千米小说网

Chapter 4 Cryptography the secret life of primes（第1页）

Chapter4Cryptography:thesecretlifeofprimes

Thereaderreciatethattheofumbershas，fromtheearliesttimes，beenreizedastherepositoryofriddlesas，manyofwhieverbeehisday。Formanyofus，thisisenoughtojustifytheuedseriousstudyofhersmaytakeadifferentattitude。Intriguinganddifficultasthesesmaybe，itmightbeimagiheyhavelittlebeariofhumanwisdom。Butthatwouldbeamistake。

&helastfewdecadesithasemergedthatordihekindswealliimetotime，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。Seesontotheswemasktheseplaihothernumbers。

&roduceyoutothefictitiouscharactersthatpopulatethevarioussariraphy，whichisthestudyofciphers（secretcodes）。WeimagineAlidBob，whowanttouheachother，withoutbeiheeavesdropper，Eve。Instinctively，wemightsympathizewithAlidBEveasuptonogood，butofcoursethereversemaybetrue，withEverepresentinganoblepoligauthtoprotectusallfromtheevilplotsofBobandAlice。

&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，ishecase。Theywouldbewelladvisedtogethatnumber，iteroffusiimebecauseiftheydoemwillbegintoleakinformationtoEve。Forexample，sayinafutureweekAlitstosendtoBobthesamemessagenumber8。EverythingwouldrunasbeforeandonEvewouldihemysteriousnumber65fromtheairwaves，butthistimeitwouldtellhersomething。Evehateverthismessageis，itisthesamemessagethatAlittoBobiweek–thisisjustthesortofthingAlidBobwouldoknow。

This，however，lookstobenobigproblemforAlidBob。Whemeetupto‘exgekeys’，insteadonumber，AlicecouldprovideBobwithaloofthousaobeusedoher，thusavoidingthepossibilityofmeaningfultheirpubliclyavailableunis。

Andthisisiisdoice。Thiskindofciphersystemisknowradeasaohesenderaheirplaihasingle-usehe‘pad’。Thatleafofthepadisthehthesenderahemessagehasbeeaheoimepadrepresentsapletelysecuresysteminthattheihattravelsinthepubliainoinformationaboutthetoftheplaiodecipherit，theiorholdofthatpadiaiioionkey。

Keysandkeyexge

Itwouldseemthenthattheproblemofseunipletelysolvedbytheoimepadand，inaway，thatistrue。Thedifficultywithciphersliketheoimepad，however，isthattheyrequirethepartitstoexgeakeyiousethem。Iakesalotofeffh-levelunis，suchasthosebetweeeHouseandtheKremlin，moneyishenecessaryexgesarecarriedoutuionsofmaximumsetheeverydayworldoherhand，allsortsofpeopleandinstitutioouhoherinatialfashioitsotaffordthetimeandeosecurekeyexd，evenifthiswerearrarustedthirdparty，itexpensivebusiness。

Theondrahersthathadbeehousandsofyearsupuntilthe1970swasthattheywereallsymmetricciphers，meaningthattheenaioiallythesame。Whetheritwasthesimplealphabet-shiftcipherofJuliusplexEnigmaCipheroftheSedWorldWar，theyallsufferedfromtheoonadversarylearnedhowyouwereenessages，theyjustaswellasyou。Iomakeuseofasymmetriccipher，theunigpartoexgethecipherkeyinasecure>

&ohavebeentacitlyassumedthatthiswasanunavoidableprincipleofsecretcodes–foraciphertobeusedthepartnersneeded，somehoworother，toexgethekeytothedtokeepitsetheehismightberegardedasmathemationsense。

Thisisthekindofassumptionthatmakesamathematisuspiciwithwhatisessentiallyamathematicalsituation，soosuciple’tobewellfoundedaedbysomeformofmathemati。Yettherewasnosu，atherewasnosurinciplesimplyisnotvalid，asthefollowingthoughtexperimentreveals。

TransmissionefromAlicetoBobdoesnotinitselfheexgeofthekeytoacipher，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）thatiscarriedoutfirstbyAlidthenbyBobunworkable。However，thatthismethodbeeffectiveublistratedbyWhitfieldDiffieandMartinHellmanin1976。

Asedrelatedapproachistheideaofasymmetricorpublickeycryptographyinwhiepublishestheirownpublickeythatistheesmeantforthatperson。Hoersonalsoholdsaprivatekey，withoutwhichthemessageseheirownuniquepubliotberead。Ihepadlockmetaphor，AliceprovidesBobwithaboxinwhichtoplacehisplaiogetheradlock（herpublickey）towhichshealohekey（herprivatekey）。

Aublickeysystemmightseemtoomuchtoaskforasthetwisofsedeaseofuseseemtoflict。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，theRSAsystemofengisbasedontheempiricalfactthatitisprohibitivelydiffidtheprimefactorsofavery，verylargehecleverpart，whichlainintheremaihechapter，liesindevisingawaythatthemessagenumbermcipheredjustusingthepublibers，inpractice，degrequirespossessionoftheprimefa。

HereishohatBobsendsthroughtheetherisheremainderwhenmeisdividedbyheakingthisremainderrandsimilarlygtheremainderwhenrdisdividedbyn。TheunderlyiisuresthattheouteforAliceistheinalmessagem，whitheoordinaryplaibyAliputersystem。Thisis，ofcourse，happeningseamlesslybehindthesesforanyreal-lifeAlidBob。

ItwouldseemthattheonlythingthatEvelacksthatreallymattersisthisdegnumberd。IfEvek，shecoulddecipherthemessagejustaswellasAlice。Itturnsoutthatdisasolutioaiion。SolviionisputationallyquiteeasyaheEuAlgorithm，publishedintheBooksofEu300BC。Thatisnotthedifficulty。Thetroubleisthatitisnotpossibletofilywhatequationtosolveunlessyoukoheprimespandq，andthatistheobstaclethatstopsEveiracks。

laihowthenumbersinvolvedinallthisworkiem。First，thereisapparentlyquiteaproblemwithBob’sinitialtask。Thenumbermisbig，thenumbernismonstrous（oftheorderof200digits）andevehatlarge，thenumbermeisgoiremelylargeaswell。Aftergit，wehavetodividemebythetheremainderr，whichrepresentsthee。Itmightseemthatthecalsaretoouobepractical。Weshouldbeawarethateventhoughmodernputersareextremelypowerful，theyyethavetheirlimitations。Whencalsinvhpowers，theyexceedtheputersystem。Welyethatanypracticalcalthatwesetforaputereinashortperiodoftime。

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。