当前位置： 文档下载 > 所有分类 > Smooth numbers and the quadratic sieveWhen faced with a large number n to factor, what do you do first? You might say “Look at the last digit, ” with the idea of cheaply pulling out possible factors of 2 and 5. Sure, and more generally, you can test for divisibility cheaply by all of the ver
SMOOTHNUMBERSANDTHEQUADRATICSIEVE
CarlPomerance
Whenfacedwithalargenumberntofactor,whatdoyoudo rst?Youmightsay“Lookatthelastdigit,”withtheideaofcheaplypullingoutpossiblefactorsof2and5.Sure,andmoregenerally,youcantestfordivisibilitycheaplybyalloftheverysmallprimes.Soitmayaswellbeassumedthatthenumbernhasnosmallprimefactors,saybelowlogn.Sinceitisalsocheaptotestforprobableprimeness,saythroughthestrongprobableprimetest,andthenactuallyproveprimalityasin[4]inthecasethatyoubecomeconvincednisprime,italsomayaswellbeassumedthatthenumberniscomposite.
Trialdivisionisafactoringmethod(andintheextreme,aprimalitytest)thatinvolvessequentiallytryingnfordivisibilitybytheconsecutiveprimes.Thismethodwasinvokedabovefortheremovalofthesmallprimefactorsofn.Theonlythingstoppingusfromcontinuingbeyondthesomewhatarbitrarycuto oflognistheenormoustimethatwouldbespentifthesmallestprimefactorofnisfairlylarge.Forexample,ifnwereamodulusbeingusedintheRSAcryptosystem,thenascurrentprotocolsdictate,nwouldbetheproductoftwoprimesofthesameorderofmagnitude.Inthiscase,factoringnbytrialdivisionwouldtakeroughlyn1/2steps.Thisalreadyisanenormouscalculationifnhasthirtydecimaldigits,andfornumbersonlyslightlylonger,thecalculationisnotpossibleatthistimebythehumanraceandalloftheircomputers.
Di erenceofsquares
Wehavelongknownhoweverthattrialdivisionisnottheonlygameintownforfactoring.Takethenumber8051forexample.Thisnumberiscompositeandnotdivisiblebyanyprimeuptoitslogarithm.Onecanseeinstantly(ifonelooks)thatitis8100 49,thatis,
8051=902 72.
Thus,wecanusealgebratofactor8051asadi erenceofsquares.Itis(90 7)×(90+7),or83×97.Everyoddcompositecanbefactoredasadi erenceofsquares(aneasyexercise),sowhydon’tweusethismethodinsteadoftrialdivision?
Letustryagainonthenumbern=1649.Again,1649iscomposite,butnotdivisiblebyanyprimeuptoitslogarithm.Whatworkedwith8051wastotakethe rstsquareabove8051,namely902,andthennoticethat902 8051=49,where49isrecognizedasasquare.Itwouldseemingeneralthatonecouldwalkthroughthesequencex2 nwithx= n1/2 , n1/2 +1,...,lookingforsquares.Withn=1649wehave
412 n=32,
422 n=115,
432 n=200,
...
withnosquaresinimmediatesight.
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