One-Time Pad Cryptography: Principles, Attacks, and Classical Cryptography - Prof. David M, Study notes of Electrical and Electronics Engineering

Download One-Time Pad Cryptography: Principles, Attacks, and Classical Cryptography - Prof. David M and more Study notes Electrical and Electronics Engineering in PDF only on Docsity! 1 Classical Cryptography CS461/ECE422 2 Reading • Chapter 2 from Security in Computing • Chapter 9 from Computer Security: Art and Science • Handbook of Applied Cryptography. http://www.cacr.math.uwaterloo.ca/hac/ 5 Example • Example: Cæsar cipher (The most basic cipher) – M = { sequences of letters } – K = { i | i is an integer and 0 ≤ i ≤ 25 } – E = { E | k ∈ K and for all letters m, E(m, k) = (m + k) mod 26 } – D = { D | k ∈ K and for all letters c, D(c,k) = (26 + c – k) mod 26 } – C = M 6 Attacks • Opponent whose goal is to break cryptosystem is the adversary – Standard cryptographic practice: Assume adversary knows algorithm used, but not the key • Three types of attacks: – ciphertext only: adversary has only ciphertext; goal is to find plaintext, possibly key – known plaintext: adversary has ciphertext, corresponding plaintext; goal is to find key – chosen plaintext: adversary may supply plaintexts and obtain corresponding ciphertext; goal is to find key 7 Basis for Attacks • Mathematical attacks – Based on analysis of underlying mathematics • Statistical attacks – Make assumptions about the distribution of letters, pairs of letters (diagrams), triplets of letters (trigrams), etc. • Called models of the language • E.g. Caesar Cipher, letter E – Examine ciphertext, correlate properties with the assumptions. 10 Transposition Cipher • Generalize to n-columnar transpositions • Example 3-columnar – HEL LOW ORL DXX – HLODEORXLWLX 11 Attacking the Cipher • Anagramming – If 1-gram frequencies match English frequencies, but other n-gram frequencies do not, probably transposition – Rearrange letters to form n-grams with highest frequencies 12 Example • Ciphertext: HLOOLELWRD • Frequencies of 2-grams beginning with H – HE 0.0305 – HO 0.0043 – HL, HW, HR, HD < 0.0010 • Frequencies of 2-grams ending in H – WH 0.0026 – EH, LH, OH, RH, DH ≤ 0.0002 • Implies E follows H 15 Attacking the Cipher • Exhaustive search – If the key space is small enough, try all possible keys until you find the right one – Cæsar cipher has 26 possible keys • Statistical analysis – Compare to 1-gram model of English – CryptoQuote techniques 16 Statistical Attack • Compute frequency of each letter in ciphertext: G 0.1H 0.1K 0.1O 0.3 R 0.2U 0.1Z 0.1 • Apply 1-gram model of English – Frequency of characters (1-grams) in English is on next slide – http://math.ucsd.edu/~crypto/java/EARLYCIP HERS/Vigenere.html 17 Character Frequencies 0.002z0.015g 0.020y0.060s0.030m0.020f 0.005x0.065r0.035l0.130e 0.015w0.002q0.005k0.040d 0.010v0.020p0.005j0.030c 0.030u0.080o0.065i0.015b 0.090t0.070n0.060h0.080a 20 The Result • Most probable keys, based on ϕ: – i = 6, ϕ(i) = 0.0660 • plaintext EBIIL TLOLA – i = 10, ϕ(i) = 0.0635 • plaintext AXEEH PHKEW – i = 3, ϕ(i) = 0.0575 • plaintext HELLO WORLD – i = 14, ϕ(i) = 0.0535 • plaintext WTAAD LDGAS • Only English phrase is for i = 3 – That’s the key (3 or ‘D’) 21 Cæsar’s Problem • Key is too short – Can be found by exhaustive search – Statistical frequencies not concealed well • They look too much like regular English letters • Improve the substitution permutation – Increase number of mapping options from 26 22 Key the Mapping • Caesar mapping (shift 3) – ABCEDFGHIJKLMNOPQRSTUVWXYZ – XYZABCEDFGHIJKLMNOPQRSTUVW • Key mapping – ABCEDFGHIJKLMNOPQRSTUVWXYZ – SECURABDFGHIJKLMNOPQTVWXYZ • Poor mapping at the end • Still only one mapping of a character across whole message – Just a crypto quote 25 Relevant Parts of Tableau G I V A G I V B H J W E L M Z H N P C L R T G O U W J S Y A N T Z B O Y E H T • Tableau shown has relevant rows, columns only • Example encipherments(?): – key V, letter T: follow V column down to T row (giving “O”) – Key I, letter H: follow I column down to H row (giving “P”) 26 Useful Terms • period: length of key – In earlier example, period is 3 • tableau: table used to encipher and decipher – Vigènere cipher has key letters on top, plaintext letters on the left • polyalphabetic: the key has several different letters – Cæsar cipher is monoalphabetic 27 Attacking the Cipher • Approach – Establish period; call it n – Break message into n parts, each part being enciphered using the same key letter – Solve each part • We will show each step • Automated in applet – http://math.ucsd.edu/~crypto/java/EARLYCIP HERS/Vigenere.html 30 Repetitions in Example 2, 36124118CH 339794SV 2, 368377NE 2, 2, 2, 2, 34811769PC 7, 74910556QO 2, 2, 2, 3, 37212250MOC 2, 2, 11448743AA 2, 2, 2, 3246339FV 2, 3, 5305424OEQOOG 552722OO 2, 510155MI FactorsDistanceEndStartLetters 31 Estimate of Period • OEQOOG is probably not a coincidence – It’s too long for that – Periomay be 1, 2, 3, 5, 6, 10, 15, or 30 – Most others (7/10) have 2 in their factors • Almost as many (6/10) have 3 in their factors • Begin with period of 2 x 3 = 6 32 Check on Period • Index of coincidence is probability that two randomly chosen letters from ciphertext will be the same • Tabulated for different periods: 1 0.066 3 0.047 5 0.044 2 0.052 4 0.045 10 0.041 Large 0.038 35 Frequency Examination ABCDEFGHIJKLMNOPQRSTUVWXYZ 1 31004011301001300112000000 2 10022210013010000010404000 3 12000000201140004013021000 4 21102201000010431000000211 5 10500021200000500030020000 1 01110022311012100000030101 Letter frequencies are (H high, M medium, L low): HMMMHMMHHMMMMHHMLHHHMLLLLL 36 Begin Decryption • First matches characteristics of unshifted alphabet • Third matches if I shifted to A • Sixth matches if V shifted to A • Substitute into ciphertext (bold are substitutions) ADIYS RIUKB OCKKL MIGHK AZOTO EIOOL IFTAG PAUEF VATAS CIITW EOCNO EIOOL BMTFV EGGOP CNEKI HSSEW NECSE DDAAA RWCXS ANSNP HHEUL QONOF EEGOS WLPCM AJEOC MIUAX 37 Look For Clues • AJE in last line suggests “are”, meaning second alphabet maps A into S: ALIYS RICKB OCKSL MIGHS AZOTO MIOOL INTAG PACEF VATIS CIITE EOCNO MIOOL BUTFV EGOOP CNESI HSSEE NECSE LDAAA RECXS ANANP HHECL QONON EEGOS ELPCM AREOC MICAX 40 One-Time Pad • A Vigenère cipher with a uniformly random key at least as long as the message – Provably unbreakable – Why? Look at ciphertext DXQR. Equally likely to correspond to plaintext DOIT (key AJIY) and to plaintext DONT (key AJDY) and any other 4 letters – Warning: keys must be random, or you can attack the cipher by trying to regenerate the key • Approximations, such as using pseudorandom number generators to generate keys, are not random 41 Book Cipher • Approximate one-time pad with book text – Sender and receiver agree on text to pull key from – Bible, Koran, Phone Book • Problem is that book text is not random – Combine English with English – Can still perform language based statistical analysis 42 Enigma - Rotor Machines • Another approximation of one-time pad • Substitution cipher – Each rotor is a substitution – Changes in rotor position change how substitutions are stacked 45 Rotor Mappings • Rotor III – ABCDEF G HIJKLMNOPQRSTUVWXYZ BDFHJL C PRTXVZNYEIWGAKMUSQO • Rotor II – AB C DEFGHIJKLMNOPQRSTUVWXYZ AJ D KSIRUXBLHWTMCQGZNPYFVOE • Rotor II – ABC D EFGHIJKLMNOPQRSTUVWXYZ EKM F LGDQVZNTOWYHXUSPAIBRCJ • Reflector B – ABCDE F GHIJKLMNOPQRSTUVWXYZ YRUHQ S LDPXNGOKMIEBFZCWVJAT 46 Lessons from Enigma • The importance of known plaintext (cribs) • Mechanical assisted key breaking – Leading to modern computers • Information in the pattern of traffic – Traffic analysis • Humans in the loop are important – Information from spies – Poor user procedures • Birthday messages – many cribs • Repeated patterns – Reluctance to believe cipher has been broken