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Problem 3 (First Encryption Key)

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.
  2. Encode the plain text message "attack at dawn" into ciphertext using the same encryption key you discovered in the first part.
  3. Are there any other $n$ that would have produced the same results as above? Be ready to fully justify your answer.

Solution

1. The ciphertext decoded is (w,e,l,l,d,o,n,e). The original message is "well done." The value of $n$ that was used is $12$.

2. The plain text message "attack at dawn" encrypted into ciphertext with the same encryption key as before is (m,f,f,m,o,w,m,f,p,m,i,z).

3. There are other $n$ that would have produced the same result. Let $n_1 = 12$. Let $n_2 = 38$. Clearly it is true that $\phi_{n_1}(a) = \phi_{n_2}(a)$. By inspection, all other 25 letters can be checked and shown that $\phi_{n_1}(m) = \phi_{n_2}(m)$, where $m$ is an arbitrary letter in the alphabet.

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