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

0 1 2 5 0 0 1 1 N O P Q R S T U V W X Y Z 0 0 1 0 3 3 0 0 1 1 3 0 2

      From this table, we deduce that since “e” likely enciphers to “I,” our fourth and final key letter is “E.”

      Putting all of this together, we have determined that the period is Four, and the corresponding keyword is the word “code.” This gives the message, “The Vigenère cipher was created in the sixteenth century and was considered by everyone to be unbreakable until the twentieth century.”

      The method used above, though simple to use, is very effective in determining the keyword of a given cipher text passage. The reader should be aware that there may be times where it may take some more work to pin the keyword down, due to multiple period choices and ambiguities that may occur in the frequencies of cipher text letters.

      Remark

In examining these methods for breaking the Vigenère cipher, we have stated many times that these methods are only probabilistic, that is, they are only likely to work, not guaranteed. It is possible that we could go through the above process, only to decipher a given cipher text incorrectly. The question of how many messages encipher to a given cipher text is discussed in Chapter 15, and it turns out that we can roughly expect there to be only one intelligible message fitting with a given cipher text when the cipher text has more than 28 letters.

2.7 The Enigma Machine and Its Mathematics

      During World War II, German troops were able to march unopposed through much of Eastern Europe. At the heart of this war machine was an encryption scheme that allowed commanders to transfer crucial planning data with near total secrecy. Before the invasion of Poland, three Polish cryptologists by the names of Marian Rejewski, Henry Zygalski, and Jerzy Róźycki were able to crack the majority of the Enigma code used by the German army. Fearing capture during the German invasion, they escaped to France, bringing with them vital secrets about the Enigma machine.

Mechanically speaking, the Enigma machine consists of three removable rotors, a keyboard, a reflector plate, and a plugboard (see Figure 2.2). Each rotor has 26 electrical contacts on each face (with each one representing a value between 0 and 25), and wires connecting contacts on opposite faces in a variety of ways. The rotors rotate clockwise and are geared in such a way that the settings of the first rotor change with each plain text character that is enciphered and cycles through the values 0 to 25. Upon the transition between 25 back to 0, the second rotor rotates 1/26th of a turn. Following the transition between 25 back to 0 on the second rotor, the third rotor rotates 1/26th of a turn. This ensures that if the same character is sent twice in a row, it will likely be enciphered as two different cipher text letters. To increase the number of possible permutations, the rotors can be interchanged with one another. The reflector plate is a device with 26 contacts on the face adjacent to the last rotor, wired in such a way that the contacts are connected in pairs. Once a signal is sent to the reflector, it is sent through the corresponding wire and back into the third rotor. The plugboard consists of a series of sockets, and the board changes the identity of the input character based on the following conventions: if the given socket contains a plug, the character's identity is changed. If the socket is empty, the character remains unchanged. This device simply provides another set of permutations, meant to increase the complexity of the enciphering scheme. A basic block diagram of the system is depicted in Figure 2.3.

Figure 2.2 The German Enigma machine. (a) the Enigma machine type‐K, (b) the German Enigma machine display at the Naval Museum of Alberta, Canada, (c) the caption on the display at the Naval Museum of Alberta, Canada, (d) the Enigma machine type‐K, power supply, and additional lamp panel.

      Source: Used with permission of the Naval Museum of Alberta, Canada.

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