I attended the ACE2012 conference over the weekend, and the issue of the stecker patching on the front of the Enigma came up. The Germans used 10 stecker cables to cross connect 20 letters in the Enigma. It was presented that the 10 cables that the Germans used was the most mathematically complex combination that can be used. However, it is not!
The optimal number of stecker cables is, in fact, 11 which would increase the complexity of the Enigma by just over 36% (as the table below shows).
It is believed that the reason the Germans chose 10 instead of 11 was in part due to the physical difficult of squeezing an eleventh cable in the confined space at the front of the Enigma; and perhaps, the increased possibility of operator error. Additionally, the Germans could have made life more interesting by increasing the combination from 150 billion to over 500 billion by having a variable number of stecker pairs instead of the fixed 10.
The formula and the mathematical results are here:
In Microsoft Excel terms, the formula looks like this:
=FACT(26)/(FACT(<n>)*FACT((26-2*<n>))*2^<n>) where <n> is the number of cables in the second column
If you want more information on this, then the excellent cryptomusem site at http://www.cryptomuseum.com/crypto/enigma/working.htm has the same maths on show, along with a lot more information on the Enigma and other crypto devices.
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