Check the previous daily post for some announcements in case you missed them.
Solutions to the last daily post are in the lecture notes and also here. Solutions to the one before are here. Compare with your work. Note: these solutions are slightly disorganized (some extra problems etc), so let me know if you have any trouble with them.
The first two boxes on the Finite Field Tools page are a calculator for any finite field you want. You may use these boxes to solve the following problems.
Let us work in the Finite Field of size $5^2$, which, in the tools page, will by $\mathbb{F}_5[x]/(x^2+4x+2)$ where $x$ (called $a$ in the calculator) is a multiplicative generator. Set this field R up in the first box on that page so you can use it in what follows.
As a test of your calculator skills, what is the inverse of $x$? You can input R(a^(-1)) into the second box. You should get that the inverse is $2x + 3$ (which appears as 2a+3).
Suppose you are Bob and your El Gamal private key is $b = 5$. The generator is $g=x$ in this same finite field. What is your public key (reduced to small degree)?
Suppose that Alice encrypts you a message using your public key. The message is a polynomial. She sends you the ciphertext $c=4x$ with ephemeral public key $K = 4x+3$. What is the plaintext polynomial?
Using the online tools/Sage, find a multiplicative generator for $\mathbb{F}_3[x]/(x^2+1)$.