参考分析密码学

CollegeofComputerScienceandElectronicconclusiononletter-basedciphers 3500

~1950

math

2015nomathematicalproofofanykindforpracticalciphers.forbruteattacks:64bitskey-hourstodays;128bitskey-longtermsecuritywithoutQ.C.;256bitskey-longtermsecurityevenwithQ.C.intextalphabet{A,B,…,Z}maptoapermutationalphabet{c,g,…,a=rmod <=>a=qm+r(quotientand(a+b)modm=((amodm)+(bmodm))mod(a-b)modm=((amodm)-(bmodm))mod(ab)modm=((amodm)(bmodm))mod(a/b)modm=a*b-1modm=((amodm)(b-1modm))modthesufficientandnecessaryconditionforb-1modmexistsis(b,m)=congruenceclass:{x|x=rmodm}forgivenrandringisasetthatmeetsthefollowingtherearetwooperationsand・botharecloseontheforoperation+andanythreeelementsa,b,cinthea+b=b (a+b)+c=a+(b+ thereisazeroelement0intheset,suchthata+0=0+a=aforanyelementaintheset,thereisanadditiveinverseelement-aintheset,suchthata+(-a)=0foroperationsandandanythreeelementsabcinthe(a+b)・c=a・c+b・ (rightmultiplicationa・(b+c)=a・b+a・ (leftmultiplicationnote:multiplicationisNOTnecessarily(somemathematiciansrequiresthereisaunityelement1intheset,suchthatforanyelementaintheset,a・1aand1・aa(othermathematicianssaysthisconditionleadstoanewalgebraicstructure:semi-note:thereisnothingaboutmultiplicativeinverseinaintheringa・00・a0somemoremath:) 群环域thatdealswithadditionandmultiplicationandwhatisadditionandwhatis加法（+）：yx1x2x3乘法（・）：yx1・x2・x3・ 对于加法y=x1+x2+x3+….+xn，哪怕我们不知道于是对于加性系统，我们（大体上）可以预计它的可靠性与稳定性（期望和故障概率预计它运行时性能的随机波动（噪声对于乘性系统，须承认有没有可能存在一种药/秘法，能让人ContentofthisOverviewonthefieldofBasicsofsymmetricAttackingcryptoSubstitutionModularShift(orCaesar)CipherandAffine Chapter1ofUnderstandingCryptographybyChristofPaarandJan Shift(orCaesar)CipherAncientcipher,allegedlyusedbyJulius ces intextletterbyanother cementruleisverysimple:TakeletterthatfollowsafterkpositionsintheNeedsmapfromletters→ABCDEFGHIJKLM0123456789NOPQRSTUVWXYZExamplefork=intext=ATTACK=0,19,19,0,2,Ciphertext=haahr=7,0,0,7,Notethattheletters”wraparound”attheendofthealphabet,whichcanbemathematicallybeexpressedasreductionmodulo26,e.g.,19+7=26≡0mod26 Chapter1ofUnderstandingCryptographybyChristofPaarandJan Shift(orCaesar)CipherElegantmathematicaldescriptionoftheLetLetk,x,yε{0,1,…,••y=ek(x)≡x+kmodx=dk(x)≡y-kmodQ;IstheshiftcipherA:No!severalattacksarepossible,Exhaustivekeysearch(keyspaceisonlyLetter ysis,similartoattackagainstsubstitution Chapter1ofUnderstandingCryptographybyChristofPaarandJan AffineCipherExtensionoftheshiftcipher:ratherthanjustaddingthekeytothe intext,wealsomultiplybythekeyWeuseforthisakeyconsistingoftwoparts:k=(a,LetLetx,yε{0,1,…,••y=ek(x)≡ax+bmodx=dk(x)≡a-1(y–b)modSincetheinverseofaisneededfordecryption,wecanonlyusevaluesforafor(a,26)=Thereare12valuesforathatfulfillthisFromthisfollowsthatthekeyspaceisonly12x26=312(cf.Sec1.4inUnderstandingAgain,severalattacksarepossible,Exhaustivekeysearchandletterfrequency ysis,similartotheattackagainstthesubstitutioncipher Chapter1ofUnderstandingCryptographybyChristofPaarandJantheorderofletters complextransposit