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Chapter4Cryptography:thesecretlifeofprimes
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Thereaderreciatethattheofumbershas，fromtheearliesttimes，beenreizedastherepositoryofriddlesas，manyofwhieverbeehisday.Formanyofus，thisisenoughtojustifytheuedseriousstudyofhersmaytakeadifferentattitude.Intriguinganddifficultasthesesmaybe，itmightbeimagiheyhavelittlebeariofhumanwisdoButthatwouldbeamistake.
&helastfewdecadesithasemergedthatordihekindsweallindulgeinfromtimetotime，becodedassebers.Thishasintopradourmostprecioussecrets，whethertheybeerilitary，personalorfinancial，politicalhtsdalous，allbeprotetheIbymaskingthemusiordinaryumbers.
&uronumbers
Hoossible?Anyinformatiobeapoemoraba，ablueprintforaonoraputerprogram，bedeswords.Wemay，however，oaugmethatisusedtomakeupourwordsbeyondtheordihealphabet.Wemayinumbersymbols，punbolsingspecialsymbolsforspaordinarywords，butitishecasethatalltheinformatiootransfer，instrusfpiddiagrams，beexpressedusingwordsfromaussay，hahousandsymbols.Wetthesesymbolsaeachsymboluniquelyasanumber.Sinumbersarediible，itmaybettousenumbersallwiththesamenumberofdigitsforthispurpose（so，forexample，everysymbolresentedusitPIringthesymbolstogetherasrequiredtogiveonebiglooldtheeory.Weworkinbinaryifwewishandsodeviseawayanyinformatioringof0sand1s.Everymessagewemighteverwanttosendbecodedasabinarystringatheotherendbyasuitablyprogrammedputer，tobepiledinethatrehend.Thistherealizatiooseweenonepersonaisenough，bothintheoryandiobeabletosendnumbersfromooanother.
Turnionumbers，however，isnotthebigidea.Tobesure，theexactprocessbywhichalltheinformationisdigitizedmaybehiddenfromthegeneralpubliethelessisnotthesourceofproteeavesdroppers.Ihepointofviewraphy，wemayidentifyaheso-calledplaihtherepresentsitahinkofthatheplaiisassumedthatanyonehasaccesstothewherewithalthatwillalloworaheother.Seesontotheslaihothernumbers.
&roduceyoutothefictitiouscharactersthatpopulatethevarioussariraphy，whichisthestudyofciphers（secretcodes）.WeimagineAlidBob，whowanttouheachother，withoutbeiheeavesdropper，Eve.Instinctively，wemightsympathizewithAlidBEveasuptonogood，butofcoursethereversemaybetrue，withEverepresentinganoblepoligauthtoprotectusallfromtheevilplotsof BobandAlice.
&hemoralstandiits，thereisanage-oldapproachthatAliploytocutEveoutoftheversationeveerceptsmessagesthatpassbetweeheycryptthedatausihatisknownonlytoAlidBob.Whattheymayarraodoistomeetinaseviroheyexgewithoheraseumber（letussay57）aurimees，AlicewillwanttoseoBoband，justtoillustratethepoint，supposethatmessageberepresentedbyasibetween1and9.Onthebigday，AlitstoseoBob.Shetakeshermessageandaddsthesegredient，thatistosayshemasksitstruevaluebyadding57ahemessagetoBob，ainseel，of8+57=65.BobreceivesthismessageandsubtractstheseumbertoretrieveAlitext65-57=8.
ThenefariousEve，hoodideawhatthesettoandindeeddoesmaercepttheencipheredmessage，65.Butwhatshedowithit?ShemaykAlitoheninepossiblemessages1，2，3，…，9toBobandalsoknowsthatshehasebyaddihemessage，whichmustthereforeliebetween65-9=55and65-1=64.However，becausesheottellwhichoftheseninemaskingnumbershasbeei），sheishewiserastotheactualplaihatAlittoBob，whichisstilljustaslikelytobeaheninepossibilities.AllsheknowsisthatAlitamessagetoBobbuthasis.
ItmightseemthatAlidBobareothemaliceofEveanduhimpunityusingthemagiumber57todisguisealltheyhavetosay.That，however，isn'tquitethecase.Theywouldbewelladvisedtogethatnumber，iteroffusiimebecauseiftheydon't，thesystemwillbegintoleakinformationtoEve.Forexample，sayinafutureweekAlitstosendtoBobthesamemessagenumber8.EverythingwouldrunasbeforeandonEvewouldihemysteriousnumber65fromtheairwaves，butthistimeitwouldtellhersomething.Evehateverthismessageis，itisthesamemessagethatAlittoBobiweek–thisisjustthesAlidBobwouldoknow.
This，however，lookstobenobigproblemforAlidBob.Whemeetupto'exgekeys'，insteadonumber，AlicecouldprovideBobwithaloofthousaobeusedoher，thusavoidingthepossibilityofmeaningfultheirpubliclyavailableunis.
Andthisisiisdoice.Thiskiemisknowradeasaohesenderaheirplaihasingle-usehe‘pad’.Thatleafofthepadisthehthesenderahemessagehasbeeaheoimepadrepresentsapletelysecuresysteminthattheihattravelsinthepubliainoinformationaboutthetoftheplaiodecipherit，theiorholdofthatpadiaiioionkey.
Keysandkeyexge
Itwouldseemthenthattheproblemofseunipletelysolvedbytheoimepadand，inaway，thatistrue.Thedifficultywithciphersliketheoimepad，however，isthattheyrequirethepartitstoexgeakeyiousetheIakesalotofeffh-levelunis，suchasthosebetweeeHouseandtheKremlin，moneyishenecessaryexgesarecarriedoutuionsofmaximumsetheeverydayworldoherhand，allsortsofpeopleandinstitutioouhoheriialfashioitsotaffordthetimeandeosecurekeyexd，evenifthiswerearrarustedthirdparty，itexpensivebusiness.
Theondrahersthathadbeehousandsofyearsupuntilthe1970swasthattheywereallsymmetricciphers，meaningthattheenaioiallythesame.Whetheritwasthesimplealphabet-shiftcipherofJuliusplexEnigmaCipheroftheSedWorldWar，theyallsufferedfromtheoonadversarylearnedhowyouwereenessages，theyjustaswellasyou.Iomakeuseofasymmetriccipher，theunigpartoexgethecipherkeyinasecure>
&ohavebeentacitlyassumedthatthiswasanunavoidableprincipleofsecretcodes–foraciphertobeusedthepartnersneeded，somehoworother，toexgethekeytothedtokeepitsetheehismightberegardedasmathemationsense.
Thisisthekindofassumptionthatmakesamathematisuspiciwithwhatisessentiallyamathematicalsituation，soosuciple’tobewellfoundedaedbysomeformofmathemati.Yettherewasnosu，atherewasnosurinciplesimplyisnotvalid，asthefollowingthoughtexperimeransmissionefromAlicetoBobdoesnotinitselfheexgeofthekeytoacipher，fortheyproceedasfollows.AlicewritesherplaimessageforBob，ainaboxthatshesecureswithherownpadlolyAlicehasthekeytothislock.ShethenpoststheboxtoBob，whoof.Bob，however，thenaddsasedpadlocktothebox，forossessesthekey.TheboxistheoAliovesherownlodsendstheboxforaseetoBob.Thistime，BobmayunlocktheboxandreadAlice’smessage，sethekhemeddlingEveothavepeekedatthetsduringthedeliveryprothisway，asecretmessagemaybesetonaninseelwithoutAlidBihisimaginarysarioshowsthatthereisnolawthatsaysthatakeymustdsintheexgees.Iem，AlidBob’s‘locks’mightbetheirownessageratherthanaphysicaldeviceseparatingthewould-beeavesdropperfromtheplai.Aliaythehisiosetupanordiriccipherthatwouldbeusedtomaskalltheirfutureuni.
&hisisthewayaseunielisofteablishedintherealwphysigdevicesbyperso，however，soeasytodo.UheengsofAliayihoher，makingtheunsg（thatis，theunlog）thatiscarriedoutrstbyAlidthenbyBobunworkable.However，thatthismethodbeeffectiveublistratedbyWhiteldDifeandMartinHellmanin1976.
Asedrelatedapproachistheideaofasymmetricorpublickeycryptographyinwhiepublishestheirownpublickeythatistheesmeantforthatperson.Hoersonalsoholdsaprivatekey，withoutwhichthemessageseheirownuniquepubliotberead.Ihepadlockmetaphor，AliceprovidesBobwithaboxinwhichtoplacehisplaiogetheradlock（herpublickey）towhichshealohekey（herprivatekey）.
Aublickeysystemmightseemtoomuchtoaskforasthetwisofsedeaseofuseseemtofiict.Fast，safeenis，however，availabletothegeneralpublitheIheybarelyrealizethatitisthere，safeguardierests.Anditisalldowntonumbers，andprimehat.
Howsecretprimesprotectoursecrets
&everyplaimessageisregardedjustasasiisnaturaltotrytomaskthisnumberusingotherhemostonwaytodothisisthroughemployingtheso-calledRSAengprocess，publishedin1978byitsfounders，Ro，AdiShamir，andLeonardAdleman.InRSA，ea’sprivatekeysistsofthreenumbers，p，q，andd，wherepandqare（verylarge）primehethirdidisAlice’ssecretdegheroleofwhichwillbeexplainedinduecourse.Aliceprovidesthepubli=pq，theproductofhertwosecretprimes，andanengnumbere（whiordinarywholenumber，inothespestaionedinChapter2）.
AsimpleexampleforthepurposesofillustrationwouldbeforAlicetohavetheprimesp=5andq=13sothatn=5×13=65.IfAlicesetsherengobee=11，thenherpublickeywouldbe（n，e）=（65，11）.Toencryptamessagem，BobonlyneedsnaodeciphertheencryptedmessageE（m）thatBobtransmitstoAlicerequiresthedegnumberd，whithissouttobed=35，asweshallshowalittlefurtheroicsthatallowsdtobecalculatedrequiresthattheprimespandqareknown.Inthistoyexample，giventhatn=65，anyonewouldsoop=5andq=13.However，iftheprimesparemelylarge（typicallytheyarehunderedsofdigitsihistaskbeesapracticalimpossibilityforalmostaem，atleastinareasonablyshorttime，suchastwoorthreeweeks.Insummary，theRSAsystemofengisbasedontheempiricalfactthatitisprohibitivelydifficulttofindtheprimefactorsofavery，verylargehecleverpart，whichlainintheremaihechapter，liesindevisingawaythatthemessagenumbermcipheredjustusingthepublibers，inpractice，degrequirespossessionoftheprimefa.
HereishohatBobsendsthroughtheetherisheremainderwhenmeisdividedbyheakingthisremainderrandsimilarlygtheremainderwhenrdisdividedbyn.TheunderlyiisuresthattheouteforAliceistheinalmessagem，whitheoordinaryplaibyAliputersysteThisis，ofcourse，happeningseamlesslybehindthesesforanyreal-lifeAlidBob.
ItwouldseemthattheonlythingthatEvelacksthatreallymattersisthisdegnumberd.IfEvek，shecoulddecipherthemessagejustaswellasAlice.Itturnsoutthatdisasolutioaiion.SolviionisputationallyquiteeasyaheEuAlgorithm，publishedintheBooksofEu300BC.Thatisnotthedifficulty.Thetroubleisthatitisnotpossibletofilywhatequationtosolveunlessyoukoheprimespandq，andthatistheobstaclethatstopsEveiracks.
laihowthenumbersinvolvedinallthisworkieFirst，thereisapparentlyquiteaproblemwithBob’sinitialtask.Thenumbermisbig，thenumbernismonstrous（oftheorderof200digits）andevehatlarge，thenumbermeisgoiremelylargeaswell.Aftergit，wehavetodividemebythetheremainderr，whichrepresentsthee.Itmightseemthatthecalsaretoouobepractical.Weshouldbeawarethateventhoughmodernputersareextremelypowerful，theyyethavetheirlimitations.Whencalsinvhpowers，theyexceedtheputersysteWelyethatanypracticalcalthatwesetforaputereinashortperiodoftime.
ThesavinggraceforBobisthatitispossibletofindtherequiredremaihoutdoingthelongdivisionatall.Iheremaidependonremainders，andhereisaoillustratethepoint.Whatarethefinaltwodigitsof739?（Thatistosay，whatistheremaihisnumberisdividedby100?）Ioahisquestiobeginbygthefirstfewpowersof7:71=7，72=49，73=343，74=2，401，75=16，807，….Itwillsoonbeeclear，however，thatthesheersizeofthesenumbersisgoingtobeanageablewellbefetanywherenear739.Oherhaeoeranother，atterhekeyobservationisthat，aswecalculatesugpowers，thefinaltwodigitsoftheanswerdependowodigitsnumber，aswhenweultipli，digitsinthehundredsnandbeyondhaveonisandtensns.
Whatismore，since74has01asthefinaldigitpair，thefourpowerswillendin07，49，43，andthen01again.Heesugpowers，thepatterwodigitswillsimplyrepeatthiscygthfour，overandaihequestioninhand，since39=4×9+3，wewillpassthroughthisfour-etimesahreemorestepsingthefinaltwodigitsof739，whichmustthereforebe43.Andthisworksquitegenerally.Iheremainderwhensomepowerabisdividedbynsay，weaketheremainderrwhenaisdividedbyraaindersaswetakesuccessivepowersofr.iththeremainderr，whichwillbeaheraoissaythatwearewmodulaiplesofnthatmayarise，astheyleavearemainderof0whendividedbyn，andsotributetothevalueofthenalremainderr.
YoumightstillsuspectthatIhaveriggedtheevidencebyganexamplewhereaverysmallpowerleftaremainderof1–inthisinstance74was1morethanamultipleofn=100.This，however，isonlypartlytrue.Itturnsoutthat，ifwetakeanytwonumbers，aandonfactoris1（wesaysuumbersaremutuallye），thenthereisalowertsuchthatatequals1modulon，thatistosayleavesaremainderof1whendividedbyn.Fromthispoint，theremaindersofsuccessivepowersfollowacygtht.It，however，behardtopredictwhatisthevalueoft，butitisknownthattmustalwaysequalorbeafabertraditionallywrittenasφ（n），thevalueoftheEulerphifun.
whichisthesameastheresultthatweobtailyfromthedefinition.Usihod，youmightliketocheckyourselfthatφ（100）=40，andso，forihenfollowsthat740equals1modulo100.However，aswehavealreadyseepowerof7thatyieldsaremainderof1isnot40butitsdivisor4.
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