Day 13, Mat194: Mathematical Sciences Seminar

• Friday seminar: Chris Christensen on "Lester Hill Revisted." 3:00 in ST 111. This concerns cryptography, which is appropriate given our topic today!

Here's a summary of our discussion:

For talks: what makes a good talk?

1. Presenter Preparedness
2. Presenter Confidence
3. Audience Awareness -- did the presenter pitch it at the right level?
4. Presenter Knowledge of Material
5. Avoid "ummms" and other verbal tics
6. If a group is presenting, each person in the group should have a role.
7. If you're involved, make it look like you're involved.
8. Look enthusiastic (never cry!)
9. If there's a Q&A, be prepared (consider planting people to move things in the right direction).
10. Enunciate.
11. Care about your topic (or at least appear to!).
12. If you include material, talk about it.
13. No excess fluff.

1. Don't "over-text": too many words is a turn-off.
2. Be ready to explain everything you put up on your poster.
3. Make sure that it's visually appealing (but don't overdo -- dozens of fonts, for example)
4. Be sensitive to color-confused; consider using line-type or -width to help those who aren't color-enabled.

You should have enough to talk about to entertain -- and I emphasize entertain -- a listener for five minutes.

• Julius Caesar had a scheme....

  It can be broken, however, using trial and error:

  USWKSJ KAEHDQ JWHDSUWV WSUZ DWLLWJ AF S EWKKSYW OALZ LZW DWLLWJ LZSL AK LZJWW
  HDSUWK XMJLZWJ VGOF LZW SDHZSTWL.
  

  (More info, and an applet.)

• More complicated schemes exist, of course: for example, we could simply do a substitution of each letter by a different letter: This method, called a Monoalphabetic Substitution Cipher, can be broken by frequency analysis and knowledge of our language structure (e.g. frequencies of first letters in words):

  

  JI KU FUFRY KHFUP BDTCA BOFBS DUF DI POFXF ZDXXJECF SFYX FVFWY XFBDUA, JP LDTCA
  PKSF WDTHOCY DUF EJCCJDU PJRFX POF CJIFPJRF DI POF TUJVFWXF PD BOFBS KCC DI
  POFR KUA IJUA POF BDWWFBP DUF.POJX XJRZCF EWTPF IDWBF KZZWDKBO BCFKWCY LJCC UDP
  LDWS.
  

  Each student will be assigned a letter or two to count, and we'll build a frequency table, and attempt to decode this message.

  We can do a little bit better by jamming the letters together, so that you can't use the structure of the language, and punctuation....

  This was the code referred to by our author, used by Mary Queen of Scots, who wrote:

  To Anthony Babington, I agree that you can murder Queen Elizabeth at the earliest time. From Mary.

  Off with her head!

• The most interesting code, however, is the RSA encryption scheme:
  • So how does it work?
    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 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!)