Day 08: Mat115

Today:

• Return Homework 2.1
  • #6: 2000 pounds or so....
  • #8: 18 446 744 073 709 551 615

• Collect 2.2 homework

• Collect Project titles and partner info

• Questions/Comments on old stuff?

• Section 2.5:
  • Cryptography -- the study of secret codes
    • One for the kids....

  • How would you encode a message?

  • The Carson City Kid

  • Clock arithmetic and big primes are at the heart of modern secret codes

  • Public key codes:
    Allow us to encode any message, but prevent us from decoding any message

  • RSA:
    • n=pq - public product of two huge primes, perhaps 200 digits each.

    • m=(p-1)(q-1) - pick a number e that is relatively prime to m, and make that public.

    • Find d and y such that ed-my=1: d will decode the message (shhhh! tell no one....)

    • To encode: message^e = encoded mod n

    • To decode: encoded^d = message mod n

    • Note: we can only encode numbers from 0 to n in size.

    • Examples: #3, 4, p. 107

    • Examples: #13-15, p. 107

  • We're going to have to raise numbers to enormous powers -- how on earth will we be able to do the calculations?

    Fermat's little theorem:

    If p is a prime number and n is any integer that does not have p as a factor, then n^p-1 is equal to 1 mod p.

    In other words, n^p-1 will always have a remainder of 1 when divided by p.

    What is 23²⁶³⁹ mod 13?

    (Use Fermat's little theorem!)