­¦ý°±·;œ œ›œ œœœR ¿F!KÀúYº¿CompObj\WordDocument ObjectPoolÄä¾ úYº!KÀúYºœœœ œœœœœœ>œ@ABCDEFGHIJKLœNOPQRSœœVœXYZ[\]^œœ ¿FMicrosoft Word 6.0 DocumentœNB6WWord.Document.6;œ ­¦ý°±·;œ LâŠ` HËË­¦ý°±·;œ ⊠r .1  @&¿ß¿Á & MathType0 ™"-}™}ï_924373491 ¿FÄä¾ úYºÄä¾ úYºOle PIC LMETA ϚĜêTimes New Romanh\- 2 ýLx® 2 Îgn¿ 2 d¿ 2 ·ik 2 ¢ñik 2 y©tk 2 ý7y® 2 ½þik 2 ýá x® 2 ½/ ik 2 ý™ n¿šÄœêSymbol-• 2 ýh=² 2 ¢v=² 2 ˜dšĜêTimes New Romanh\-• 2 ýd(~ 2 ýß)~ 2 ý mod+¿¿ 2 ¢ò1¿ & šº"Systemn-•­¦ý°±·v*x x*Ýå *x"0 °dPPNTTimes New Roman,ŽTimes New RomanŽ .+x(3n( 0d+i(%i( t+% y+i(Lx+i(pn°dPPNTSymbol, Symbol( =+=(°dPPNTTimes New RomanŽ(+()))mod(%%1Ýç°dPPNT"System­¦ý°±·œ ¿FMicrosoft Equation 2.0 DS EqPICT  vCompObjfObjInfo Equation Native úuation Equation.2Ù9¾q­¦ý°±·;ý°±·;œ ‹ÄG:`þ<<þ< ÉxÜ=Ç(ÉnÉdÉiÇ)ÉiÜ=à1ÉtÜÂÉyÉiÉxÉiÇmÇoÇdÉn& MathTy­¦ý°±·;œ ••••••••••••••••33ffffff33ff33ÃÃff33ôôff33ffff3333ff33ffffÃÃffôôffffff33ff3333ÃÃ33ôô33ff33333333ÃÃ33ÃÃÃÃ33ÃÃôô33ÃÃff33ÃÃ3333ÃÃ33ôô33ôôÃÃ33ôôôô33ôôff33ôô3333ôô33ffECS253 Notes on April 9, 97 Modular Arithmetic: solve ax mod n = 1 by the Extended Euclidean algorithm e.g. to solve: g=gcd(5,7) g x y q 7 0 1 5 1 0 1 2 -1 1 2 1 3 -2 2 0 gcd(5,7)=3*5+(-2)*7=1 solve x in: ax mod n = b if we can find x0 s.t. ax0 mod n = 1, then we have a(x0b mod n) = b thus x = x0b is the solution Chinese method: solve x in x mod di = xi ( i ([1,t] solution:  EMBED Equation.2  where  EMBED Equation.2  yi can be solved by the above methods Cipher: transposition cipher: diffusion effect (suggested by Shannon) can be attacked by frequency count (digram, trigram etc.) substitution cipher: confusion effect (suggested by Shannon) simple substitution: one mapping applied to all components e.g. Caesar cipher f(m)=(m+k) mod 26 where k is the key length is affine transformation: f(m)=(km + kí) mod n where gcd(k,n) =1 can be attacked by comparing the shift with frequency distribution of normal English text example to attack Caesar cipher is run through homophonic substitution (1 mapping for the components |HHŽ(··˜EG(¸HHŽ(d'`ê­¦ý°±·;œ ­¦ý°±·;œ L ا®ËË­¦ý°±·;œ  Ø ±; ._924373490¿FÄä¾ úYºÄä¾ úYºOle <PIC  =LMETA ?l1  @Ý &¿µ` ž & MathType• ™"-› ›*šÄœêTimes New Roman - 2 `4(~ 2 `<)~ 2 `;mod+¿¿šÄœêTimes New Roman-• 2 kn¿ 2 âÞd¿ 2 `Íy® 2 `"d¿š êTimes New Roman -• 2 Èßi> 2 ¿êi> 2 ¿Îi>šÄœêSymbol-• 2 `=²šÄœêTimes New Roman -• 2 `à 1¿;&lMathTypeUU` Ç(ÉnÉd Éi Ç)Éy Éi ÇmÇoÇdÉd Éi Ü=à1 & šº"Systemn-•"­¦ý°±·é"U U"Ýå "U" °dPPNTTimes New Roman,ŽTimes New RomanŽ .+()))mod°dPPNTTimes New Roman( PICT MéObjInfoTEquation Native U|SummaryInformation(W” n(d(y)"d°dPPNTTimes New Roman ( i(i)"i°dPPNTSymbol, Symbol (F=°dPPNTTimes New RomanŽ)1Ýç°dPPNT"System­¦ý°±·;œ ý°±·;œ ‹`ý¨J—ø Ç(ÉnÉd Éi Ç)Éy Éi ÇmÇoÇdÉd Éi Ü=à1­¦ýœ ÚüÖýO˜h´ë+'„0¥òÌ%9 ? K Wck sö ¢¨­¦ý°±·MMacintosh HD:applications:Microsoft Office:Microsoft Word 6:Templates:NormalECS253 Notes on April 9, 97 Matt Bishop Matt Bishop'@2.æúYº@Äv²ÚÃÁ@2.æúYº@Microsoft Word 6.01œ ЀhA¿ e pppppppÑÑÑÑÑÑ éÑ"Lû(™¸¸¸¸¸¸&nXE"p"ppûpp™ÑÑpppp™4ECS253 Notes on April 9, 97 Modular Arithmetic: solve ax mod n = 1 by the Extended Euclidean algorithm e.g. to solve: g=gcd(5,7) g x y q 7 0 1 5 1 0 1 2 -1 1 2 1 3 -2 2 0 gcd(5,7)=3*5+(-2)*7=1 solve x in: ax mod n = b if we can find x0 s.t. ax0 mod n = 1, then we have a(x0b mod n) = b thus x = x0b is the solution Chinese method: solve x in x mod di = xi ( i ([1,t] solution:  EMBED Equation.2  where  EMBED Equation.2  yi can be solved by the above methods Cipher: transposition cipher: diffusion effect (suggested by Shannon) can be attacked by frequency count (digram, trigram etc.) substitution cipher: confusion effect (suggested by Shannon) simple substitution: one mapping applied to all components e.g. Caesar cipher f(m)=(m+k) mod 26 where k is the key length is affine transformation: f(m)=(km + kí) mod n where gcd(k,n) =1 can be attacked by comparing the shift with frequency distribution of normal English text example to attack Caesar cipher is run through homophonic substitution (1 mapping for the components |HHŽ(··˜EG(¸HHŽ(d'`êÔ`öHW/ N†ÙPHoHo / N†Ùö jÇ PJ(OÔgHo?/ N†—ƒOÔ ``Ho?/ N†˜*OÔ `NJ*qgHHW/ N†ÙHoHo /Ùž›œ23ijnoqruvÜáöõúù§€½¼†ª‡æ›œù™–Ò–Ò–Ò–Ò–Ò–Ò–Í–“–¤–¦Ÿ¤–¤–†Ø¤–Ò–®–¶u J„cIuDÚ‹7ce’KIuDÚ‹7cvKIuDÛ‹7ceŽKIuDÛ‹7acvKIuDcI J‘cI J"cI chIcI UÅc I$1hÇåìú¶ƒ„ ”(E|§º’“Î)cxÝœœœðœœœœœœœð½ó½ð½½½œœð½½ð½ òœ 4h… 8òœ 4h…­ hòœ 4h…ݵ¤Ó5\›þææææùææþæõ òœ 4h… òœ 4h… 8òœ 4h… K @Ò Normal ]a c"A@Ú°"Default Paragraph FontùÝÜöú§½†:îïÑ:îïÑ4 Matt Bishop%Macintosh HD:Desktop Folder:0407n.doc@Ä >LêMTimes New Roman êSymbol MArial êMTimesêMPalatino"1Å­hg-&g-&É$EECS253 Notes on April 9, 97 Matt Bishop Matt Bishop­¦ý°±·;œ œ