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Cryptography

Learn the fundamentals of cryptography, including symmetric and public-key systems. Explore simple algorithms like transposition, shift cipher, and Vigenere cipher. Discover the principles behind public-key encryption using RSA.

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Cryptography

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  1. Cryptography Taehyung Kim HPC Lab. POSTECH

  2. Goals • To know • What is the cryptography? • What is a key in cryptography system? • What is a symmetric key system? • What is a public key system? • What is the base of the Public-key cryptography? • To exercise simple cryptography algorithms

  3. Cryptography • Definition • The art of secret writing • Crypto(hidden) + graphy(writing) • Objective • To enable two people to communicate over an insecure channel in such a way that opponent cannot understand what is being said

  4. Cryptography • The mostly used tool for securing information and services • Relies on ciphers: mathematical functions used for encryption and decryption of a message • Encryption – the process of disguising a message in such a way as to hide its substance • Ciphertext – an encrypted message • Decryption – the process of returning an ecrypted message back into plaintext

  5. Ciphers • For some message M • Let’s denote • the encryption of that message into cipher text as Ek(M) = C • the decryption into plain text as Dk(C) = M • Notice, • Symmetric-key algorithm Dk(Ek(M)) = M • Public-key algorithm Dk1(Ek2(M))= M

  6. Cryptosystem • Algorithm + Key • The reason for having a key to an algorithm • Difficult to devising new algorithms • Difficult to quickly explain a newly devised algorithm to the person with whom you’d like to start communication securely • A good cryptosystem • Perfectly OK to have everyone know the algorithm • Knowledge of the algorithm without the key does not help unmangle the information

  7. Classical Cryptography • Transposition (permutation) cipher • Simple transposition • Substitution cipher • shift cipher • Polyalphabetic cipher

  8. Simple Transposition • Grouping the plaintext into blocks of t characters • Applying to each block a single permutation  on the numbers 1 through t • Encryption • C = E(M) = m(1) …m(t), (M=m1…mt) • Decryption • M = D(C) = E-1(C) = c-1(1) …c-1(t), (C = c1…ct)

  9. Simple Transposition Example • T = 6,  = (6 4 1 3 5 2) • M = CAESAR • m1 = C, m2 = A, m3 = E, m4 = S, m5 = A, m6 = R • C = m6m4m1m3m5m2 = RSCEAA • Exercise • C = OPCYTRYPGAHR , M = ? -1 = (3 6 4 2 5 1) M = D(C) = D({OPCYTR})D({YPGAHR}) = CRYPTOGRAPHY

  10. Shift Cipher • A kind of simple substitution ciphers • Example • M = CAESAR • m1 = C, m2 = A, m3 = E, m4 = S, m5 = A, m6 = R • Ci = (mi + 3) % 26 • C = FDHVDU • General shift cipher • Ci = (a·mi + k) % 26

  11. Polyalphabetic (Vigenere) cipher • Grouping the plaintext into blocks of n characters -> block size = 5 = n • Let E1,E2,…,En be distinct substitution ciphers. • To encrypt a plaintext message M, apply the Ei( i=1,2,…,n) cyclically to the message characters • Encryption • C = E(M) = E1(m1)E2(m2)…En(mn)E1(mn+1)…. (M=m1…mt) • Decryption • M = D(C) = E1-1(c1)E2-1(c2)…En-1(cn)E1-1(cn+1)…(C = c1…ct)

  12. Polyalphabetic Cipher Example • Block size = 3 • E1, E2, E3 are shift ciphers with Key = { 2, 16, 5 } • M = CRYPTOGRAPHY • C = E1(C)E2(R)E3(Y)E1(P)E2(T)E3(O) E1(G)E2(R)E3(A)E1(P)E2(H)E3(Y) • =EHDRJTIHFRXD

  13. Cryptanalysis of a substitution cipher • Letter-frequency in English can be broken into 5 groups: • e (0.127 most common occurrence); • t, (0.091), a (0.082), o (0.075), i (0.070), n (0.067), s (0.063), h (0.061), r (0.060); • d (0.043), l (0.040) ; • c (0.028), u (0.028), m (0.024), w (0.023), f (0.022), g (0.020), y (0.020), p (0.019), b (0.015); • v (0.010), k (0.008), j (0.002), x (0.001), q (0.001), z (0.001 least common)

  14. Exercise • Decipher a given ciphertext • Key = { 5, 17, 4, 20 } • C = IVGCUYILFXMPJEGCUYILYVBN

  15. Exercise • Decipher a given ciphertext. ( you first find the block size. Look at the letters applied same substitution cipher, and see which letter occurs most frequently And using letter frequency in English, decipher. ) CHREEVOAHMAERATBIAXXWTNXBEEOPHBSBQMQEQERBWRVXUOAKXAOSXXWEAHBWGJMMQMNKGRFVGXWTRZXWIAKLXFPSKAUTEMNDCMGTSXMXBTUIADNGMGPSRELXNJELXVRVPRTULHDNQWTWDTYGBPHXTFALJHASVBFXNGLLCHRZBWELEKMSJIKNBHWRJGNMGJSGLXFEYPHAGNRBIEQJTAMRVLCRREMNDGLXRRIMGNSNRWCHRQHAEYEVTAQEBBIPEEWEVKAKOEWADREMXMTBHHCHRTKDNVRZCHRCLQOHPWQAIIWXNRMGWOIIFKEE

  16. Public Key Cryptography • Problem with symmetric key algorithms • A secure method of telling your partner the key • A separate key for everyone you might communicate with • Public-Key algorithms use a public-key and private-key pair over a message

  17. Public Key Cryptography Encryption

  18. Public Key Cryptography Authentication

  19. RSA Public-key encryption • Proposed by Rivest, Shamir, and Adleman in 1977 • Mostly widely used public-key cryptosystem • The security is based on the intractability of the integer factorization problem

  20. RSA Public Key Cryptography Key Generation • Generate two large random primes p and q. • Compute n = pq and  = (p-1)(q-1). • Select a random integer e, 1 < e < , such that gcd(e, ) = 1. • Compute the unique integer d, 1 < d < , such that ed  1 (mod ). Public key : (n, e) Private key : d

  21. RSA Public Key Cryptography Encryption • Bob encrypts a message m for Alice • Obtain Alice’s public key (n, e) • Represent the message as an integer m in the interval [0, n-1] • Compute C Me (mod n) • Alice recovers plaintext m from c as follows: • M  Cd (mod n)

  22. RSA Public Key Cryptography Theorem: C  Md (mod n) • Proof C d  (Me)d (mod n)  Med (mod n)  Mk+1 (mod n)  MMk (mod n)  MMk(p-1)(q-1) (mod n)  M (mod n)

  23. RSA Public Key Cryptography Example • p = 2357, q = 2251 • n = pq = 6012707,  = (p-1)(q-1) = 6007800 • Choose e = 3674911, then d = 422191 • Public key (n = 6012707, e = 3674911) • Private key d = 422191 • Plaintext m = 5234673 C = me mod n = 52346733674911 mod 6012707 = 3650502

  24. RSA Public Key Cryptography Exercise • Public key (119, 5), private key d = 77 • Plaintext M = 19 • Ciphertext C = ? • What about manual decryption ?

  25. RSA Public Key Cryptography Exercise • Find five prime integer more than 1000 • Make a public key and a private key • Two primes, p and q, must be less than 1000 • Select a plaintext and cipher it using your public key • Decipher the other group’s ciphertext

  26. Cryptography

  27. How to determine a block size • Kasiski's method: 1. Find all repeated strings in ciphertext of length at least 3 2. Compute distances between them 3. Block size will be a common divisor of those numbers

  28. Letter-frequency • e (0.127 most common occurrence); • t, (0.091), a (0.082), o (0.075), i (0.070), n (0.067), s (0.063), h (0.061), r (0.060); • d (0.043), l (0.040) ; • c (0.028), u (0.028), m (0.024), w (0.023), f (0.022), g (0.020), y (0.020), p (0.019), b (0.015); • v (0.010), k (0.008), j (0.002), x (0.001), q (0.001), z (0.001 least common)

  29. Answer • A:7 B:6 C:6 D:4 E:1 F:2 G:0 H:0 I:1 J:2 K:1 L:0 M:2 • N:4 O:0 P:1 Q:4 R:3 S:0 T:2 U:1 V:1 W:9 X:5 Y:0 Z:0 • A:3 B:1 C:0 D:3 E:9 F:2 G:3 H:5 I:3 J:0 K:0 L:0 M:2 • N:6 O:6 P:1 Q:0 R:3 S:7 T:5 U:1 V:1 W:1 X:0 Y:0 Z:0 • A:5 B:2 C:1 D:0 E:2 F:0 G:4 H:3 I:3 J:3 K:0 L:2 M:0 • N:5 O:1 P:2 Q:4 R:13 S:1 T:2 U:2 V:4 W:0 X:1 Y:2 Z:0 • A:1 B:1 C:1 D:0 E:10 F:0 G:1 H:3 I:4 J:2 K:4 L:4 M:8 • N:0 O:0 P:3 Q:2 R:3 S:1 T:1 U:0 V:3 W:3 X:4 Y:1 Z:2 • A:3 B:5 C:0 D:0 E:2 F:2 G:7 H:6 I:0 J:0 K:4 L:6 M:5 • N:0 O:0 P:1 Q:0 R:2 S:0 T:4 U:0 V:1 W:3 X:10 Y:0 Z:1 • THEALMONDTREEWASINTENTATIVEBLOSSOMTHEDAYSWERELONGEROFTENENDINGWITHMAGNIFICENTEVENINGSOFCORRUGATEDPINKSKIESTHEHUNTINGSEASONWASOVERWITHHOUNDSANDGUNSPUTAWAYFORSIXMONTHSTHEVINEYARDSWEREBUSYAGAINASTHEWELLORGANIZEDFARMERSTREATEDTHEIRVINESANDTHEMORELACKADAISICALNEIGHBORSHURRIEDTODOTHEPRUNINGTHEYSHOULDHAVEDONEINNOVEMCALORBUSINESSRIVALSE

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