For this set of letters, the most frequently occurring letter is H. Therefore, we make the assumption that “e” enciphers to “H.” This corresponds to a key letter of “D.”
Finally, for the
| A | B | C | D | E | F | G | H | I | J | K | L | M |
| 1 | 0 | 1 | 0 | 2 | 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
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.
