为了算出攻击者依然可以赶上的概率，我们要把每一个攻击者已有的进展的帕松密度乘以他可以从那一点能够追上来的概率：
Rearrangingtoavoidsummingtheinfinitetailofthedistribution...
为了避免对密度分布的无穷级数求和重新整理
ConvertingtoCcode...
转换为C语言程序
#include<math.h>doubleAttackerSuccessProbability(doubleq,intz){doublep=1.0-q;doublelambda=z*(q/p);doublesum=1.0;inti,k;for(k=0;k<=z;k++){doublepoisson=exp(-lambda);for(i=1;i<=k;i++)poisson*=lambda/i;sum-=poisson*(1-pow(q/p,z-k));}returnsum;}
Runningsomeresults,wecanseetheprobabilitydropoffexponentiallywithz.
获取部分结果，我们可以看到概率随着z的增加指数级下降：
q=0.1z=0P=1.0000000z=1P=0.2045873z=2P=0.0509779z=3P=0.0131722z=4P=0.0034552z=5P=0.0009137z=6P=0.0002428z=7P=0.0000647z=8P=0.0000173z=9P=0.0000046z=10P=0.0000012q=0.3z=0P=1.0000000z=5P=0.1773523z=10P=0.0416605z=15P=0.0101008z=20P=0.0024804z=25P=0.0006132z=30P=0.0001522z=35P=0.0000379z=40P=0.0000095z=45P=0.0000024z=50P=0.0000006
SolvingforPlessthan0.1%...
若是P小于0.1%
P<0.001q=0.10z=5q=0.15z=8q=0.20z=11q=0.25z=15q=0.30z=24q=0.35z=41q=0.40z=89q=0.45z=340
Wehaveproposedasystemforelectronictransactionswithoutrelyingontrust.Westartedwiththeusualframeworkofcoinsmadefrom digital signatures,whichprovidesbcontrolofownership,butisincompletewithoutawaytopreventdouble-spending.Tosolvethis,weproposedapeer-to-peernetworkusingproof-of-worktorecordapublichistoryoftransactionsthatquicklybecomescomputationallyimpracticalforanattackertochangeifhonestnodescontrolamajorityofCPUpower.Thenetworkisrobustinitsunstructuredsimplicity.Nodesworkallatoncewithlittlecoordination.Theydonotneedtobeidentified,sincemessagesarenotroutedtoanyparticularplaceandonlyneedtobedeliveredonabesteffortbasis.Nodescanleaveandrejointhenetworkatwill,acceptingtheproof-of-workchainasproofofwhathappenedwhiletheyweregone.TheyvotewiththeirCPUpower,expressingtheiracceptanceofvalidblocksbyworkingonextendingthemandrejectinginvalidblocksbyrefusingtoworkonthem.Anyneededrulesandincentivescanbeenforcedwiththisconsensusmechanism.
我们提出了一个不必依赖信任的电子交易系统；起点是一个普通的使用数字签名的硬币框架开始，虽然它提供了健壮的所有权控制，却无法避免双重支付。为了解决这个问题，我们提出一个使用工作证明机制的点对点网络去记录一个公开的交易记录历史，只要诚实节点能够控制大多数CPU算力，那么攻击者就仅从算力方面就不可能成功篡改系统。这个网络的健壮在于它的无结构的简单。节点们可以在很少协同的情况下瞬间同时工作。它们甚至不需要被辨认，因为消息的路径并非取决于特定的终点；消息只需要被以最大努力为基本去传播即可。节点来去自由，重新加入时，只需要接受工作证明链，作为它们离线之时所发生之一切的证明。它们通过它们的CPU算力投票，通过不断为链添加新的有效区块、拒绝无效区块，去表示它们对有效交易的接受与否。任何必要的规则和奖励都可以通过这个共识机制来强制实施。
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W.Dai,b-money, http://www.weidai.net/bmoney.txt,1998.H.Massias,X.S.Avila,andJ.-J.Quisquater,Designofasecuretimestampingservicewithminimaltrustrequirements,In20thSymposiumonInformationTheoryintheBenelux,May1999.S.Haber,W.S.Stornetta,Howtotime-stampa digital document,InJournalofCryptology,vol3,no2,pages99-111,1991.D.Bayer,S.Haber,W.S.Stornetta,Improvingtheefficiencyandreliabilityofdigitaltime-stamping,InSequencesII:MethodsinCommunication,SecurityandComputerScience,pages329-334,1993.S.Haber,W.S.Stornetta,Securenamesforbit-strings,InProceedingsofthe4thACMConferenceonComputerandCommunicationsSecurity,pages28-35,April1997.A.Back,Hashcash-adenialofservicecounter-measure, http://www.hashcash.org/papers/hashcash.pdf,2002.R.C.Merkle,Protocolsforpublickeycryptosystems,InProc.1980SymposiumonSecurityandPrivacy,IEEEComputerSociety,pages122-133,April1980.W.Feller,Anintroductiontoprobabilitytheoryanditsapplications,1957.
b-money DaiWei(1998-11-01) http://www.weidai.net/bmoney.txt
Designofasecuretimestampingservicewithminimaltrustrequirements HenriMassias,XavierSerret-Avila,Jean-JacquesQuisquater 20thSymposiumonInformationTheoryintheBenelux (1999-05) http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.13.6228
Howtotime-stampadigitaldocument StuartHaber,W.ScottStornetta JournalofCryptology (1991) https://doi.org/cwwxd4 DOI: 10.1007/bf00196791
ImprovingtheEfficiencyandReliabilityofDigitalTime-Stamping DaveBayer,StuartHaber,W.ScottStornetta SequencesII (1993) https://doi.org/bn4rpx DOI: 10.1007/978-1-4613-9323-8_24
Securenamesforbit-strings StuartHaber,W.ScottStornetta Proceedingsofthe4thACMconferenceonComputerandcommunicationssecurity-CCS’97(1997) https://doi.org/dtnrf6 DOI: 10.1145/266420.266430
Hashcash-ADenialofServiceCounter-Measure AdamBack(2002-08-01) http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.15.8
ProtocolsforPublicKeyCryptosystems RalphC.Merkle 1980IEEESymposiumonSecurityandPrivacy (1980-04) https://doi.org/bmvbd6 DOI: 10.1109/sp.1980.10006
AnIntroductiontoProbabilityTheoryanditsApplications WilliamFeller JohnWiley&Sons (1957) https://archive.org/details/AnIntroductionToProbabilityTheoryAndItsApplicationsVolume1
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