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Chapter 4 Cryptography the secret life of primes（第2页）

Whatismore，since74has01asthefinaldigitpair，thefourpowerswillendin07，49，43，andthen01again.Heesugpowers，thepatterwodigitswillsimplyrepeatthiscygthfour，overandaihequestioninhand，since39=4×9+3，wewillpassthroughthisfour-etimesahreemorestepsingthefinaltwodigitsof739，whichmustthereforebe43.

Andthisworksquitegenerally.Iofindtheremainderwhensomepowerabisdividedbynsay，weaketheremainderrwhenaisdividedbyraaindersaswetakesuccessivepowersofr.iththeremainderr，whichwillbeaheraoissaythatwearewmodulaiplesofnthatmayarise，astheyleavearemainderof0whendividedbyn，andsotributetothevalueofthefinalremainderr.

AllthisservestogiveanindithatthebyBobtoAlice，memodulon，deedbecalculatedwithouttoomucheffortonbehalfofBob’sputer.Allthesame，thenumbersinvolvedareiybig，somoreexplanatiooshowthattheydled.Thelargepowersinvolvedinebedealtwithinstagesbyaproasfastexpoion.Withoutgoihemethodinvolvessuccessivesquaringandmultiplyioarriveatmemodulonwiththebinaryfuidihmthroughtoquidtherequiredremaiivelyfewsteps.

EuclidshowsAlicehowtofindherdegnumber

&erddusioolthatisover2，300yearsold，theEuAlgorithm，whichwillbeexplainedi.Eve’sputercouldofethingifitjustknewwhichequationtosolve.However，sindqareprivatetoAlice，sois（p-1）（q-1）akobegin.

&otheEuAlgorithm，thisbeginsfromtheobservationthatitispossibletofionfabersa＞bbysuccessivesubtra.（Thehcfisalsoknownasthegcd–greatestondivisor.）Wejustr=a-bhasthepropertythatanyonfaytwoofthethreenumbersa，b，andrwillalsobeafactorofthethird.Forexample，ifonfactorofaandb，sothata=db=cb1say，weseethatr=a-b=ca1-cb1=c（a1-b1），givingusafactorizationofrinvolvingthedivisorparticular，thehdbisthesameasthehdr.Sihesenumbersarelessthana，wehesameproblembutappliedtoasmallernumberpair.Repetitionofthisideatheuallyleadtoapairwherethehcfisobvious.（Iwonumbersinhauallybethesame，forifnotwecouldproorestep;theirohenthenumberweseek.）

&ofindthehcfofa=558andb=396，thefirstsubtrawouldgiveusr=558-396=162，soournewpairwouldbe396and162.Since396-162=234，ourthirdpairbees234and162，aihefulllistofnumberpairsis:

andsothehcfof558and396is18.

Itispossibletowritedownthehberpairfromtheprimefactorizationsofthenumbersiion.Inthisexample，558=2×32×31，while396=22×32×11;takingtheonporimeeothefactorizatiohehcfas2×32=18.heless，ferakesmuchlessworktouseEuclid’sAlgorithmasitisgeoperformsubtrasthantofiorizations.

AnotherbonusoftheEuAlgorithmisthatitisalossibletoworkitbadinsodoihehtermsinaltwooseethisinathepreviousexample，itisbesttopressthecalwhenthesamenumberappearsseveraltimesoverinthecourseofthesubtras，representingthisasasiionasfollows:

558=396+162

396=2×162+72

162=2×72+18

72=4×18.

Beginningwiththesedtolastline，weleequatioetheieremaiime.Inthisexample，byusingfirstthepeioheo>

18=162-2×72=162-2×（396-2×162）=5×162-2×396

andfinallyusiequatioethefirstieremainderof162:

=5×（558-396）-2×396=5×558-7×396=18.

48=4×11+4

11=2×4+3

4=1×3+1ingthatthehdeisindeed1.Reversihm>

1=4-3=4-（11-2×4）=3×4-11=3（48-4×11）-11=3×48-13×11.

Thisgivesaninitialvalueofd=-13asthesolutiontotherequirementthat11dleavesremainder1upondivisionby48，setapositivevalueofdintherequiredraothisd=48-13=35.

AndinpassingitiswelltopointoutthattheEuAlgorithmprovidesthemissinglinkinourproofoftheuniquenessofprimefactorizationasitallowsustoverifytheeupropertythatifaprimepisafactoroftheproductab，sothatab=pcsay，thenpisafactorofatleastoneofaandb.Thereasonforthisisthatifpisnotafactorofathen，sincepisprime，thehdpis1.ByreversingtheEuAlgorithmliedtothepairaahenfindintegersrandssaysuchthatra+sp=1.Thisisenoughtoshoisthenafactorofbfor，sinceab=pc，

b=b×1=b（ra+sp）=r（ab）+psb=r（pc）+psb=p（rc+sb）.

Thisistherequiredfactorizationofbthatfeaturestheprimepasafactor.

In，theheRSAengmakesthesystemsound，althoughvariousprotocolsthathavenotbeenexplaiberespeuardtheiyofthesystem.Thereareissuesofauthentifi（whatifEvetactsAlidingtobeBob?），ion（whatifBobpretendsthatitwasEvewhoseoAlididentityfraud（whatifAliceabusestialidentifittoherbyBobaoimpersonatehimonliherweakhesystembeexposedwheableorrepeatedmessagesproliferate.However，thesedifficultiesmaypotentiallyariseinanypublickey.Theyeandinthemaiotheunderlyieiquesthatensurehighqualityandrobusten.

ThischapterhasdemonstratedamajorappliehetheoryofdivisibilityandremaimathematicsofEudthe18th-turytributionofEulerallowsthistobeexplaionlyinbroadpriiail.

&partofourbookclosester5whiespecialclassesofintegersassociatedwiththeeiourallyroupings.