Due Friday August 26th

For Friday:

  • Make sure you understand the ins and outs of the course setup (you’ve read all the top bar info on this site, heard me discuss in class, and have asked any questions if you have some), and have the textbook.
  • Please make sure you have set a nickname/alias in discord that is your first name with last initial or last name.  Discord is meant to be a continuance of class, so we build community (so we like to know who is who).
  • Take a look at the multiplication table modulo $26$, and figure out when a function $\alpha x+\beta \pmod{26}$ is suitable for encryption (this is affine cipher).  In other words, for which pairs $(\alpha,\beta)$ is the function $f(x)=\alpha x+\beta \pmod{26}$ invertible?  Write out a complete answer (some way of describing which pairs work and which don’t).  (If needed, remind yourself about invertible, bijective, injective, surjective.)
  • Come up with a short (one word is best) answer to the question “What’s the coolest math?”  This is your plaintext.
  • Choose a pair $(\alpha,\beta) \in (\mathbb{Z}/26\mathbb{Z})^2$ to use as a key.   Make sure you choice is suitable as described above.
  • Encrypt the plaintext with affine cipher (by hand, using the cryptography tools sheet if you like).  If you need some notes on affine cipher, it’s Section 2.2 in the text.  This gives you your ciphertext.
  • Post your ciphertext on the discord channel #ciphertexts, along with the key.
  • Choose the most recent user’s post (besides yours) from #ciphertexts, and decrypt it (by hand).  Post the answer in the form “So-and-so thinks the coolest math is….”
  • 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.
  • Write a proof for the Main Theorem of modular arithmetic that we did in class today (here are the lecture notes).  This theorem can be found everywhere (including in your textbook), but try to write your own proof first.  If you can’t, then look at a proof to understand the idea and then try to write your own with notes closed.
  • 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.
  • Check out this tool for making addition and multiplication tables modulo $n$.  Try it for $n=5$ and $n=6$ with different $a$’s.
  • Form a conjecture about which values of $a$ result in the function $f(x)=ax$ being invertible modulo n.  You might want to try different $n$’s.
  • What do you think is the essential reason that $f(x)=3x$ is not invertible modulo $6$?
  • OPTIONAL:  If you would like a longer discussion of how and why modular arithmetic is a well-defined mathematical system, and all the formal mathematics underlying it, then I’ve got a video for you (17:26). But watching it is optional; in this course we focus on the intuition, properties and applications, not the mathematical formalism in terms of equivalence relations.