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Mathematics of Internet Security Brian McEnnis Department of Mathematics

Mathematics of Internet Security Brian McEnnis Department of Mathematics. Godfrey Harold “G. H.” Hardy 1877-1947. From G. H. Hardy’s “A Mathematician’s Apology” (1940):

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Mathematics of Internet Security Brian McEnnis Department of Mathematics

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  1. Mathematics of Internet Security Brian McEnnis Department of Mathematics
  2. Godfrey Harold “G. H.” Hardy 1877-1947
  3. From G. H. Hardy’s “A Mathematician’s Apology” (1940): I have never done anything ‘useful’. No discovery of mine has made, or is likely to make, directly or indirectly, for good or ill, the least difference to the amenity of the world.
  4. I have never done anything ‘useful’. Hardy ignores his contribution to biology: Hardy-Weinberg equilibrium
  5. But here I must deal with a misconception. It is sometimes suggested that pure mathematicians glory in the uselessness of their work, and make it a boast that it has no practical applications. The imputation is usually based on an incautious saying attributed to Gauss, to the effect that, if mathematics is the queen of the sciences, then the theory of numbers is, because of its supreme uselessness, the queen of mathematics—I have never been able to find an exact quotation.
  6. Carl Friedrich Gauss 1777-1855
  7. Gauss (Wikiquote): Mathematics is the queen of sciences and number theory is the queen of mathematics.
  8. Number Theory • The Queen of Mathematics (Gauss) • Supremely useless (Hardy, 1940)
  9. Number Theory • The Queen of Mathematics (Gauss) • Supremely useless (Hardy, 1940) Number Theory (now) • The foundation of internet commerce
  10. Internet Commerce The internet is not a secure channel for communication. If you use the internet to send confidential information, such as a credit card number or a password, then it is possible that your information may be intercepted. To secure your information, it must be encrypted.
  11. Encryption • An encryption key is used to “scramble” the message • The resulting cipher text is sent over the internet Decryption • The intended recipient receives the cipher text • A decryption key is used to recover the original message
  12. Public Key Cryptography • Encryption key is public information • Decryption key is kept private Important Features: • One way function of encryption • “Trap door” for decryption
  13. RSA Algorithm (1977) Ron Rivest Adi Shamir Leonard Adleman
  14. American Standard Code for Information Interchange (ASCII) Examples: A typical message: 01000010011100100110100101100001011011100010000001101101 011000110100010101101110011011100110100101110011 Prior to encryption, the message is “padded” to more than 1000 binary digits
  15. RSA Encryption • The message M is repeatedly multiplied by itself (Amazon uses M65537) • The encrypted cipher text has approximately the same number of digits as M
  16. RSA Encryption • The message M is repeatedly multiplied by itself (Amazon uses M65537) • The results of all calculations are reduced in size so that the encrypted cipher text has approximately the same number of digits as M
  17. The size of the numbers in the calculations is reduced by using Mod Results of all calculations are required to be smaller than a specified number: The modulus
  18. Mod 60
  19. Mod 60
  20. Amazon’s modulus in binary form (1024 bits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n conjunction with the exponent of 65537, this is Amazon’s public key
  21. The Trapdoor…
  22. 10
  23. 10 2 x 5
  24. Prime numbers 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59…
  25. 15
  26. 15 3 x 5
  27. 644,710,213,393
  28. 644,710,213,393 747647 x 862319
  29. Mod 10 All numbers are less than 10 Single digit decimal numerals
  30. Calculations done mod 10 give single digit results The result is the last digit of the “regular” answer
  31. Encryption by squaring mod 10
  32. Encryption by squaring mod 10 If the cipher text is 4, what was the original message?
  33. Encryption by cubing mod 10
  34. Decryption by cube root mod 10
  35. Decryption by cube root mod 10
  36. Repeated multiplication mod 10
  37. Repeated multiplication mod 10
  38. Repeated multiplication mod 10
  39. Repeated multiplication mod 10 C3 = M
  40. Encryption and decryption by cubing mod 10
  41. Repeated multiplication mod 10
  42. Repeated multiplication mod 10 10 = 2 x 5
  43. Repeated multiplication mod 10 10 = 2 x 5
  44. Repeated multiplication mod 10 10 = 2 x 5 The units mod 10 are the numbers that are not multiples of 2 or 5: 1, 3, 7, and 9
  45. Cycles of Units mod 10 Multiplication by 3 Multiplication by 7 Multiplication by 9 Multiplication by 1
  46. Cycles mod 10 Multiplication by 7 Multiplication by 7
  47. Cycles mod 10 Multiplication by 7 Multiplication by 7 or Multiplication by 2
  48. Any message repeats in a cycle whose length is the number of units How many units are there for a given modulus?
  49. Leonhard Euler 1707-1783
  50. Any message repeats in a cycle whose length is the number of units How many units are there for a given modulus? (decrease each factor by 1) There are 4 units mod 10 (The totient of 10 is 4)
  51. James Joseph Sylvester 1814-1897
  52. What is the totient of 15?
  53. What is the totient of 15? (decrease each factor by 1) The totient of 15 is 8 There are 8 units mod 15: 1, 2, 4, 7, 8, 11, 13, 14
  54. What is the totient of 644,732,633,687?
  55. What is the totient of 644,710,213,393? (decrease each factor by 1)
  56. An Example
  57. An Example • Select two prime numbers: 2 and 11
  58. An Example • Select two prime numbers: 2 and 11 • Calculate the modulus and its totient: Modulus = 2 x 11 = 22 Totient = 1 x 10 = 10
  59. An Example • Select two prime numbers: 2 and 11 • Calculate the modulus and its totient: Modulus = 2 x 11 = 22 Totient = 1 x 10 = 10 • Select a unit mod 10: 3
  60. An Example • Select two prime numbers: 2 and 11 • Calculate the modulus and its totient: Modulus = 2 x 11 = 22 Totient = 1 x 10 = 10 • Select a unit mod 10: 3 • The public encryption key is C = M3 mod 22
  61. An Example • Find the inverse of 3 mod 10 (3 x ? = 1 mod 10)
  62. An Example • Find the inverse of 3 mod 10 (3 x ? = 1 mod 10) • Using the extended Euclidean algorithm, the inverse of 3 is found to be 7 (3 x 7 = 1 mod 10)
  63. An Example • Find the inverse of 3 mod 10 (3 x ? = 1 mod 10) • Using the extended Euclidean algorithm, the inverse of 3 is found to be 7 (3 x 7 = 1 mod 10) • The private decryption key is M = C7 mod 22
  64. Suppose the message is 9: M = 9 Then the cipher text is 93 = 3 mod 22 The cipher text 3 is sent over the internet
  65. The cipher text received is 3: C = 3 The message is decrypted: 37 = 9 mod 22 The original message was 9
  66. Digital Signatures RSA can be employed in reverse: • Send an (unencrypted) message to Amazon • Amazon encrypts it with their private key and returns it • You decrypt the cipher text with Amazon’s public key • If it matches your original message, then you know you are dealing with Amazon (and not a fake site)
  67. Certificates • A trusted certificate authority (e.g., VeriSign) produces a certificate with information about the company (e.g., Amazon): its name, address, web sites, etc. and the company’s public key • VeriSign encrypts it with their private key and gives the encrypted certificate to Amazon. • Amazon sends the certificate when you log into their secure website • You decrypt the certificate with VeriSign’s public key and verify that the information matches
  68. Repeated Multiplication The Amazon public key requires the calculation ofM65537, with the given modulus. M65537 can be calculated with only 17 multiplications. How is that possible?
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