Cryptology Cryptanalysis of the Enigma Without Plugboard

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The Commercial Enigma

The types C and D of Enigma had a reflecting rotor but no plugboard. They were sold on the free market and could be comprehensively analyzed by everyone.

In the Spanish civil war all parties used the Enigma D. All big powers broke it.

The formula for the substitution is in the mathematical part.

Cryptanalytic Approach

Searching for Isomorphs

In a section where only rotor 1 moves the two inner rotors together with the reflecting rotor effect a constant involution. If the plaintext for this section is known, then we encrypt this plaintext and the corresponding ciphertext with the last rotor only. This gives isomorphic intermediate texts, if we use the correct last rotor in correct position. Otherwise we get isomorphic texts only by chance.

Therefore pattern search identifies the fast rotor and its state by testing all rotors and all initial states. For this we have to test all three rotors with all 26 start positions. This exhaustion comprises 3×26 = 78 different constellations, each of which has to be tested for a matching pattern. Probably there are several but few possible solutions.

The next sieving step uses the fact that the two intermediate texts differ by a fixed involution. Therefore each letter, no matter in which of the two intermediate texts it occurs, must pair with the same letter in the other text. If not, we discard the solution. If yes, we even identified a 2-cycle of the involution, reducing the number of 26² = 676 possible states of the two inner rotors. A useful tool for this is a precomputed table of length 676 for the 6 different combinations of these two rotors that contains the cycle decomposition of this involution for all states, making a total of 6×676 = 4056 involutions.

The location of known plaintext, if not known a priori, may be narrowed down by negative pattern search.

Example

Conclusion

The introduction of the reflecting rotor aimed at significantly increasing the security of Enigma by doubling the number of rotor passages. This turned out to be an illusory complication. The attack by isomorphs reduces the cryptanalysis in a substantial manner.

To prevent this attack the Wehrmacht (= army) introduced the plugboard when adopting the Enigma. Then the intermediate texts can not be gotten from the ciphertext and plaintext because this transformation depends on (fixed but) unknown permutation.