For Wednesday:
- Remember that you only need to spend 1 hour. If you get stuck on a problem, don’t spin your wheels; try some others instead and sleep on it. You can come back to things. I will post full solutions for these.
- Here are some solutions to the last daily post, which you should compare to your own answers from last time.
- Compute the Euler phi functions $\varphi(12)$, $\varphi(26)$, $\varphi(27)$. For each of these $n$, verify your answer by computing the set $(\mathbb{Z}/n\mathbb{Z})^*$.
- Another ciphertext exchange on discord!
- Use Hill cipher to encrypt the answer to the question “What is your favourite vegetable?” You can use some Hill Cipher tools here, to avoid by-hand computations. Or you can write your own encryption/decryption program. You can choose your own (valid!) key.
- Post your key and ciphertext. Use someone else’s key & ciphertext to get their plaintext and announce the result. There’s a #ciphertexts channel for this.
- Can you find the Hill cipher key? The Hill cipher (with 2×2 matrices) was used to encrypt the plaintext SOLVED to get the ciphertext GEZXDS.
- Consider Hill cipher with the matrix $$\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$$ modulo $26$. Can you find two plaintexts that encrypt to the same ciphertext? (The plaintexts don’t need to be english, they can just be any letters.) What’s wrong with this matrix?
- Prove that
- if you take successive powers of $a$ mod $n$, i.e. $a,a^2,a^3,a^4,a^5,\cdots$, that eventually you will get a repeat (the same residue will appear more than once, e.g. $a^2$ might equal $a^7$)
- if $a$ is invertible, then eventually you will get a $1$
- Remind yourself how to use the Multiplicative Dynamics tools. Can you find an example where you never get a $1$ in that list?