Here is my second post to test how well I can use Latex in my blog. Fingers crossed!!! 

(P.S. This is the first write up I turned in for my Abstract Algebra class, now there is something "to see" as my Padre would say.)

    Suppose you encounter some ciphertext $(i,q,x,x,p,a,z,q)$ that you know has been encrypted using a simple shift permutation $\phi_n$ for some $n$. 

$1$.  Decode the ciphertext and state the original message. State the value of $n$ that was used to encode the original message.

The message found after decoding the ciphertext said, $(w,e,l,l,d,o,n,e)$. The original message was encoded using $\phi_{12}$. We know this because $\phi_{12}(w)=i$ and so forth.

$2$.  Encode the plain text message "attack at dawn" into cipertext using the encryption key you discovered in the first part.

Using the encryption key we discovered in the first part, we would encode the message $(a,t,t,a,c,k,a,t,d,a,w,n)$ using $\phi_{12}$. In the end we get the new ciphertext $(m,f,f,m,o,w,m,f,p,m,i,z)$.

$3$.  Are there any other $n$ that would have produced the same results as above? Be ready to fully justify your answer.

There are other $n$ that would have produced the same results as above. Another shift permutation $\phi_n$ that would work is $\phi_{38}$. We know that this will work because $12 + 26 = 38$. We can see then, that $\phi_{38}$ works because we will shift right $26$ times and then an additional $12$ times landing at the same place we did when we just moved $\phi_{12}$.