Cryptography, Information Theory, and Error-Correction. Aiden A. Bruen

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Figure 2.3 Block diagram of the Enigma machine.

To use the machine, an operator inputs the desired plain text into a keyboard, one character at a time. An electrical signal is passed from the keyboard through the rotors which are connected in series, until the charge reaches the reflector plate. Then, the signal is passed back from the plate through the rotors and back into the keyboard, where a separate panel consisting of light bulbs is illuminated. Each light bulb corresponds to a cipher text letter, which is recorded by the operator. As the signal passes through each rotor, the plain text character is continually substituted, depending on the daily settings of the rotor and the specific wiring between contacts, which govern the permutations of substitutions that are possible. When the enciphering process is complete, the operator sends the cipher text via radio to the intended receiver, who also possesses an Enigma machine. The receiver can then decode the message, given that the initial settings and the permutation sets of the machines are coordinated, by simply typing in the cipher text into the machine. The plain text message then appears on the illuminated keyboard.

      It is worth noting some of the deficiencies in the machine design, as they made it possible for Allied cryptanalysts to eventually break the cipher. There is a very nice YouTube video, “Flaw in the Enigma Code ‐ Numberphile,” [Gri13], that talks about flaws in the Enigma machine. They note that a given letter of the alphabet might be mapped to any other letter, i.e. a letter is never encoded as itself. The number of permutations of objects is , but if we insist that no object is mapped to itself, then the number of such permutations, i.e. the number of derangements of objects, is reduced to approximately , (where is the base of the natural logarithm). See Ryser [Rys63, p. 23]. Thus, some information about the encoding is revealed along with a coded message!

      We will now investigate the machine from a mathematical standpoint. Each rotor is represented by a set of permutations containing all letter values between 0 and 25. The transition of each set runs left to right, with each bracket representing a wrap‐around or cycle. The first, second, and third rotors have unique permutation sets denoted , , and , respectively (each representing the possible transitions between letters). To aid in our analysis, we introduce the variables , , and to represent the initial rotor settings (taken to be the character currently located at the top of the rotor). For the purposes of this analysis, we will be ignoring the role of the plugboard. Finally, the reflector plate is modeled as a set of permutations between pairs of characters, denoted by . The goal is to track a signal as it leaves the input keyboard, travels though the rotors and reflector, and back to the illuminated display. To determine the appropriate cipher text for each given plain text letter, we will calculate the shift of each rotor, the resulting reflector permutation and reflected signal shifts until we end up with a final cipher text character.

To show how the enciphering process works, consider the modified system shown in Figure 2.4.

      The idea is to keep track of each intermediate substitution, in order to determine the final cipher text character. To illustrate the encoding process, consider the following example:

      Example 2.1 Suppose the permutation sets of each rotor and reflector are defined as follows:

Figure 2.4 Simplified Enigma model.

So, with , 0 gets moved to 15, 15 gets moved to 6, 11 moves to 0, etc.

       Each permutation set possesses an inverse, which “undoes” the action of said permutation, as follows:

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