Question about Enigmas involutional property and its reduction of possible ciphers I'm currently learning cryptography at a university. I can't wrap my mind around how the involutional property of Engima reduces its possible ciphers while being poly-alphabetic. Can someone please explain this?

The German Wikipedia link says that if the plaintext is ABCD, then the ciphertext CADB is impossible. 

We're looking at all possible permutations of ABCD and apparently the only possible permutations are:


BADC
CDAB
DCBA
The number of involutions of size $n$ satisifies the recurrence relation:

$$I(n) = I(n-1) + (n-2)I(n-2)$$

(This recurrence is easy to deduce).

Given than $I(1) = 1$ and $I(2) = 2$, we can easily compute $I(4) = 8$, which is strictly smaller than the number of permutations of size 4 (which is 24).

This recurrence allows us to quickly compute the number of involutions of size 26 as 158,432,124,870,784; much smaller than the number of permutations 26! = 604,937,191,689,908,372,404,961,280

BTW: the involutions of size 4 are:

$$ABCD$$

$$ABDC$$

$$ACBD$$

$$ADCB$$

$$BACD$$

$$BADC$$

$$CBAD$$

$$CDAB$$

$$DBCA$$

$$DCBA$$