Day 15, Mat194: Mathematical Sciences Seminar

Today:

Announcements:

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New Homework:

Wed

4/28

RSA Encoding

Next time: our poster session will be Wednesday, May 5th, from 1-3 (that's the closest I could find to our class in the exam schedule). Come prepared to show off your stuff! I'll probably have everyone go around the room, and say a little about their poster. Then we'll turn folks loose in waves (so a third, say, of the posters will be available for review with their presenters at any one time).

Upcoming talks and events:
- I've got none -- you?

Today:
- Last time: we had a statistician/marketing analyst in. Any comments I should pass along? If there's anything too personal to share with his brother, you can pass them along to the chair.
- One more (and the most important and interesting) code: the RSA encryption scheme
  - How does it work? Computers can't factor quickly yet.
  - 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!)
- Here's the recipe:
  1. Choose two "humongous primes", p and q. They might have 200 digits each.
  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. After that they're aliased.
- In pairs: 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 your partner, and see if they can decode it.
  - Use your calculators! I can help with any calculations that you need to do....

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Updated on 04/28/2010 15:21:00