The number usually quoted for Enigma is 158,962,555,217,826,360,000 possible settings. It is roughly the number of grains of sand in a cube two kilometres on each side, and it is correct.
It is also almost irrelevant, and understanding why requires following the current through the machine one keypress at a time.
What Was Inside the Box
A military Enigma looks like a heavy typewriter in a wooden case. It has no screen, no printer and no power supply beyond a battery. There are five components in the signal path.
- The keyboard, 26 letters, no numbers and no punctuation. Numbers were spelled out and full stops were sent as XX
- The plugboard (Steckerbrett), on the front, where cables swap pairs of letters. Normally ten cables, so twenty letters were swapped in pairs and six were left alone
- Three rotors, removable discs with 26 contacts on each face and a scramble of internal wiring joining one face to the other. The army and air force chose three from a set of five after December 1938
- The reflector (Umkehrwalze), a fixed disc at the far end that does not rotate and does not pass current through. It connects the 26 contacts to each other in thirteen pairs and sends the current back
- The lampboard, 26 bulbs. Press a key and one lamp lights. That letter is the ciphertext
Press A. The current leaves the key, goes through the plugboard, where it may be swapped to another letter, passes into the right hand rotor, then the middle, then the left, gets reflected, comes back through the left, middle and right rotors by a different path, passes through the plugboard again, and lights a lamp.
Nine passes through wiring for one letter, and the rotors move before the current flows, so pressing A twice gives two different letters.
The Stepping, and the Bug Nobody Fixed
The right hand rotor advances one position on every keypress. Each rotor has a notch, and when the right hand rotor passes its notch it carries the middle rotor forward one step, and when the middle rotor passes its notch it carries the left one. This is a car odometer.
Except that it is not, quite. Because of how the pawl and ratchet mechanism engages, when the middle rotor is stepped by the right hand rotor it also steps itself again on the very next keypress. This is the double stepping anomaly, and the practical effect is that the middle rotor occasionally advances twice in a row, shortening the period of the whole machine from 26 cubed to 26 times 25 times 26.
It was a mechanical side effect rather than a design intention, and it was never corrected. Cryptanalysts had to know about it, because a machine that behaves unexpectedly is a machine whose behaviour can be predicted once the unexpectedness is understood.
Where the Big Number Comes From
The settings an operator had to get right, and what each contributes:
- Which three rotors, in which order: choosing three from five and arranging them gives 60 possibilities
- The ring settings, which rotate the alphabet ring relative to the internal wiring and therefore change where the notches fire: 26 times 26 times 26, or 17,576
- The starting positions of the three rotors: another 17,576
- The plugboard, with ten cables among 26 letters: 150,738,274,937,250 arrangements
Multiply and the total is around 1.59 times ten to the twentieth. The plugboard supplies almost all of it, and the Germans knew this, which is why they were confident. Their confidence was reasonable given a brute force attack, because testing that many settings at one per microsecond would take longer than the age of the universe.
Nobody attacked it by brute force.
The Reflector Was the Mistake
The reflector exists for operational convenience, and the convenience is real. Because the current goes in, turns around and comes back through a different route, the machine is reciprocal: if at some setting A encrypts to G, then at that same setting G encrypts to A. So the receiving operator sets his machine identically, types the ciphertext, and reads the plaintext off the lamps. One procedure, one machine, no switch between encrypt and decrypt modes, nothing to get wrong in the field.
The price is catastrophic, and it comes in two parts.
No letter can ever encrypt to itself. The current cannot come back along the same wire it went out on, so the lamp that lights is never the key that was pressed. This sounds trivial. It is the hole the entire British attack went through.
Suppose you suspect a message contains the word WETTERBERICHT, the weather report, which German operators sent at the same time every morning. Slide that guess along the ciphertext. Wherever the guess puts a W above a W, or any letter above itself, that alignment is impossible and can be discarded instantly, without any knowledge of the settings. On a long crib, almost all alignments die immediately, and the few survivors are where the crib probably sits.
Reciprocity constrains the plugboard. If you hypothesise that A is plugged to G, reciprocity means G is plugged to A, so one guess propagates. Gordon Welchman’s diagonal board wired this fact into the Bombe physically, so that a single assumption fanned out into a chain of consequences and any contradiction anywhere killed the whole hypothesis.
That is the crucial inversion. The plugboard contributed 150 trillion of the 158 quintillion settings, and the diagonal board turned it from the machine’s strongest feature into an additional source of logical constraints.
How the Attack Actually Ran
The Bombe did not search for the right answer. It eliminated wrong ones. Given a crib and a hypothesis about the rotor order and position, it followed the implications electrically at high speed and stopped whenever it found no contradiction. A stop was a candidate to be checked by hand, not a solution.
This required cribs, which required predictable German behaviour, and German behaviour was extremely predictable:
- Weather reports at fixed times, in fixed format, containing the word Wetter
- Keine besonderen Ereignisse, nothing to report, sent by quiet outposts day after day
- Formal military courtesies and ranks, and Hitler’s name and title in full
- Operators choosing message keys that were their girlfriends’ initials, or three adjacent keys on the keyboard, or the same three letters they used yesterday. Bletchley called these cillies
- The same message retransmitted on a different network with different settings, which hands over a matched pair
None of this is mathematics. It is the recognition that a cipher system includes the humans operating it, and that they will produce regularities under pressure.
The One Change That Nearly Worked
In February 1942 the German navy introduced the M4, with a fourth thin rotor and a matching thinner reflector for its Atlantic U boat traffic. It multiplied the work by roughly twenty six and Bletchley lost the network almost completely for ten months, a blackout named Shark, during which convoy losses rose sharply.
It was recovered partly through captured material and partly through a procedural weakness: the M4 could be set to behave exactly like a three rotor machine so that it could talk to ordinary units, and weather short signal codebooks gave cribs back.
The fourth rotor was the right kind of fix. Adding a rotor genuinely increases the search. What the Germans never did was remove the reflector, and the reflector was the vulnerability, because it was the one component that guaranteed a mathematical property an attacker could rely on regardless of how the machine was set.
Reading It in Turing’s Own Words
Turing wrote all of this down during the war, in a long typewritten treatise known at Bletchley Park as the Prof’s Book: the structure of the machine, the properties of the permutations, the statistical weighing of evidence and the logic the Bombe automated, with his handwritten corrections in the margins.
Our remastered edition of Turing’s treatise on the Enigma reproduces that manuscript with the annotations traced and the typography reset, which makes it legible in a way the archival scans are not. There is a walkthrough of its contents in our post on what the treatise says, and the earlier Polish reconstruction of the same wiring is covered in our post on the mathematicians who broke Enigma first.
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