumption.C2: tx v and tx u are appended to their correspondingblockchains BC v and BC u in two different epochs e m and e n . Let tx ji also denote the transaction generated for the firsttime in position tx ji and appended to the blockchains ofat least f + 1 nodes in e i . Since a leader cannot broadcasttwo different blocks in the same epoch, the nodes whoappend tx ji to their local blockchain in e i should have anidentical view of tx ji . Without loss of generality, assume i < m < n . Using contradiction, we assume tx v (cid:54) = tx ji .Since i < m , v should crash before e m and recover in e m sothat tx v is appended to BC v when v updates its blockchainby applying update () . The leader who sends the updateinformation in e m has an identical view with at least f nodes, which means that at least f + 1 nodes have the sameview on tx ji in e m . Since there are also at least f + s nodeswho agree on tx ji in e i and the network size N = 2 f + 1 , wehave a contradiction saying that N > f +1+ s > N . That is,one can only have tx ji = tx v . By applying the same proof, we can obtain tx ji = tx u . Hence tx ji = tx v = tx u , whichcontradicts the assumption that tx v and tx u are different.In a nutshell, all the nodes queried for a transaction ina specific position should report the same result or reporterror messages. Theorem 3.
Liveness.
If a non-faulty node generates a trans-action and contends to send it, wChain can add it to theblockchains within T slots w.h.p., where the upper bound of T is O (log N log Γ) when crash failures happen in a low frequency,and the worst-case upper bound of T is O ( f log N log Γ) .Proof. In a specific epoch, the best case for wChain occurs when no crash failures happen for all nodesthroughout the epoch so that the leader executing the
Reaggregation ( data (cid:96) ) subroutine can broadcast m (cid:96) ← MSG ( stop (cid:96) ) without the need of waiting for the aggregateddata from a new collector. By Theorem 1, the view mes-sages and the transactions can be fully aggregated within O (log N log Γ) w.h.p. Besides, the DECIDE phase takes O (1) slots. Hence the upper bound of T is O (log N log Γ) . In anormal case, failures happen in a low frequency so thatdata aggregation can be executed in O (1) time during thedata reaggregation process, and the PREPARE and
COMMIT phases take O (log N log Γ) slots in total. Therefore, theupper bound of T is still O (log N log Γ) .From the worst-case perspective, we assume that be-fore epoch e i , nodes are all non-faulty, and f nodes crashduring e i . If a leader crashes, the worst-case runtime ofthe current epoch is still O (log N log Γ) when it crashesin the DECIDE phase. If a follower v crashes, some datafrom v ’s children cannot be collected to the leader suchthat the leader must execute SpannerConstruction and
DataAggregation ( data v ) , which take O (log N log Γ) slots.Thus the extra runtime brought by a one-time crash failureis O (log N log Γ) . If f nodes crash, the total time spenton waiting for a successful DECIDE phase is boundedby O ( f log N log Γ) . This gives the normal and worst-caseupper bound of T as O (log N log Γ) and O ( f log N log Γ) ,respectively. IMULATION R ESULTS
In this section, we conduct simulation experiments to val-idate the performance of wChain . The impacts of variousparameters are investigated, including the SINR model pa-rameters, network size N , and Γ , the ratio of the maximumdistance to the minimum distance between nodes. If notstated otherwise, we adopt the following parameter settings: α = 3 , β = 3 , s = 100 , λ (cid:48) = 25 , N = 1 , P i = 2 N β iα . Thefrequency of node crashes is set to be × N nodes persecond. To evaluate the performance, we adopt two metrics,namely epoch length and throughput . The epoch length is thenumber of slots within an epoch; given that the unit slottime for IEEE 802.11 is set to be µs , one can calculatethroughput asThroughput = The number of transactionsEpoch length × µs , (7)hence the unit of throughput is transactions per second(TPS). OURNAL OF L A TEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 9
Network size E po c h l e ng t h ( s l o t s ) uniformgaussexponential (a) × plane Network size Th r oughpu t ( T PS ) uniformgaussexponential (b) × plane
200 400 600 800 1000123 E po c h l e ng t h ( s l o t s ) uniformgaussexponential (c) N = 2000
200 400 600 800 10001000200030004000 Th r oughpu t ( T PS ) uniformgaussexponential (d) N = 2000 Fig. 2. The performance of wChain vs. the network size N and Γ (under uniform, normal, and exponential distributions). The simulation program is written in C and all theexperiments are performed under a CentOS 7 operatingsystem running on a machine with an Intel Xeon 3.4 GHzCPU, 120 GB RAM, and 1 TB SATA Hard Drive. Over 20runs are carried out to get the average for each result. Γ Since liveness is mainly determined by N and Γ , we firststudy the impacts of N and Γ on the performance of wChain . We consider three types of distributions, namelyuniform, normal, and exponential. To investigate the im-pacts of N , we adopt parameters α = 3 , β = 3 for the SINRmodel, and the plane is of size × . The results arereported in Fig. 2.One can observe from Fig. 2(a) that the epoch length in-creases with N . With a uniform distribution and N = 5000 ,it is 49364 slots (about 2.47s). Under normal and exponentialdistributions, the nodes have larger epoch lengths since theyare denser in the center or the corner. They may suffer fromheavier contention and spend more time transmitting a mes-sage. This result is consistent with our model assumptionwhich states that the network density should be limited.Fig. 2(b) indicates that under uniform distributions wChain has the highest throughput, which only increaseswith N . When N = 5000 , the throughput with uniformdistributions reaches 2546 TPS and is about 28% higher than that for normal distributions. Under normal distributions,the throughput can reach 1986 TPS when N = 5000 . Notethat that the throughput under normal or exponential distri-butions has a small decrease from N = 3000 . This is becausethe density is so high in some areas with a large number ofnodes that contend heavily, negatively affecting throughput.Then we investigate the impact of Γ and set N = 2000 .In Fig. 2(c), with an increasing Γ , the epoch length increasessince the spanner has more levels. The epoch lengths undernormal and exponential distributions are still larger thanthe one under uniform distributions, which is caused by thesame reason as Fig. 2(a).Concerning that the throughput is a function of Γ ,our protocol running under uniform distributions yieldsthe largest throughput. However, the throughput decreaseswith a larger Γ because a larger Γ indicates that the spannerhas more levels, and the data aggregation process takes alonger time. We perform four experiments to explore how the SINRmodel parameters, namely α and β , impact the performanceof wChain . Assume that nodes are uniformly distributed inthe plane. The combinations of α = 3 , , and β = 2 , aretested with different N and Γ in our experiments.As shown in Fig. 3(a), the epoch lengths scale from about × to × slots with N under different settings of OURNAL OF L A TEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 10
Network size E po c h l e ng t h ( s l o t s ) =3, =2=3, =3=4, =2=4, =3=5, =2=5, =3 (a) × plane Network size Th r oughpu t ( T PS ) =3, =2=3, =3=4, =2=4, =3=5, =2=5, =3 (b) × plane
200 400 600 800 100001234 E po c h l e ng t h ( s l o t s ) =3, =2=3, =3=4, =2=4, =3=5, =2=5, =3 (c) N = 2000
200 400 600 800 1000010002000300040005000 Th r oughpu t ( T PS ) =3, =2=3, =3=4, =2=4, =3=5, =2=5, =3 (d) N = 2000 Fig. 3. The performance of wChain with various α and β (under a uniform distribution). α and β but they differ very little for different α and β and the same N . In 3(b), even though we observe a smallthroughput decrease when α = 3 and β = 3 , the epochlength and throughput are independent of the values of α and β with different N . In Fig. 3(c) and 3(d), the epochlength and throughput slightly change with different α and β . This is because α , as the path-loss exponent, works closelywith the distance between nodes as well as Γ . The impactsof α and β on throughput are in an allowable range so thatone can claim that the performance of wChain is insensitiveto them. ONCLUSION AND F UTURE R ESEARCH
In this paper, we propose a fast fault-tolerant blockchainprotocol, namely wChain , which can ensure the fast dataaggregation leveraging a spanner as the communicationbackbone. The runtime upper bound of our protocol is O (log N log Γ) when crash failures happen in a low fre-quency, and the worst-case upper bound of wChain is O ( f log N log Γ) . Besides, wChain tolerates at most f = (cid:98) N (cid:99) faulty nodes and is capable of handling node recoverywhile satisfying persistence and liveness, the two crucialproperties for a blockchain protocol to function well. Boththeoretical analysis and simulation studies are conducted tovalidate our design. On the last point, we only grapple withcrash failures in this paper, but it would be interesting to consider Byzantine fault-tolerance in wireless networks. Itis also worthy of investigating the impacts of mobility in adhoc wireless networks. A CKNOWLEDGMENT
This study was partially supported by the National Nat-ural Science Foundation of China under Grants 61771289,61871466, 61832012, and 61672321, the Key Science andTechnology Project of Guangxi under Grant AB19110044,and the US National Science Foundation under grants IIS-1741279 and CNS-1704397. R EFERENCES [1] J. Feng, F. R. Yu, Q. Pei, J. Du, and L. Zhu, “Joint optimizationof radio and computational resources allocation in blockchain-enabled mobile edge computing systems,”
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PPENDIX
Lemma 2. (Chernoff Bound). Given a set of independent bi-nary random variables X , X , · · · , X n , let X = (cid:80) n X i and µ = (cid:80) n p i , where X i = 1 with probability p i . If E (cid:2)(cid:81) i ∈ S X i ≤ (cid:81) i ∈ S q i (cid:3) , where S ⊆ { , , · · · , n } , then it holdsfor any δ > that P r [ X ≥ (1 + δ ) µ ] ≤ e − δ µ δ/ b ) . If E (cid:2)(cid:81) i ∈ S X i ≥ (cid:81) i ∈ S q i (cid:3) , where S ⊆ { , , · · · , n } , then forany δ ∈ (0 , , we have P r [ X ≤ (1 − δ ) µ ] ≤ e − δ2
Lemma 2. (Chernoff Bound). Given a set of independent bi-nary random variables X , X , · · · , X n , let X = (cid:80) n X i and µ = (cid:80) n p i , where X i = 1 with probability p i . If E (cid:2)(cid:81) i ∈ S X i ≤ (cid:81) i ∈ S q i (cid:3) , where S ⊆ { , , · · · , n } , then it holdsfor any δ > that P r [ X ≥ (1 + δ ) µ ] ≤ e − δ µ δ/ b ) . If E (cid:2)(cid:81) i ∈ S X i ≥ (cid:81) i ∈ S q i (cid:3) , where S ⊆ { , , · · · , n } , then forany δ ∈ (0 , , we have P r [ X ≤ (1 − δ ) µ ] ≤ e − δ2 µ2