Difference between revisions of "Workings of the Enigma machines"

Latest revision as of 14:25, 15 December 2017

This is discussion of the workings of an Enigma machine, see Logical workings of the Enigma machines for a mathematical overview.

Input and output

The Enigma has an input of 26 distinct values, which are mapped to Roman letters [ilmath]\mathrm{A}-\mathrm{Z} [/ilmath], only in upper-case, there is no space or punctuation.

Major components

In order of a conceptual signal passing through:

1. Keyboard
2. Plugboard [optional]
3. Rotors (wheels) of which there are 3 to 5)
   1. Through the 3-5 rotators
   2. Through the reflector rotor
   3. Back through the 3 to 5 rotor wheels (this time in the opposite direction) of step 3.1
4. Back through the plugboard (this time in the opposite direction to 2) [optional]
5. To the output lamps.

The keyboard simply takes the input, and the lamps simply display the output. Specifically there is one lamp for each symbol the Enigma machine offers as able to input, a lamp lights up when you press a key which under the current configuration represents a symbol which is mapped. All other parts will be discussed below.

Plugboard

This is simply a board which with nothing plugged in maps every letter to itself, an identity map. If it is absent in a particular model of Enigma machine, just think of it as having an empty plugboard for the purposes of analysis.

Each plug is a female connector for 2 pins, one large (top) and one small (bottom), each plugboard cable end has a male connector which is 2 pins, a large and a small one to match the receptor, both ends are the same.

Caveat:Not sure if the cable maps big-pin-to-small-pin (cross-over cable) or if it's two parallel wires.

Connecting a cable between two letters, say [ilmath]A[/ilmath] and [ilmath]C[/ilmath] has the effect of creating a transposition between those letters^{[Note 1]}, when the [ilmath]A[/ilmath] key is pressed and the "signal hits the plugboard" it is transferred to enter the wheels as the [ilmath]C[/ilmath] signal, and the [ilmath]C[/ilmath] key corresponds to an [ilmath]A[/ilmath] signal entering the wheels

On the return path, any signals which leave the rotors

Any signals coming out of rotors that pass through a plugboard have the same map applied, but in reverse.

• Which should be the same as the forward map by^{[Note 1]} Caveat:However I've not proven this yet

Rotors

All rotors except the reflector are a wheel with 26 conductive pads on one side, and 26 spring loaded (meaning "pushed outwards by springs all the time" - this maintains connector contact) pins on the other side. All pads are connected electrically to exactly 1 pin, and which one it is connected to is specific for each wheel, representing an arbitrary permutation of the 26 symbols.

Reflector

The reflector is key to the property that an Enigma machine never maps a letter to itself. Electrically it cannot do this, it must forward the signal somewhere else. Electrically each pad is wired to exactly 1 other pad (so there are 13 wires in total, for the 26 symbols), this signal then goes back through the rotors.

It is trivial to prove (and indeed is proved in Logical workings of the Enigma machines) that it is impossible to chain rotor wheels (that are permutations, obviously if they're not it's quite easy) then have a reflector as described above, and not have unique pathways through the wheels, that is there is "no crossover" the input signal (1 out of 26) cannot output as itself.

Other major components

In addition to the above there is another "rotor" called a "stator" (static rotor) which has pins and never moves, it's is the electrical collection between the rotors and the plugboard, it serves only to connect them and doesn't represent a permutation itself.

Mechanics

When a key goes down it causes a rotor Template:Caveat:Rightmost, I believe - one of the end ones to advance one position. This has markers (steppers) on it, called the Caveat:ring setting - confirm this is not initial rotor position when this stepper reaches a certain position, rotation of the wheel will cause rotation of it's neighbour.

• Critically this is to point out that the rotors advance ON KEY DOWN, not afterwards.

Warning:

• I've read about "Enigma stepping", which rotates the neighbouring wheel twice, once when the mark is in position and once either on the next or previous advancement of the wheel... not sure which
• Not sure if the reflector rotates or not.

Notes

1. ↑ ^{1.0 1.1} Let [ilmath]T:\eq (a,b)[/ilmath] be a transposition we claim that [ilmath]TT\eq \text{Id} [/ilmath].

   Proof:

   • consider [ilmath]a[/ilmath], then [ilmath]T(T(a))\eq T(b)\eq a[/ilmath] as required
   • consider [ilmath]b[/ilmath], then [ilmath]T(T(b))\eq T(a)\eq b[/ilmath] as required
   • consider any other item, say [ilmath]x[/ilmath], then [ilmath]T(T(x))\eq T(x)\eq x[/ilmath] - as required

   So we see that the inverse of a transposition is the transposition

References