English Work to Crack Enigma
Due to the work of the Poles, along with captured Enigma machines, the English were aware of the wirings of all of the rotors. Their work concerned trying to determine the initial rotor setting (initial state) of the machine, along with the plugboard settings, for a given day. Here are some annotated quotes from the book Alan Turing, The Enigma, by Andrew Hodges (ISBN: 0-671-52809-2), that explains how the English approached the problem.
Suppose now that the letters LAKNQKR are known to be the encipherment of GENERAL on the full Enigma with plugboard. This time there is no point in trying out LAKNQKR on the basic Enigmas, and looking at what emerges, for some unknown plugboard swapping must be applied to LAKNQKR before it enters the Enigma rotors. Yet the quest is not hopeless. Consider just one letter, the A. There are only 26 possibilities for the effect of the
plubgoard on A, and so we can think about trying them out. We may start by taking the hypothesis (AA), i.e. the supposition that the plugboard leaves the letter A unaffected.
(What we are doing here is basically using the idea of a proof by contradiction. We assume that the plugboard leaves the letter A alone, and we try to reach an absurdity, allowing us to conclude that, in fact, the plugboard does not leave A alone.)
What follows is an exploitation of the fact that there is only one plugboard, performing the same swapping operation on the letters going into the rotors as on the letters coming out. (If the Enigma had been fitted with two different plugboards, one swapping the ingoing letters and one the outcoming, then it would have been a very different story.) It also exploits the fact that this particular illustrative "crib" contains a special feature - a closed loop. This is most easily seen by working out the deductions that can be made from (AA). Looking at the second letter of the sequence, we feed A into the Enigma rotors, and obtain an output, say O. This means that the plugboard must contain the swapping (EO).
Now looking at the fourth letter, the assertion (EO) will have an implication for I, say (NQ); now the third letter will give off an implication for K, say (KG).
(In other words: the Enigma machine transforms the fourth letter E into N. First the plugboard turns E into O. Then the rotors turn O into some unknown letter, say Q. So, it must be that the plugboard turns Q into N. Similarly, we can conclude based on the third letter, that K is transposed with G on the plugboard.)
Finally we consider the sixth letter: here the loop closes and there will either be a consistency or a contradiction between (KG) and the original hypothesis (AA). If it is a contradiction, then the hypothesis must be wrong, and can be eliminated.
(The Enigma machine transposes A and K in the sixth letter. If we have the KG plug and we have no plug on A, then the rotors must change A to G and G to A. So, we would begin by guessing the initial state, i.e. the initial rotor settings, of the machine. If the sixth state of the machine, without plugboard, does not change A to G, then it must be that either the initial rotor settings were wrong, or the guess that A is not changed by the plugboard was wrong, or both.)
So, the approach basically was first to find loops in the message. Once these were found, the English would use the process of elimination to get rid of machine state/plugboard pairs until found pairs that did not lead to a contradiction.
Here is another example: Suppose the coded message read A A R C B O N O and the original message read COOK DUCK. Then we have the following loop: C to A (1st letter), A to O (2nd letter), O to K (8th letter), and K to C (4th letter). With a loop, we know that the process described above will lead end with either a contradiction or not.
We now choose a plug and an initial state. Let’s guess that the initial rotor settings were AQF and that there was a plug (C D). We now use an Enigma machine set up with AQF and no plugs, and we think this through. Starting with the first letter of the message COOK DUCK, the plug would change C to D. Then the machine would change D to some other letter (we try the machine to see what it does). Let’s say it changes it to E. Then we know that (A E) must also be a plug, given our assumptions. We continue with the 2nd, 4th, and 8th letters until we get a contradiction or not.
In practice, there were millions of assumptions to check, and the British couldn’t just use the machines they had acquired. Instead, they created some of the first computers in the world, called bombes, to test many assumptions quickly. They had to find loops by hand, however, in order to figure out what to program the bombes to check.