1Historical CiphersECE 646 - Lecture 6Why (not) to study historical ciphers?AGAINSTFORNot similar to modern ciphersLong abandonedBasic components becamea part of modern ciphersUnder special circumstancesmodern ciphers reduceto historical ciphersInfluence on world eventsThe only ciphers you can break!2Secret WritingSteganography(hidden messages)Cryptography(encrypted messages)SubstitutionTransformationsTranspositionCiphers(change the orderof letters)Codes SubstitutionCiphers(replacewords)(replaceletters)Selected world events affected by cryptology1586 - trial of Mary Queen of Scots - substitution cipher1917 - Zimmermann telegram, America enters World War I1939-1945 Battle of England, Battle of Atlantic, D-day -ENIGMA machine cipher1944 – world’s first computer, Colossus -German Lorentz machine cipher1950s – operation Venona – breaking ciphers of soviet spiesstealing secrets of the U.S. atomic bomb – one-time pad3Ciphers used predominantly in the given period(1)Electromechanical machine ciphers(Complex polyalphabetic substitution ciphers)1919Vigenère cipher(Simple polyalphabetic substitution ciphers)CryptographyCryptanalysis1586 Invention of the Vigenère CipherMonoalphabetic substitution cipherHomophonic ciphersInvention of rotor machinesXVIII c.Black chambers1863Kasiski’s method1918Index of coincidenceWilliam FriedmanShift ciphers100 B.C.IX c.Frequency analysisal-Kindi, Baghdad1926 Vernam cipher (one-time pad)Ciphers used predominantly in the given period(2)CryptographyCryptanalysisDES19772001AESTriple DES193219772001Rejewski, PolandReconstructing ENIGMA19391949Shennon’s theoryof secret systemsPolish cryptological bombs,and perforated sheetsPublication of DES1945British cryptologicalbombs, Bletchley Park, UKBreaking Japanese “Purple” cipher1990DES crackersone-time padStream CiphersS-P networks4Substitution Ciphers (1)1. Monalphabetic (simple) substitution cipherM = m1m2m3 m4 . . . . mNC = f(m1) f(m2) f(m3) f(m4) . . . . f(mN)Generally f is a random permutation, e.g.,f = a b c d e f g h i j k l m n o p q r s t u v w x y zs l t a v m c e r u b q p d f k h w y g x z j n i oKey = fNumber of keys = 26! ≈ 4 ⋅ 1026Monalphabetic substitution ciphersSimplifications (1)A. Caesar Cipherci= f(mi) = mi+ 3 mod 26No keyB. Shift Cipherci= f(mi) = mi+ k mod 26Key = kNumber of keys = 26mi= f-1(ci) = ci- 3 mod 26mi= f-1(ci) = ci- k mod 265Coding characters into numbersA ⇔ 0B ⇔ 1C ⇔ 2D ⇔ 3E ⇔ 4F ⇔ 5G ⇔ 6H ⇔ 7I ⇔ 8J ⇔ 9K ⇔ 10L ⇔ 11M ⇔ 12N ⇔ 13O ⇔ 14P ⇔ 15Q ⇔ 16R ⇔ 17S ⇔ 18T ⇔ 19U ⇔ 20V ⇔ 21W ⇔ 22X ⇔ 23Y ⇔ 24Z ⇔ 25Caesar Cipher: ExamplePlaintext:Ciphertext:I C A M E I S A W I C O N Q U E R E D8 2 0 12 4 8 18 0 22 8 2 14 13 16 20 4 17 4 311 5 3 15 7 11 21 3 25 11 5 17 16 19 23 7 20 7 6L F D P H L V D Z L F R Q T X H U H G6Monalphabetic substitution ciphersSimplifications (2)C. Affine Cipherci= f(mi) = k1⋅ mi+ k2mod 26Key = (k1, k2)Number of keys = 12⋅26 = 312gcd (k1, 26) = 1mi= f-1(ci) = k1-1⋅ (ci- k2) mod 26Most frequent single lettersAverage frequency in a long English text:E — 13%T, N, R, I, O, A, S — 6%-9%D, H, L — 3.5%-4.5%C, F, P, U, M, Y, G, W, V — 1.5%-3%B, X, K, Q, J, Z — < 1%= 0.038 = 3.8%Average frequency in a random string of letters:1267Digrams:TH, HE, IN, ER, RE, AN, ON, EN, ATTrigrams:THE, ING, AND, HER, ERE, ENT, THA, NTH, WAS, ETH, FOR, DTHMost frequent digrams, and trigrams02468101214A B C D E F G H I J K L M N O P Q R S T U V W X Y ZRelative frequency of letters in a long English textby Stallings7.251.253.54.2512.75323.57.750.250.53.752.757.757.52.750.58.569.2531.51.50.52.250.25802468101214a b c d e f g h i j k l m n o p q r s t u v w x y z02468101214a b c d e f g h i j k l m n o p q r s t u v w x y zCharacter frequencyin a long English plaintextCharacter frequencyin the corresponding ciphertextfor a shift cipher02468101214a b c d e f g h i j k l m n o p q r s t u v w x y zCharacter frequencyin a long English plaintextCharacter frequencyin the corresponding ciphertextfor a general monoalphabeticsubstitution cipher02468101214a b c d e f g h i j k l m n o p q r s t u v w x y z902468101214a b c d e f g h i jk lm n o p q r s tu vw x y z02468101214a b c d e f g h i jk lm n o p q r s tu vw x y z02468101214a b c d e fg h I j k lm n o p q rs tu vw x y z02468101214a b c d e f g h I j k lm n o p q rs t u v w x y zLong English text TCiphertext of the long English text TShort English message MCiphertext of the short English message MFrequency analysis attack: relevant frequenciesCiphertext:FMXVE DKAPH FERBN DKRXR SREFM ORUDSDKDVS HVUFE DKAPR KDLYE VLRHH RHA B C D E F G H I J K L M N O P Q R S T U V W X Y ZR - 8D - 7E, H, K - 5Frequency analysis attack (1)Step 1: Establishing the relative frequency of letters in the ciphertext10f(E) = Rf(T) = Df(4) = 17f(19) = 3Frequency analysis attack (2)Step 2: Assuming the relative frequency of letters in the corresponding message, and derivingthe corresponding equationsAssumption: Most frequent letters in the message: E and TCorresponding equations:E → RT → D4 → 1719 → 3f(4) = 17f(19) = 3Frequency analysis attack (3)Step 3: Verifying the assumption for the case of affine cipher4⋅k1+ k2≡ 17 (mod 26)19⋅k1+ k2≡ 3 (mod 26)15⋅k1≡ -14 (mod 26)15⋅k1≡ 12 (mod 26)11Substitution Ciphers (2)2. Polyalphabetic substitution cipherM = m1m2… mdKey = d, f1, f2, …, fdNumber of keys for a given period d = (26!)d≈ (4 ⋅ 1026)dmd+1md+2… m2dm2d+1m2d+2… m3d…..C = f1(m1) f2(m2) … fd(md)f1(md+1) f2(md+2) … fd(m2d )f1(m2d+1) f2( m2d+2) … fd(m3d )…..d is a period of the cipher02468101214a b c d e f g h i j k l m n o p q r s t u v w x y zCharacter frequencyin a long English plaintextCharacter frequencyin the corresponding ciphertextfor a polyalphabeticsubstitution cipher02468101214a b c d e f g h i j k l m n o p q r s t u v w x y z126⋅⋅⋅⋅ 100% ≈≈≈≈ 3.8 %12Polyalphabetic substitution ciphersSimplifications (1)A. Vigenère cipher: polyalphabetic shift cipherInvented in 1568ci= fi mod d(mi) = mi + ki mod dmod 26Key = k0, k1, … , kd-1mi= f-1i mod d(mi) = mi - ki mod d mod 26Number of keys for a given period d = (26)dVigenère Squarea b c d e f
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