Rings and Encryption

Find an inverse of $f(x)=x+2$ when $N=2$ and $p=4$.

Show that a polynomial in $\mathcal{L}(k,k)$, where $k$ is some positive integer smaller than $\frac{N}{2}$, does not have an inverse modulo $p$ for any $p$.

Let $N=6$, $p=5$, and $q=16$, and suppose Bob chooses $f(x)=x^{3}+x^{2}-1$ and $g(x)=x^{4}-x^{3}-x+1$. Given $F_{5}(x)=3x^{5}+4x^{4}+x^{3}+2x^{2}+2x+4$ and $F_{16}(x)=7x^{5}+5x^{4}+14x^{3}+9x^{2}+12x+2$, what is Bob’s public key?

Express the character ‘U’ as a representative polynomial.

Let $N=8$, $p=3$, and $q=64$. Let Bob’s public key be

Alice wants to transmit the character ‘U’ to Bob. Suppose Alice encodes ‘U’ using the ASCII conversion table (1.1) and obtains an ‘encoding fuzz’ polynomial of $x-1$. What is the encrypted message?

Suppose $N=8$, $p=3$, and $q=64$. Bob received an encrypted message from Alice,

that was encrypted using his private key

which has an inverse modulo 3 of

What was the character that Alice sent to him?(Note: although the private key and its inverse look familiar, we used a different $g(x)$ and $\phi(x)$ from the previous exercises to encrypt the message.)