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Problem 9(Simple Shift Repetition)

Let's now devise a way to not only encrypt a message, but also keep track of who has seen the message? There are several ways to do this. Let's look at an example that involves repeated application of the same encrpytion key. For this example, let's use the encryption key $\phi_4:S\to S$ (the simple shift permutation that shifts right 4).

A group of military commanders decide to send messages using a message chain. The message passes sequentially from one person to the next, where at each stage they apply the encryption key to the ciphertext before sending it on. If I want to send the plain text message $attackatdawn$, then I'd send the message $exxegoexhear$ to the person after me in the chain. This person would decipher the text using $\phi_4^{-1}$ (shift left 4), and then apply $\phi_4$ to the cipher text $exxegoexhear$ and then send the ciphertext $ibbiksibliev$ to the person after them in the chain. This would repeat until the message made it to every commander.

  1. You receive the cipher text $skkzgzkomnz$. What is the plain text message? How many people have seen this message?
  2. How many commanders can we include in this encryption scheme and still tell who sent the message?
  3. If we use $\phi_5$ instead of $\phi_4$, how many commanders can we include in this encryption scheme and still tell who sent the message?
  4. Is there an encryption key $\phi_n$ so that $\phi_n^2=\phi_0$. In other words, is there an encryption key $\phi_n$ that will only allow up to two commands to be in the chain?

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