Day 08: Mat115

Today:

• Hand in problems 2.3

• Questions/Comments on old stuff?

• 2.2 problems returned:
  • 12 people didn't turn homework in! This is the last time I'm going to emphasize it: you won't survive this class if you don't turn in the homework. It's 20% of your grade.
  • II.18: 50 = 34 + 13 + 3. Take 3 sticks.
  • III.28
    • Each pair reproduces twice.
    • I assumed that new pairs were counted three times before dying.

• Section 2.5:
  • Cryptography -- the study of secret codes
    • UPCs are in a sense "secret codes" -- it's just that we now know how to break them.
    • "3 1 3 1 3 1 3 1 3 1 3" -- interesting that 3: what do you notice about the digits 0 through 9 and their values*3 mod 10? Why does this make sense?

      digit	0	1	2	3	4	5	6	7	8	9
      digit*3 mod 10

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

  • RSA: (developed by Rivest, Shamir, and Adleman in 1977)
    • Is an example of a Public key code: these allow us to encode any message, but prevent us from decoding any message

    • RSA depends on the following fact:

      Computers can't quickly and efficiently factor humongous numbers.

    • So how does it work?
      1. Choose two "humongous primes", p and q. They might have 200 digits each, so they're even bigger than that number 25²⁴ that we looked at the other day, when we were talking about UPCs.

      2. Compute n=pq - public product of p and q

      3. Compute m=(p-1)(q-1)

      4. Pick a number e that is relatively prime to m (that means they share no common prime factor), and make that public, too.

      5. So we shout to the whole world "e, n!"

      6. To encode: message^e = encoded mod n

      7. To decode: encoded^d = message mod n
         • But how do we find d? It's top secret, of course, because it allows us to decode any message.
         • Find d and y such that ed-my=1: d will decode the message (shhhh! tell no one....)
         • Why can't others find d? They don't know m!

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

    • Examples: #4, p. 107

    • Examples: #13-15, p. 107

    • In groups: sending and receiving messages
      • Use p=7, q=11, e=13, d=37
      • Use the numbers 1-26 for the alphabet, a-z.
      • Encode your messages letter-by-letter.
      • Send a secret message -- a word of length at least 8 -- to another group, and see if they can decode it.
      • Use your calculators! I can help with any calculations that you need to do....

  • 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!)