Editing Enigma machine (section)

=== Mathematical analysis ===
The Enigma transformation for each letter can be specified mathematically as a product of [[permutation]]s.{{sfn|Rejewski|1980}} Assuming a three-rotor German Army/Air Force Enigma, let {{mvar|P}} denote the plugboard transformation, {{mvar|U}} denote that of the reflector (<math>U=U^{-1}</math>), and {{mvar|L}}, {{mvar|M}}, {{mvar|R}} denote those of the left, middle and right rotors respectively. Then the encryption {{mvar|E}} can be expressed as

:<math>E=PRMLUL^{-1}M^{-1}R^{-1}P^{-1}.</math>

After each key press, the rotors turn, changing the transformation. For example, if the right-hand rotor {{mvar|R}} is rotated {{mvar|n}} positions, the transformation becomes

:<math>\rho^nR\rho^{-n},</math>

where {{mvar|ρ}} is the [[cyclic permutation]] mapping A to B, B to C, and so forth. Similarly, the middle and left-hand rotors can be represented as {{mvar|j}} and {{mvar|k}} rotations of {{mvar|M}} and {{mvar|L}}. The encryption transformation can then be described as

:<math>E=P\left(\rho^n R\rho^{-n}\right)\left(\rho^j M\rho^{-j}\right)\left(\rho^{k}L\rho^{-k}\right)U\left(\rho^kL^{-1}\rho^{-k}\right)\left(\rho^j M^{-1}\rho^{-j}\right)\left(\rho^n R^{-1}\rho^{-n}\right)P^{-1}.</math>

Combining three rotors from a set of five, each of the 3 rotor settings with 26 positions, and the plugboard with ten pairs of letters connected, the military Enigma has 158,962,555,217,826,360,000 different settings (nearly 159 [[Names of large numbers|quintillion]] or about 67 [[bit]]s).<ref name="158,962,555,217,826,360,000"/>

* Choose 3 rotors from a set of 5 rotors = 5 x 4 x 3 = 60
* 26 positions per rotor = 26 x 26 x 26 = 17,576
* Plugboard = 26! / ( 6! x 10! x 2^10) = 150,738,274,937,250
* Multiply each of the above = 158,962,555,217,826,360,000