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An introduction by Michael Nüsken

Universität Paderborn FB17 Mathematik & Informatik AG Algorithmische Mathematik. Cryptography. Mathematics for James Bond & Co. or. or. How to conceal Her Majesty‘s secret?. An introduction by Michael Nüsken.

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An introduction by Michael Nüsken

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  1. Universität Paderborn FB17 Mathematik & Informatik AG Algorithmische Mathematik Cryptography Mathematics for James Bond & Co. or or How to conceal Her Majesty‘s secret? An introduction by Michael Nüsken

  2. Universität Paderborn FB17 Mathematik & Informatik AG Algorithmische Mathematik Cryptography Mathematics for James Bond & Co. or or How to conceal Her Majesty‘s secret? An introduction by Michael Nüsken

  3. Universität Paderborn FB17 Mathematik & Informatik AG Algorithmische Mathematik Workshop Cryptography Mathematics for James Bond & Co. or or How to conceal Her Majesty‘s secret? An introduction by Michael Nüsken

  4. Universität Paderborn FB17 Mathematik & Informatik AG Algorithmische Mathematik Pupils' crypto 2001 Mathematics for James Bond & Co. or or How to conceal Her Majesty‘s secret? An introduction by Michael Nüsken

  5. Universität Paderborn FB17 Mathematik & Informatik AG Algorithmische Mathematik Pupils' crypto 2001 Mathematics for James Bond & Co. or or How to conceal Her Majesty‘s secret? An introduction by Michael Nüsken

  6. Secret talking KOKALOLLOLE KOKALOLLOLE KOKALOLLOLE KALLE

  7. Secret talking JOJAMOMESOS BOBONONDOD JOJAMOMESOS BOBONONDOD JAMES BOND

  8. Secret talking • Encrypt: difficult to learn • Decrypt: easy • Break: easy, even without knowledge

  9. 3 Caesar‘s Encryption ABCDEFGHIJKLMNOPQRSTUVWXYZ ABCDEFGHIJKLMNOPQRSTUVWXYZ D

  10. 3 Caesar‘s Encryption ABCDEFGHIJKLMNOPQRSTUVWXYZ D E

  11. 3 Caesar‘s Encryption ABCDEFGHIJKLMNOPQRSTUVWXYZ DE EFGHIJKLMNOPQRSTUVWXYZ

  12. Caesar‘s Encryption ABCDEFGHIJKLMNOPQRSTUVWXYZ D EFGHIJKLMNOPQRSTUVWXYZ ABC

  13. Caesar‘s Encryption ABCDEFGHIJKLMNOPQRSTUVWXYZ D EFGHIJKLMNOPQRSTUVWXYZABC Encrypt: JAMES

  14. Encrypt: Caesar‘s Encryption ABCDEFGHIJKLMNOPQRSTUVWXYZ D EFGHIJKLMNOPQRSTUVWXYZABC JAMES M

  15. Encrypt: Caesar‘s Encryption ABCDEFGHIJKLMNOPQRSTUVWXYZ D EFGHIJKLMNOPQRSTUVWXYZABC JAMES MD

  16. Encrypt: Caesar‘s Encryption ABCDEFGHIJKLMNOPQRSTUVWXYZ D EFGHIJKLMNOPQRSTUVWXYZABC JAMES MDP

  17. Encrypt: Caesar‘s Encryption ABCDEFGHIJKLMNOPQRSTUVWXYZ D EFGHIJKLMNOPQRSTUVWXYZABC JAMES MDPH

  18. Encrypt: Caesar‘s Encryption ABCDEFGHIJKLMNOPQRSTUVWXYZ D EFGHIJKLMNOPQRSTUVWXYZABC JAMES MDPHV

  19. Encrypt: Decrypt: Caesar‘s Encryption ABCDEFGHIJKLMNOPQRSTUVWXYZ D EFGHIJKLMNOPQRSTUVWXYZABC JAMES MDPHV VHFUHW SECRET

  20. ABCDEFGHIJKLMNOPQRSTUVWXYZ ABCDEFGHIJKLMNOPQRSTUVWXYZ Caesar‘s Encryption ABCDEFGHIJKLMNOPQRSTUVWXYZ D EFGHIJKLMNOPQRSTUVWXYZABC Ring

  21. Caesar‘s Basic Situation I want to write to Cleopatra ...

  22. Caesar‘s Basic Situation But Brutus must not known what ...

  23. Caesar‘s Basic Situation ?

  24. 3 Caesar‘s Basic Situation !

  25. 3 Caesar‘s Basic Situation Now I can write to Cleopatra and Brutus will not know what ... I hope Caesar‘s writing soon! Ha ha ha! If they knew how easy ...

  26. Giovanni Batista Porta (1563) De Furtivis Literarum Notis Enigma (invented 1918) Cipher machine of the German Forces Modern Times‘ Cryptography • Encryption methods are • refined, • mechanized and • remain symmetric. • Encryption methods are • refined, • mechanized • Encryption methods are • refined,

  27. + = Modern Times‘ Cryptography The secret keyword An example: words as keys TOHERMAJESTYTHEQUEEN The original message CROOK CROOK CROOK CROOK VFVSBORXSCVPHVOSLSSX

  28. Coat hanger of a Stasi spywith hidden One-Time-Pad (From: Spiegel Spezial 1/1990) One-Time-Pad • Random sequence instead of words as key. • Absolutely secure! • Provably. • Therein unique. • Problem: large key lengths.

  29. Modern Times‘ Situation Symmetric keys!

  30. Mathematics 1 Numbers instead of letters ABCDEFGHIJKLMNOPQRSTUVWXYZ Why letters? With numbers we could calculate ...

  31. Z -5 -1 0 1 5 10 15 20 25 Mathematics 1 Numbers instead of letters Numbers instead of letters ABCDEFGHIJKLMNOPQRSTUVWXYZ Advantage: Numbers can be • added 17 + 10 = 27 • multiplied 10 · 5 = 50 There you are ...

  32. Mathematics 1 Hum, we rolled up letters ...

  33. -5 -1 0 1 5 10 15 20 25 Z Mathematics 1 „Rolling up“: We begin with the number axis.

  34. Z -5 -1 0 1 5 10 15 20 25 26 Z Mathematics 1 „Rolling up“: 13 14 12 15 11 This is a ring! 10 16 9 17 18 8 7 19 6 20 5 -5 21 -4 4 22 3 -3 23 2 -2 1 -1 24 0 27 25 50 26 53 51 52

  35. 13 14 12 11 15 10 16 9 17 18 8 Z 7 19 26 6 20 -5 5 21 -4 4 22 -3 3 23 -2 2 -1 24 1 0 27 25 50 26 53 51 52 Mathematics 1 In the ring Z26 we can: • Add 17 + 10 = 1 • Add 17 + 10 = ? • Multiply 10 · 5 = 24 • Multiply 10 · 5 = ? ! Calculating in rings is easy! 50 = 24 27 = 1

  36. 13 14 12 11 15 10 16 9 17 18 8 Z 7 19 26 6 20 -5 5 21 -4 4 22 -3 3 23 -2 2 -1 24 1 0 27 25 50 26 53 51 52 Mathematics 1 In the ring Z26 we can: • Add 17 + 10 = 1 • Multiply 10 · 5 = 24 Mathematicians call this:Calculating „modulo 26“.

  37. Alan M. Turing (1912-1954) Turing Bombe Bletchley Park Manson NCR Bombe (Dayton,USA) Cryptography until 1950 All broken! Enigma

  38. Cryptography Today • Euro cheque cards, ATMs

  39. Cryptography Today • Euro cheque cards, ATMs • Interbank money transfer

  40. Cryptography Today • Euro cheque cards, ATMs • Interbank money transfer • Satellite communication, PayTV

  41. Cryptography Today • Euro cheque cards, ATMs • Interbank money transfer • Satellite communication, PayTV • Telephone, mobile phone

  42. Cryptography Today • Euro cheque cards, ATMs • Interbank money transfer • Satellite communication, PayTV • Telephone, mobile phone • Internet shopping

  43. ?? Cryptography Today • Euro cheque cards, ATMs • Interbank money transfer • Satellite communication, PayTV • Telephone, mobile phone • Internet shopping • and much more ...

  44. ?? Cryptography Today • Euro cheque cards, ATMs • Interbank money transfer • Satellite communication, PayTV • Telephone, mobile phone • Internet shopping • and much more ... A lot of the methods applied here use symmetric keys! Is this necessary?

  45. Let‘s reconsider the situation: Is Symmetry Inevitable? Where is the symmetry?

  46. Let‘s reconsider the situation: Is Symmetry Inevitable? Hey, the situation is indeed not symmetric!

  47. Is Symmetry Inevitable? Answer of modern cryptography: No, there is another way! This answers has already been given by • 1970-74: the British Secret Service CESG * • 1976: Diffie & Hellman RSA • 1978: Rivest, Shamir & Adleman:

  48. messages cipher texts Is Symmetry Inevitable? We can do without symmetry?! And how? Using a one-way function: Using a one-way function with trapdoor: Encryption is easy. Breaking is difficult. Decryptionis difficult!??? Decryptionis easy given the trapdoor!

  49. Z 35 17 18 16 19 15 20 14 21 13 22 ·4 6 5 1 times 2 3 4 12 23 11 24 25 10 9 26 8 27 7 28 6 29 5 30 4 31 3 32 2 33 1 34 0 RSA . . . ? Mathematics 2 Exponentiation is repeated multiplication: Oops, now it goes round!

  50. Z 35 17 18 16 19 15 20 14 21 13 22 ·4 6 times 12 23 11 24 ·5 4 5 3 1 times 2 6 25 10 9 26 8 27 7 28 6 29 5 30 4 31 3 32 2 33 1 34 0 Mathematics 2 Exponentiation is repeated multiplication: Repeated „·5“ also goes round!

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