Performance study of Byzantine Agreement Protocol with artificial neural network

Performance study of Byzantine Agreement Protocolwith artificial neural networkKok-Wah Leea,*,Hong-Tat Ewe b a Faculty of Engineering &Technology,Multimedia University,Jalan Ayer Keroh Lama,75450Bukit Beruang,Melaka,Malaysiab Faculty of Information Technology,Multimedia University,Jalan Multimedia,63100Cyberjaya,Selangor,MalaysiaReceived 23December 2005;received in revised form 10April 2007;accepted 16April 2007AbstractSince 1982,numerous Byzantine Agreement Protocols (BAPs)have been developed to solve arbitrary faults in the Byz-antine Generals Problem (BGP).A novel BAP,using an artificial neural network (ANN),was proposed by Wang and Kao.It requires message exchange rounds similar to the traditional BAP and its suitability,in the context of network size,has not been investigated.In the present study,we propose to adopt Nguyen–Widrow initialization in ANN training,which modifies message communication and limits the message exchange rounds to three rounds.This modified approach is referred to as BAP-ANN.The BAP-ANN performs better than the traditional BAP,when the network size n is greater than nine.We also evaluate the message exchange matrix (MEM)constructed during the message exchange stage.For a fixed number of faulty nodes and remainder cases of (n mod 3),the study shows that the mean epoch for ANN training decreases as the network size increases,which indicates better fault tolerance.Ó2007Elsevier Inc.All rights reserved.Keywords:Byzantine Agreement;Communication protocol;Artificial neural networks;Cryptology;Fault tolerance1.IntroductionWith the emergence of the Internet,the consensus issue or Byzantine Generals Problem (BGP)of the dis-tributed system has become more significant in achieving fault tolerance.The essential fault tolerance [3,4,22]necessary to reach common agreement enables the importance of earlier investigations [14,16,20,32].Since the study of Lamport et al.[21],various Byzantine Agreement Protocols (BAPs)for fault tolerance have been developed [10,17,27,31,39].In their studies,numerous assumptions were made to construct variations of the traditional BAP by Lamport et al.[21].Wang and Kao [43]used a feedforward neural network (FFNN)with a backpropagation learning algorithm (BPLA)to construct a BAP.The message communication mode of Wang and Kao’s BAP is similar to the traditional BAP and its suitability,in the context of the network size,has not been investigated.0020-0255/$-see front matter Ó2007Elsevier Inc.All rights reserved.doi:10.1016/j.ins.2007.04.011*Corresponding author.Tel.:+6062523320;fax:+6062316552.E-mail addresses:kwlee@.my (K.-W.Lee),htewe@.my (H.-T.Ewe).Information Sciences 177(2007)4785–47984786K.-W.Lee,H.-T.Ewe/Information Sciences177(2007)4785–4798The BAP approach[43]improved with artificial neural network(ANN)by Lee and Ewe[24]has been termed,ANN-based BAP[23,25].Here,ANN-based BAP has been renamed BAP with ANN(BAP-ANN). In BAP-ANN,we demonstrate a modified message communication mode and Nguyen–Widrow initialization. Then,we evaluate the message exchange matrix(MEM)constructed during the message exchange stage to investigate the relationship between the network size and fault tolerance.From these studies,we infer that the larger the network size,the better the fault tolerance.The rest of the paper is organized as follows.Sections2and3discuss the issues of BGP and BAP-ANN, respectively.In Sections4and5,the MEM and its analysis are explained,respectively.Conclusions are pro-vided in Section6.2.Byzantine Generals Problem(BGP)Internet activities have become very important and popular among users throughout the world.This is due to the emergence of electronic data interchange(EDI),email,mobile phone and electronic commerce.These Byzantine communications are subject to the increasing trend of threat risks such as software errors and mali-cious human and computing machine attacks.These failures are related to the fault tolerance and reliability issues of distributed systems[3,4,22].When a processor within a distributed network is infected with virus,worm or spyware,it becomes a faulty processor and tries to confuse and attack the other processors by sending out arbitrary messages.The distrib-uted network is then said to become malicious and suffers from Byzantine fault or universal fault.An ordered fault classification scheme is shown in Fig.1[34,38].When a distributed network problem is caused by arbitrary message communication in a malicious net-work,it is called Byzantine Generals Problem(BGP)[21].The historical basis of the term BGP is named after a martial situation in an ancient war[14,20,21],where a group of Byzantine army generals surrounded an enemy city.The Byzantine generals had to achieve a consensus on a common battle plan via the communi-cation of messengers only.The risk existed that one or more of the Byzantine generals might be traitors and would try to confuse the other loyal Byzantine generals.The loyal Byzantine generals had tofind an algo-rithm to ensure that all the loyal Byzantine generals would obtain the same agreement after the communi-cation of the messengers among each other.The success of solving BGP ensures a secure and trustable communication among the loyal processors within a network.Due to the advancement in cryptology [13,36,42],the BGP solution becomes possible and is commonly called Byzantine Agreement Protocol (BAP)[6,9,15,33,37].In analogy,this situation is the same as the security scenario we encounter in the distributed systems and Internet nowadays.Both the loyal and faulty processors coexist within a malicious network.It has been shown that for BGP of an unsigned message,the Byzantine Agreement(BA)can be reached if,and only if,more than two-thirds of the Byzantine generals are loyal[21].In the context of a distributed computer network,an analogy between malicious attack,software errors, and the situation of the Byzantine generals exists.For a distributed network in a malicious environment,both the loyal and faulty processors exist.In this Byzantine environment,the faulty processors can transmit arbi-trary information to other processors.Such arbitrary information is called Byzantine fault,and it is a fault class that can be considered a universal fault set[12,34,38].In a system model,the class of Byzantine fault consists of every possible arbitrary or malicious fault such as timing fault,omission fault and crash fault.A faulty processor can then confuse other loyal processors within a distributed system.A distributed system can be imagined as a set of interconnected processors through a data communication network[21,43].A processor can be treated as a node,while a link between any two processors can be treated as a path.The combination of these processors and paths will collectively form a graph that symbolizes the distributed system.Therefore,in the context of graphs,both loyal and faulty nodes exist.The malicious attacks and software errors can cause the faulty node to behave in an arbitrary manner.In turn,the faulty node can send a totally opposite message to all other nodes in the network.For the loyal node to tolerate this Byzantine fault,all the loyal nodes have to arrive at a common value for the BGP.When applying BAP to get a common value,two interactive consistency conditions(ICCs)must be fulfilled. They include:ICC1:All loyal nodes obey the same order.ICC2:If the commanding general is loyal,then every loyal lieutenant obeys the order sent by the command-ing general;or adopts the DEFAULT value if no order is received or achieved.These ICCs consider all types of malicious situations.‘‘No order is received or achieved’’means no com-mon agreement(BA)can be made from the commanding general’s messages due to failed reception or due to lack of simple majority at any lieutenant node.For this case,the DEFAULT value will be adopted.The worst case scenario happens when the n-node communication network possesses a maximum number of tolerable faulty nodes,m,as in Eq.(1).The number of exchanged messages to achieve the BA for traditional BAP,Msg Trad,is in(2).Meanwhile,the number of rounds for message exchange is(m+1)[2,16]: m¼bðnÀ1Þ=3cð1ÞMsgTrad ¼X mk¼0ðnÀ1Þ!ð2Þ3.BAP-ANNSince1982[21],various BAPs have been proposed based on different assumptions[20,28,37]that include considerations of network topology[28,29,35,51],processor fault,link fault[39,40,48,49],node mobility [11,44,46,47]and faulty node detection[5,26,37,45,50].The main purpose of these BAPs is to reduce the com-putational load by specializing in the generalized model of traditional BAP[21].Wang and Kao[43]proposed an innovative BAP by using the artificial neural network[8,19,41].This concept has been further developed [23–25].ANN is applied to every processor to speed the process of BA achievement.The application of ANN shown in manyfields[1,7]has three significant advantages over currently available BAPs.These advan-tages include parallel processing capability of each node,reduction of memory space requirement and the dynamic learning ability of the neural network.The block diagram of BAP-ANN is depicted in Fig.2.The operations of this BAP and its implementations were described in detail in[24,43].From the information of[2,16,51],its performance is improved by intro-ducing a new scheme of message communication mode as shown below.The formation of message exchange matrix(MEM)[25]will be discussed in Section4:K.-W.Lee,H.-T.Ewe/Information Sciences177(2007)4785–479847874788K.-W.Lee,H.-T.Ewe/Information Sciences177(2007)4785–4798Message communication mode{Round1:One message is sent from the commander node to each lieutenant node.Round2:Each lieutenant node sends its received message to other lieutenant nodes.Round3:Each lieutenant node sends its(nÀ1)messages to other lieutenant nodes.}This step reduces the round of message exchange to three rounds.Due to fewer exchanged messages,the computation time and memory storage requirements are reduced.Consequently,the time for fault tolerance and fault detection is decreased.The efficiency of BAP-ANN is increased for the network size larger than nine (n P10).The illustration of this improvement is depicted in Fig.3.Thefive stages of BAP-ANN includes:(i)initialization stage,(ii)message exchange stage,(iii)training stage,(iv)application stage and(v)compromise stage.In the initialization stage,a commander node deter-mines which source message is delivered.Then,this node broadcasts the source message to all the lieutenant nodes to start the message exchange stage.At the end of the message exchange stage,MEM becomes the train-ing data set for the designed ANN.After training the neural network,the ANN model can be applied for any BGP as long as the condition in (1)is fulfilled.In this paper,we adopt an ANN model with the architecture of two-layer feedforward neural network(FFNN)using a backpropagation learning algorithm(BPLA).For the approach in[43],conventional BPLA was adopted.Here,the modified BPLA applies the additional parameters of momentum,learning rate and slope parameter.The advanced technique of initialization called Nguyen–Widrow initialization[30]is also applied in our BPLA to help reduce early stage oscillation problems and gain faster convergence in ANN training time.The ranges of these parameters are listed in Table1.The settings as explained in Table1include momentum=(0.2,0.5,0.9),learning rate=(0.5,1.0)and slope parameter=(2.0).For Nguyen–Widrow initialization,the settings are as follows:K.-W.Lee,H.-T.Ewe/Information Sciences177(2007)4785–47984789 Table1Ranges of parameters for the BPLABPLA parameter Standard BPLA Modified BPLAWeight&bias initialization Random Random/Nguyen–Widrow Bias Optional Compulsory Momentum,l0.0]0,1[or0<l<1 Adaptive learning rate,a 1.0]0,1]or0<a61Slope parameter 1.0P1Convergence speed Slower FasterLet:p=number of input neurons X i,where i=1,2,3,...,p,q=number of hidden neurons Y j,where j=1,2,3,...,q,r=number of output neurons Z k,where k=1,2,3,...,r,b=scale factor,where b=0.7(q)1/p.For each hidden neuron Y j:Initialize the weights w ij from input neuron X i to hidden neuron Y j:w ij(old)=random number fromÀ0.5to+0.5.k w ij(old)k=[w1j(old)2+w2j(old)2+ÁÁÁ+w ij(old)2+ÁÁÁ+w nj(old)2]1/2.Re-initialize weights w ij between input layer and hidden layer:w ij(new)=b·w ij(old)/k w ij(old)k.Set bias b j on hidden neurons Y j for j=1,2,...,q:b j=random number betweenÀb and+b.For each output neuron Z k:Initialize the w jk from hidden neuron Y j to output neuron Z k:w jk=random number betweenÀ0.5and+0.5.Set bias b k on output neurons Z k for k=1,2,...,r:b k=random number betweenÀ0.5and+0.5.For an n-processor distributed system,the number of input neurons,p,at the input layer of ANN will be (nÀ1).There are three output neurons at the output layer indicating the value of bit‘0’,bit‘1’and the ‘DEFAULT’value.In the hidden layer,the number of hidden nodes,q,is initialized to be the ceiling of (p+r)/2=(n+2)/2.Genetic algorithm can be used to optimize the value of q.This method is called evolu-tionary neural networks.The weights between any two layers of FFNN map every node to a node across the layers.Hence,location of the faulty node is independent of its connected input neurons in the ANN model of BAP-ANN.The crucial factor is the number of faulty processors or the number of faulty nodes,m,within the distributed network system.In the application stage,the weight matrix from the training stage is used for new BAP application to reach the ICCs.This application stage is the same with the feed forward process in the training stage.Received mes-sages are fed to the input layer and the outputs are collected from the output layer.These outputs are then used in thefinal stage,known as the compromise stage,where the majority function is applied to achieve BA among all the loyal nodes within the network.4.Message exchange matrix(MEM)To train the neural network,a set of training data is needed.This training data is obtained from the mes-sage exchange stage.It is then specially arranged into a form of(nÀ1)·(nÀ1)matrix,which is called the message exchange matrix(MEM).In this section,we discuss how MEM is constructed using the example4790K.-W.Lee,H.-T.Ewe/Information Sciences177(2007)4785–4798of a4-processor distributed system.Figs.4and6show two kinds of faulty situations:faulty commander node and faulty lieutenant node.From the loyal bit density data depicted in Figs.5and7,we show that the situ-ation of faulty lieutenant node is worse than the situation of faulty commander node.Hence the worse case of fault tolerance will occur when the tolerable faulty node is at its maximum and the faulty node belongs to the lieutenant node.We performed a mathematical analysis of MEM to explain the experimental phenomena of a reduction in mean epoch for ANN training.These phenomena take place when the network size increases with afixed num-ber of faulty nodes and(n mod3),which is the remainder of n/3.From the analysis,we can observe the suit-ability of BAP-ANN for different sized networks.This is important for efficient BAP implementation so as to be able to choose between the traditional BAP and BAP-ANN for different scenarios.An n-processor distributed computer network is simulated as a graph of n-node of fully connected network (FCN).In thefirst round,the commander node sends its order or value to all the other(nÀ1)lieutenant nodes.At this point,each lieutenant node has one bit of the message.In the second round,each lieutenant node sends its one bit of message to the other(nÀ2)lieutenant nodes.Now,we have(nÀ1)bits of messages at every lieutenant node.In the third round,which is also thefinal round,each lieutenant node delivers its(n À1)bits of message to all the other (n À2)lieutenant stly,we have the total bits of message (Msg ANN )at all the (n À1)lieutenant nodes as in the following equation:Msg ANN ¼ðn À1Þ½1þðn À2Þþðn À1Þðn À2Þ ¼ðn À1Þ3ð3ÞComparatively,the complexities of traditional BAP and BAP-ANN are O(n m +1)and O(n 3),respectively.As m is equal to or more than three,the efficiency of the BAP-ANN will surpass the traditional BAP [25].This phe-nomenon is shown in Fig.3.The message exchange phase of faulty commander node is shown in Figs.4and 5.Since the worst case of FCN-4is one faulty lieutenant node,we have the message exchange phase of this example explained in detail in Figs.6and 7.Here,there are one commander node C and three lieutenant nodes L 1,L 2and L 3.In the worst case,one of the three lieutenant nodes is faulty.Hence,we have three faulty situations here,i.e.either L 1,L 2or L 3is a malicious node.Due to the symmetrical property of the ANN model,these three faulty situations are equivalent to each other.For example,let the lieutenant node L 3be the faulty node,sending out arbitrary messages to confuse other loyal lieutenant nodes.Fig.6a shows the first round of message exchange stage,where the commander node delivers its order bit ‘1’to the other three lieutenant nodes.In Fig.7a,it depicts the received message bit ‘1’from the commander node that is saved in the MEM of each lieutenant node.In the second round,the loyal lieutenant nodes L 1and L 2send their saved value in the first round to the other lieutenant nodes.On the other hand,the faulty node L 3will send bit ‘0’to other lieutenant nodes to confuse the agreement.The message flow in the second round is shown in Fig.6b.Meanwhile Fig.7b depicts the update of the MEM of each lieutenant node after the second round of message exchange.Fig.6c shows the delivery of message string with a length of three bits from a lieutenant node to the other lieutenant nodes.The exchange of message strings allows the complete formation of MEM at each lieutenant node.Fig.7c shows the final update of MEM.MEM is now ready to be used as the training data set for the designed ANN at every node.Among the MEM at each lieutenant node,it is only necessary to consider the MEM of the loyal lieutenant node.In fact,we can neglect the MEM of the faulty lieutenant node.As long as condition (1)is satisfied,consideration of MEM at the loyal nodes is sufficient to solve the BGP.All the loyal lieutenant nodes will use the MEM formed to train the weight matrix and the bias matrix of the designed ANN model.Once the ANN model is trained,it can be used for future incoming BGP to achieve BA among all loyal nodes.As in this example,the MEMs at the loyal lieutenant nodes,L 1and L 2,are already sufficient to guarantee the achievement of the BA.MEM is a (n À1)·(n À1)matrix formed after the message exchange stage.This MEM is used as the train-ing data set during the training stage of BAP-ANN.After training the neural network,the MEM is used as the testing data set in the application stage.For an n -processor distributed network,a maximum of m faulty pro-cessors can be tolerated as in (1).Assume the commander processor to be loyal,then there are m faulty lieu-tenant processors.Hence as we take the distributed network as a graph of n nodes,we then have one Fig.5.MEM at each lieutenant node during the FCN-4message exchange stage with a faulty commander node C.L 11L 2 1 L 3 0(a)After first round of message exchange.L 1 1 1 0 L 2 1 1 0 L 3 1 1 0(b)After second round of message exchange.L 11 1 11 1 10 0 0 L 2 1 1 1 1 1 1 0 0 0 L 3 1 1 11 1 10 0 0(c)After third round of message exchange.K.-W.Lee,H.-T.Ewe /Information Sciences 177(2007)4785–47984791commander node and (n À1)lieutenant nodes.Among the (n À1)lieutenant nodes,m are faulty nodes.These (n À1)lieutenant nodes are represented by (n À1)columns of the MEM with m columns representing the faulty nodes.Fig.7.MEM at each lieutenant node during the FCN-4message exchange stage with a faulty lieutenant node L 3.L 11L 2 1 L 3 1(a)After first round of message exchange.L 1 1 1 0 L 2 1 1 0 L 3 1 1 1(b)After second round of message exchange.L 11 1 01 1 00 0 0 L 2 1 1 0 1 1 0 0 0 0 L 3 1 1 11 1 10 0 1(c)After third round of message exchange.4792K.-W.Lee,H.-T.Ewe /Information Sciences 177(2007)4785–4798The column of MEM represents the receiver nodes and the row of MEM represents the sender node.Hence we have m rows of faulty message senders and m columns of faulty message receivers.Due to the matrix prop-erty and the symmetrical property of ANN architecture,the sequences of the numbers of faulty rows and faulty columns are independent of BA achievement.As long as the number of faulty nodes is at a maximum of m nodes,the BGP of the unsigned message can be solved with the BAP-ANN.As the sequence of rows sent by the lieutenant nodes is independent from the neural network training,we can gather the rows sent by the faulty nodes together.For example,let us take the worst case of n =4.For n =4,the maximum value of m is 1.Therefore,either C,L 1,L 2or L 3is faulty.We consider the faulty L situation instead of faulty C situation since faulty L is worse than faulty C.However,these three faulty cases (L 1,L 2or L 3)are equivalent to each other.Hence,we can always assume that L 1is the faulty node.To make the analysis easier,we arrange the MEM so that the rows with faulty mes-sages are always at the top of the MEM.In other words,the first m rows of the MEM are rows of messages sent by the faulty lieutenant nodes.In addition,the first m columns from the left side of the MEM are arranged to be columns with faulty messages.Therefore,we have an MEM as in Fig.8for analysis.5.Analysis of message exchange matrixIn Fig.8,we observe that MEM contains both loyal and faulty bits.Out of the ðn À1ÞÂðn À1Þbits,a total of ðn Àm À1ÞÂðn Àm À1Þare loyal bits.To help analyze the MEM,we have some denotations as follows:n =number of nodes in the network,m =number of faulty nodes in the network,npr =total number of bits in the MEM =(n À1)2,nloy =number of bits of the loyal lieutenant nodes,nfau =number of bits of the faulty lieutenant nodes,v loy =number of row vectors sent by loyal lieutenant nodes,v fau =number of row vectors sent by faulty lieutenant nodes,v npr =total number of row vectors.m * m m * (n -m -1) (n -m -1) * m (n -m -1) * (n -m -1) m bits(n -m -1) bits(n -1) bits(n -1) bitsFig.8.Message exchange matrix for an n -processor distributed system.K.-W.Lee,H.-T.Ewe /Information Sciences 177(2007)4785–47984793Assuming the commander node is loyal,we have m =number of faulty lieutenant nodes.In addition,we can have some parameter relationships as in the following equations:n ¼f 4;5;6;7;...gm ¼f m P 0&m 6floor ððn À1Þ=3Þgfloor ðx Þ¼a function to round the x value to nearest integer smaller than xnpr ¼ðn À1Þ2nloy ¼ðn Àm À1ÞÂðn Àm À1Þ¼ðn Àm À1Þ2nfau ¼m Âm þm Âðn Àm À1Þþðn Àm À1ÞÂm ¼2mn À2m Àm 2npr ¼nloy þnfauv npr ¼v loy þv fau ð4ÞFor an n -node distributed network,tolerance in the number of faulty nodes,m ,ranges from 0to floor((n À1)/3).To analyze the effect of m on the MEM,we set the value of m to the worst situation,where m =floor((n À1)/3).For every critical value of m ,there exist three corresponding values of n where m ={1,2,3,4,...},we can have three cases of n as follows in (5):Case 1:n ¼3m þ1ðC1ÞCase 2:n ¼3m þ2ðC2ÞCase 3:n ¼3m þ3ðC3Þð5ÞFor the three above cases,we calculate some ratios to help the MEM analysis.These ratios are (nloy/npr),(nfau/npr),(v loy /v npr )and (v fau /v npr ).The computation results of the ratio for every case are listed in Table2.From the table,we notice the trend of the loyal bit composition of a MEM as the size of the network increases.As we compute cases 1–3,we have the conclusions as in the following inequalities:nloy npr C1<nloy npr C2<nloy npr C3v loy v npr C1<v loy v npr C2<v loy v npr C3nfau npr C1>nfau npr C2>nfau npr C3v fau v npr C1>v fau v npr C2>v fau v npr C3ð6ÞTable 2Various ratios for MEM analysis on the worst case of mCaseC1:n =3m +1C2:n =3m +2C3:n =3m +3nloynpr492m þ122m þ22nfaunpr55m 2þ2m 9m 2þ6m þ15m 2þ4m 9m 2þ12m þ4v loyv npr22m þ12m þ2v faunpr 1m m 4794K.-W.Lee,H.-T.Ewe /Information Sciences 177(2007)4785–4798From cases1–3,the ratios of(nloy/npr)andðv loy=v nprÞincrease,while the ratios of(nfau/npr)andðv fau=v nprÞdecrease.Consequently,as the network size,n,grows,cases2and3will converge to case1.For a larger net-work,the loyal bit density in a MEM is about4/9or44.4%and the density of row vectors with loyal messages is about2/3or66.7%.In this paper,the BAP-ANN is studied by using an FCN model of an n-processor distributed system for cases of n=4,7,10,and n=10,11,12.The performance evaluation gives an analysis on the MEM formed after the message exchange stage and the required mean epoch for neural network training.The MEM is used as both training and testing data sets.The analysis implies that as the network size grows larger,a smaller number of epochs will be required to train the neural network.For the case offixed number of faulty node, this is illustrated in Fig.9for n=10,11,12.Meanwhile for the case of(n mod3),it is shown in Fig.10.In what follows include the explanations of these phenomena.As the number of node n increases,the number of bits in the MEM,which isðnÀ1ÞÂðnÀ1Þ,increases in the order of n2.The pattern of bits in the MEM is classified into its corresponding classes by using the ANN.A trained ANN performs its pattern classification task by feeding the row vectors of the MEM into its input layer row by row.In an MEM,all the loyal bit positions will share a common value.Therefore,there are always a minimum ofðnÀmÀ1Þ2bits out of a total ofðnÀ1Þ2sharing the same bit value.Meanwhile,for faulty bit positions,all have randomly generated arguments.If the loyal bit locations are all bit‘1’,then the faulty bit locations can be either bit‘0’or bit‘1’.As the distributed network size,n,increases,the bit‘0’and bit‘1’in the faulty bit position will be evenly distributed with a probability of occurrence at0.5for each.At the same time,the loyal bit positions always carry the same bit value of either bit‘0’or bit‘1’.Hence,as n increases,the number of bit value with a loyal bit value will tend to be more.In other words,the sample moves away from the worst case scenario,where all the faulty bit positions carry a value different from the loyal bit value at the loyal bit positions.If n is big enough and nloybit represents the number of bits carrying the loyal bit value,we have a ratio as in the following equation:ðnloybit=nprÞ¼½nloyþfloorð0:5ÂnfauÞ =nprð7ÞAs n is sufficiently large,cases2and3will converge to case1.Hence the consideration of case1where ðn¼3mþ1Þwill be sufficient to see the effect of increasing the size of the distributed network on the bit com-position of MEM.We will then generate a simpler form for(7)as in(8)below:ðnloybit=nprÞC1¼ðnloy=nprÞC1þfloorð0:5ÂnfauÞ=npr¼4=9þfloorð0:5ÂnfauÞ=nprð8ÞTherefore,it becomes easier to recognize a pattern of bit composition for the message exchange matrix as the value of n increases.Indirectly,it means that it is easier to train the neural network for faster detection of any Byzantine fault.Hence,we canfinally deduce that the mean epoch required to train the neural network re-duces,as the distributed network grows larger.This performance analysis has the implication to ease the com-putational workload for fault tolerance within a malicious distributed computer network.6.ConclusionsOur MEM analysis proves that the loyal bit density of MEM increases with the size of the distributed net-work.The increment of loyal bit density allows less mean epoch to train the neural network.Hence,the BAP-ANN has an advantage over the traditional BAP as the network size becomes bigger since BA can be achieved at a faster rate.This implies that the secure communication within the distributed network can be improved by replacing traditional BAP with BAP-ANN.Due to lesser-exchanged messages,our improved BAP-ANN achieves BA faster than the traditional BAP when n P10as shown in Fig.3.The network size can thus be used as a criterion to determine appropriate BAP to be used for different cases of BGP application.The evaluation study indicates that for afixed number of faulty nodes and(n mod3),which is the cases of n/3remainder,the required mean epoch for neural network training decreases as the network size n increases. These phenomena imply that larger networks are safer and more tolerable to Byzantine fault.In the future, BAP-ANN may be used for multipartite protocol or distributed cryptography[18]in view of the advancement of distributed artificial intelligence(DAI).AcknowledgementsThis research was supported by Telekom Internal Research Funding,Malaysia under the Financial Grants of EUCG02/2000/RF19and PR/2001/0025.The authors would like to thank anonymous reviewers,proof-readers and language editing service providers for helping improve the readability of this article.References[1]A.G.Adami,D.A.C.Barone,A speaker identification system using a model of artificial neural networks for an elevator application,Information Sciences138(1–4)(2001)1–5.[2]M.K.Aguilera,S.Toueg,A simple bivalency proof that t-resilient consensus requires t+1rounds,Information Processing Letters71(3-4)(1999)155–158.[3]T.Anderson,P.A.Lee,Fault Tolerance Principles and Practice,Prentice Hall,Englewood Cliffs,NJ,USA,1981.[4]A.Avizienis,Fault-tolerance systems,IEEE Transactions on Computers C-25(12)(1976)1304–1312.[5]B.Ayeb,A.Farhat,Aflexible formal framework for masking/demasking faults,Information Sciences159(1–2)(2004)29–52.[6]O.Babaoglu,R.Drummond,Streets of Byzantium:network architectures for fast reliable broadcasts,IEEE Transactions onSoftware Engineering11(6)(1985)546–554.[7]Z.Boger,Experience in industrial plant model development using large-scale artificial neural networks,Information Sciences101(3–4)。