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It is well known the number of Enigma combinations with plugboard: $${5!\over(5-3)!}\cdot26^3\cdot{26!\over(26-20)!\cdot2^{10}\cdot10!}=158,962,555,217,826,360,000$$.

So, $(26-20)!\cdot2^{10}\cdot10!$ just decreases the overall number of combinations. Indeed, the commercial machine was not supplied with the board. On the other hand, the military version had the board to mix the electrical circuits with less level of combinations. I guess to prevent it's easy breaking.

My questions is about pros and cons of the switch board.

Do we want to increase the number of combinations and the strength of the code at the same time and how no board situation affects the breaking the code?

Thank you for the explanations or estimations.

PS My initial question was at Math.Stack forum but I have not got any answer (just a recommendation to ask Crypto).

Mikhail Gaichenkov
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Your interpretation of the number of combinations is slightly off. The number $$\frac{5!}{(5-3)!}26^3\frac{26!}{(26-10)!\cdot 2^{10}\cdot 10!}$$ is the number of combinations (not counting Ringstellung) of an Enigma machine issued with five possible rotors, of which three are chosen, a single Umkehrwalze, and a Steckerbrett with ten cables all of which are used. This would be a common choice for German Army and Air Force communications*. The term $5!/(5-3)!$ represents the number of ways of choosing three rotors; the $26^3$ term represents the choices for the initial position of those rotors and the term $26!/(26-10)!2^{10}10!$ represents the number of different ways that the Steckerbrett can be wired. This last term equals 150,738,274,937,250 and so represents a considerable proportion of the unknown parameter space.

The Steckerbrett should be considered as strictly adding security. I know of no attack on a Steckered Enigma that does no work on the obvious unsteckered variation.

That, said, the Stecker was applied as a conjugate permutation leading to properties of the other parts of the parameter space being deducible from structure independent of the Stecker. The Poznan and Bletchley Park attacks' ingenuity depended on finding characteristics of the encryption that were independent of the Steckerbrett, recovering the $5!26^3/(5-3)!$ other parameters and then recovering the Stecker settings (given the other parameters, the Stecker settings are not much harder than a substitution cipher to solve).

  • As implied in the question, there were many variations of the Enigma and one should not fall into the trap of thinking that it was a single cryptosystem with a single attack.
Daniel S
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