So the involution is a permutation that is its own inverse.

And again it's still susceptible to the character frequency attack but

it's proven useful as a component in a cipher machine because it can greatly

multiply the number of possibilities that an eavesdropper has to consider and

that's how it was used in the Enigma.

So for example if you want to know how many different Enigma settings you

have find, then you have to know something about enumerating involutions.

And there's a very famous story about the crypt analysis of the Enigma

involving Alan Turing and so forth.

And if you don't know that story, definitely worthwhile to look it up and

read about it.

So, as a warm up, let's take a look at the number of permutations

that are composed entirely of 2 cycles.

So, there's only one permutation size 2 that's all 2-cycles.

There's three different ones of size 3 that are all 2-cycles and so forth.

And actually, an example of this is called ROT-13, so, it's the world's

weakest cryptosystem, where we just take the letters and rotate them 13 positions.

So A goes to N, N goes to A, B goes to O, O goes to B and so forth.

And you can read about this one on the web, it's a hacker's delight because

anyone can Encode or decode and people get so they can read in this and so forth.

So, it's a very light crypto-system that hackers use to just lightly

put stuff out on the web that maybe is inappropriate content, but

you have to be at least able to decrypt in this way and

nobody would get offended if they ran across it accidentally.

So again, since this 2-cycles it's a reciprocal cypher.

So you use the same table to decrypt and encrypt.