accidental海贼王644

Op.
52ConstructingDigitalSignaturesfromaOneWayFunctionLeslieLamportComputerScienceLaboratorySRIInternational18October1979CSL-98333RavenswoodAve.
MenloPark,California94025(415)326-6200Cable:SRIINTLMPKTWX:910-373-124611.
IntroductionAdigitalsignaturecreatedbyasenderPforadocumentmisadataitemOp(m)havingthepropertythatuponreceivingmandap(m),onecandetermine(andifnecessaryproveinacourtoflaw)thatPgeneratedthedocumentm.
Aonewayfunctionisafunctionthatiseasytocompute,butwhoseinverseisdifficulttocompute[1].
Morepreciselyaonewayfunctionisafunctionfromasetofdataobjectstoasetofvalueshavingthefollowingtwoproperties:1.
Givenanyvaluev,itiscomputationallyinfeasibletofindadataobjectdsuchthat(d)=v.
2.
Givenanydataobjectd,itiscomputationallyinfeasibletofindadifferentdataobjectdfsuchthat(d!
d)Ifthesetofdataobjectsislargerthanthesetofvalues,thensuchafunctionissometimescalledaonewayhashingfunction.
Wewilldescribeamethodforconstructingdigitalsignaturesfromsuchaonewayfunction.
OurmethodisanimprovementofamethoddevisedbyRabin[2].
LikeRabin's,itrequiresthesenderPtodepositapieceofdataocinsometrustedpublicrepositoryforeachdocumenthewishestosign.
Thisrepositorymusthavethefollowingproperties:-otcanbereadbyanyonewhowantstoverifyPfssignature.
-ItcanbeproveninacourtoflawthatPwasthecreaterofoc.
Onceochasbeenplacedintherepository,Pcanuseittogenerateasignatureforanysingledocumenthewishestosend.
Rabin'smethodhasthefollowingdrawbacksnotpresentinours.
1.
ThedocumentmmustbesenttoasinglerecipientQ,whothenrequestsadditionalinformationfromPtovalidatethesignature.
Pcannotdivulgeanyadditionalvalidatinginformationwithoutcompromisinginformationthatmustremainprivatetopreventsomeoneelsefromgeneratinganewdocumentmfwithavalidsignatureap(mf).
2.
Foracourtoflawtodetermineifthesignatureisvalid,itisnecessaryforPtogivethecourtadditionalprivateinformation.
Thishasthefollowingimplications.
.
P—oratrustedrepresentativeofP—mustbeavailabletothecourt,-Pmustmaintainprivateinformationwhoseaccidentaldisclosurewouldenablesomeoneelsetoforgehissignatureonadocument.
Withourmethod,Pgeneratesasignaturethatisverifiablebyanyone,withnofurtheractiononPfspart.
Aftergeneratingthesignature,Pcandestroytheprivateinformationthatwouldenablesomeoneelsetoforgehissignature.
TheadvantagesofourmethodoverRabin'sareillustratedbythefollowingconsiderationswhenthesigneddocumentmisacheckfromPpayabletoQ.
1.
ItiseasyforQtoendorsethecheckpayabletoathirdpartyRbysendinghimthesignedmessage"makempayabletoRlf.
However,withRabin'sscheme,RcannotdetermineifthecheckmwasreallysignedbyP,sohemustworryaboutforgerybyQaswellaswhetherornotPcancoverthecheck.
Withourmethod,thereisnowayforQtoforgethecheck,sotheendorsedcheckisasgoodasacheckpayabledirectlytoRsignedbyP.
(However,someadditionalmechanismmustbeintroducedtoprevent0fromcashingtheoriginalcheckafterhehassigneditovertoR.
)2.
IfPdieswithoutleavingtheexecutorsofhisestatetheinformationheusedtogeneratehissignatures,thenRabin'smethodcannotpreventQfromundetectablyalteringthecheckm—forexample,bychangingtheamountofmoneypayable.
Suchposthumousforgeryisimpossiblewithourmethod.
3.
WithRabin'smethod,tobeabletosuccessfullychallengeanyattemptbyQtomodifythecheckbeforecashingit,Pmustmaintaintheprivateinformationheusedtogeneratehissignature.
Ifanyone(notjustQ)stolethatinformation,thatpersoncouldforgeacheckfromPpayabletohim.
OurmethodallowsPtodestroythisprivateinformationaftersigningthecheck.
2.
TheAlgorithmWeassumeasetMofpossibledocuments,asetICofpossiblekeys,1TheelementsofKarenotkeysintheusualcryptographicsense,butarearbitrarydataitems.
WecallthemkeysbecausetheyplaythesameroleasthekeysinRabin'salgorithm.
andasetV^ofpossiblevalues.
Let2denotethesetofallsubsetsof{1,.
.
.
,40}containingexactly20elements.
(Thenumbers40and20arearbitrary,andcouldbereplacedby2nandn.
WeareusingthesenumbersbecausetheywereusedbyRabin,andwewishtomakeiteasyforthereadertocompareourmethodwithhis.
)Weassumethefollowingtwofunctions.
1.
AfunctionF:IC->V_withthefollowingtwoproperties:a.
GivenanyvaluevinVfitiscomputationallyinfeasibletofindakeykinKsuchthatF(k)=v.
b.
Foranysmallsetofvaluesv1f.
.
.
,vffl,itiseasytofindakeyksuchthatF(k)isnotequaltoanyofthevi2.
AfunctionG:M^->2withthepropertythatgivenanydocumentminM,itiscomputationallyinfeasibletofindadocumentm1imsuchthatG(mf)=G(m).
ForthefunctionF,wecanuseanyonewayfunctionwhosedomainisthesetofkeys.
ThesecondpropertyofFfollowseasilyfromthesecondpropertyoftheonewayfunction.
WewilldiscusslaterhowthefunctionGcanbeconstructedfromanordinaryonewayfunction.
Forconvenience,weassumethatPwantstogenerateonlyasinglesigneddocument.
Later,weindicatehowhecansignmanydifferentdocuments.
ThesenderPfirstchooses40keysk^suchthatallthevaluesFCk.
^)aredistinct.
(OursecondassumptionaboutFmakesthiseasytodo.
)Heputsinapublicrepositorythedataitemat=(F(k.
F(kjj0)).
NotethatPdoesnotdivulgethekeys^,whichbyourfirstassumptionaboutFcannotbecomputedfroma.
Togenerateasignatureforadocumentra,PfirstcomputesG(m)toobtainasetli-j,.
.
.
,i2o^°^integers.
Thesignatureconsistsofthe20keysk,L.
Moreprecisely,wehaveap(m)=(k_.
k_.
),i1i2Qri1i20wherethei-aredefinedbythefollowingtworequirements:(i)G(m)=Ult.
.
.
,i20}.
(ii)i1computationallyinfeasible.
)Suchfunctionsaredescribedin[1]and[2].
TheobviouswaytoconstructtherequiredfunctionGistolet$besuchaonewayfunction,anddefineG(m)toequalR((m)),whereR:{0,.
.
.
,2n-1}-2.
ItiseasytoconstructafunctionRhavingtherequiredrangeanddomain.
Forexample,onecancomputeR(s)inductivelyasfollows:1.
Dividesby40toobtainaquotientqandaremainderr2.
Usertochooseanelementxfrom{1,.
.
.
,40}.
(Thisiseasytodo,since0rjtobesurethattheresultingfunctionGhastherequiredproperty.
Wesuspectthatformostonewayfunctions,thismethodwouldwork.
However,wecannotprovethis.
ThereasonconstructingGinthismannermightnotworkisthatthefunctionRfrom{0,.
.
.
,2n}into2isamanytoonemapping,andtheresulting"collapsing11ofthedomainmightdefeattheonewaynatureof.
However,itiseasytoshowthatifthefunctionRisonetoone,thenproperty(ii)ofimpliesthatGhastherequiredproperty.
ToconstructG,weneedonlyfindaneasilycomputableonetoonefunctionRfrom{0,.
.
.
,2n-1}into2,forareasonablylargevalueofn.
WecansimplifyourtaskbyobservingthatthefunctionGneednotbedefinedontheentiresetofdocuments.
Itsufficesthatforanydocumentm,itiseasytomodifyminaharmlesswaytogetanewdocumentthatisinthedomainofG.
Forexample,onemightincludeameaninglessnumberaspartofthedocument,andchoosedifferentvaluesofthatnumberuntilheobtainsadocumentthatisinthedomainofG.
Thisisanacceptableprocedureif(i)itiseasytodeterminewhetheradocumentisinthedomain,and(ii)theexpectednumberofchoicesonemustmakebeforefindingadocumentinthedomainissmall.
Withthisinmind,weletn=MOanddefineR(s)asfollows:ifthebinaryrepresentationofscontainsexactly20ones,thenR(s)={i:theitjibitofsequalsone},otherwiseR(s)isundefined.
Approximately13%ofall40bitnumberscontainexactly20ones.
Hence,iftheonewayfunctionissufficientlyrandomizing,thereisa.
13probabilitythatanygivendocumentwillbeinthedomainofG.
Thismeansthatrandomlychoosingdocuments(ormodificationstoadocument),theexpectednumberofchoicesbeforefindingoneinthedomainofGisapproximately8.
Moreover,after17pchoices,theprobabilityofnothavingfoundadocumentinthedomainofGisabout1/10^.
(Ifweuse60keysinsteadof40,theexpectednumberofchoicestofindadocumentinthedomainbecomesabout10,and22pchoicesareneededtoreducetheprobabilityofnotfindingoneto1/10p.
)Iftheonewayfunctionkiseasytocompute,thenthesenumbersindicatethattheexpectedamountofefforttocomputeGisreasonable.
However,itdoesseemundesirabletohavetotrysomanydocumentsbeforefindingoneinthedomainofG.
WehopethatsomeonecanfindamoreelegantmethodforconstructingthefunctionG,perhapsbyfindingaoneto.
onefunctionRwhichisdefinedonalargersubsetof{0,.
.
.
,2n}.
Note;WehavethusfarinsistedthatG(m)beasubsetof{1,.
.
.
,40}consistingofexactly20elements.
ItisclearthatthegenerationandverificationprocedurecanbeappliedifG(m)isanypropersubset.
AnexaminationofourcorrectnessproofshowsthatifweallowG(m)tohaveanynumberofelementslessthan40,thenourmethodwouldstillhavethesamecorrectnesspropertiesifGsatisfiesthefollowingproperty:-ForanydocumentmfitiscomputationallyinfeasibletofindadifferentdocumentmfsuchthatG(mf)isasubsetofG(m).
BytakingtherangeofGtobethecollectionof20elementsubsets,weinsurethatG(mf)cannotbeapropersubsetofG(m).
However,itmaybepossibletoconstructafunctionGsatisfyingthisrequirementwithoutconstrainingtherangeofGinthisway.
REFERENCES[1]Diffie,W.
andHellman,M.
"NewDirectionsinCryptography".
IEEETrans,^nInformationTheoryIT-22_(November1976),544-654.
[2]Rabin,M.
"DigitalizedSignatures",inFoundationsofSecureComputing,AcademicPress(1978),155-168.