Bombe

In the history of cryptography, the bombe was an electromechanical device used by British and American cryptologists to help break German Enigma machine signals during World War II. The bombe was designed by Alan Turing, with an important refinement subsequently contributed by Gordon Welchman.

The Enigma machine

:Main article: Enigma machine

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The German Army and Air Force Enigma machines used a stack of three rotors with 26 electrical contacts on each end. The wiring between the input and output contacts within each rotor was scrambled. The three rotors were connected to a non-rotating reflecting drum, or reflector, which redirected current back in reverse order through the rotors. The set of rotors and the reflector is termed the scrambler, denoted by S in this article. Each rotor could be set into one of 26 positions, resulting in 26 × 26 × 26 = 17,576 possible ways the rotor stack could rearrange the letters of the alphabet. The initial positions of the rotors formed part of the secret key of the Enigma, and purpose of the bombe was to recover these positions of the rotors. At each step of the encryption, at least one of the rotors (the "fast rotor") advanced a position. At certain points the other rotors were also advanced, but when using the bombe, it was, for a small stretch of letters, assumed that only the fast rotor moved, and that the others remained stationary. We denote this by writing S1 for some given position of the scrambler, and S2 for the same position but with the fast rotor advanced one position, and similarly S3, S4 and so forth.

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An additional complication in the German military Enigma machines was a plugboard (Steckerbrett in German, shortened to "Stecker") that further scrambled the letters. The large number of possible stecker wirings made cryptanalysis much more difficult. Letters were swapped in pairs: if A was transformed into R then R was transformed into A. This regularity was exploited by Welchman's "diagonal board" enhancement to the bombe. Here, we denote the plugboard by P. Because the plugboard simply swapped pairs, applying P twice restored the original, so that P(P(x)) = x.

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The encryption can be viewed as first applying P, then S, then P again. Mathematically, the Enigma encryption E can be written: E(x)=P(S(P(x))). The Enigma also has a "self-reciprocal" property: decryption is the same as encryption, so that E(E(x))=x.

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