Due on Wednesday September 6th:

  1. Check out this tool for drawing pictures of the function $f(x)=ax \pmod{n}$.  Try it for $n=5$ and $n=6$ with different $a$’s.  Figure out what it is doing.
  2. Check out this tool for making addition and multiplication tables modulo n.  Try it for n=5 and n=6.  Notice some patterns.  Keep it in mind as a useful tool for future.
  3. Our first ciphertext chain!!
    1. Come up with a short (one word) answer to the question “What’s the coolest math?”  This is your plaintext.
    2. Choose a key $(\alpha,\beta) \in (\mathbb{Z}/26\mathbb{Z})^2$ .   Make sure you choice is suitable as a key, as we discussed in class.
    3. Encrypt the plaintext with affine cipher (by hand, using the cryptography tools sheet if you like).  This gives you your ciphertext.
    4. Post your ciphertext on the discord channel #ciphertexts, along with the key.
    5. Choose the most recent user’s post (besides yours) from #ciphertexts, and decrypt it (by hand).
    6. Post the answer in the form “So-and-so thinks the coolest math is….”.  Use discord’s “spoiler” feature to hide your answer, in case other students want to practice, or end up using the same ciphertext (this never works quite perfectly).
  4. Suppose you eavesdrop on your little sister’s communications, and she is using affine cipher.  Her ciphertext is CRWWZ.  You know she starts every message she writes with “HA” (she’s weird that way).  Decrypt the message.  Explain how you did it.
  5. Come up with the fastest way to compute $3^{519}$ modulo $11$ by hand.  That is, count the multiplications you need to do, and try to minimize that number.  It should be doable by hand (i.e., don’t do 500+ multiplications!!) (Hint: think back to our puzzle at the end of class.)
  6. Hand items #4,5 in on canvas as your record for today’s daily post.
  7. If needed, review injective, surjective, bijective, invertible for functions.  Here’s a discussion that gets right to the point.  Know the definitions, know examples, know a function is invertible if and only if it is bijective, and know that for a function from a finite set to itself, injective, surjective and bijective are equivalent.  (To really see why this is true, try to build a function from $\mathbb{Z}/5\mathbb{Z}$ to itself that is only injective or only surjective.)
  8. The following video (17:26) I made discusses the proof of our “main theorem” from today that multiplication and addition “work” mod n.   The goal of this video for you, here and now, is to understand why it works, and how math formalism guarantees the math we are using is “safe and correct”.  If time remains, watch this; otherwise save for later when you have time.