The Bombe is the famous machine. What almost nobody knows is that for three years it could not have coped with naval Enigma on its own, and the thing that made it possible was a stack of perforated paper sheets, a light, and a statistical method Turing invented that has no counterpart anywhere else.
He called it Banburismus, after the town where the sheets were printed.
Why the Navy Was the Hard Problem
Army and air force Enigma traffic was broken with reasonable regularity from 1940. German naval traffic, the messages that mattered for the Atlantic convoys, was far worse, for two specific reasons.
First, the navy used a different way of sending the message setting. Instead of enciphering the three letter key twice at the start of the message, which was the weakness the Polish attack had lived on, the navy encoded it through printed bigram tables, the Kenngruppenbuch. Without a copy of those tables the indicator gave an attacker nothing.
Second, the navy had more rotors. Where the army chose three from five, the navy chose three from eight, and rotors VI, VII and VIII were naval only. Three from five gives 60 possible wheel orders. Three from eight gives 336.
That number is the whole problem. A Bombe run tested one wheel order at a time and took something like fifteen to twenty minutes. There were very few Bombes in 1940 and 1941, they were shared between every network Bletchley was attacking, and the traffic was only useful if it was broken the same day. Working through 336 wheel orders by machine was out of the question.
Somebody had to narrow the field before the machinery was committed, using nothing but the intercepts and arithmetic.
Counting Coincidences
Banburismus rests on a property of language that sounds too weak to be useful.
Take two passages of German text and lay them side by side. At each position, ask whether the two letters happen to be the same. If the texts were random sequences of 26 letters, the answer would be yes about one time in 26, which is 3.85 per cent. Real language is not random: E and N are common, Q and X are rare, so two real German texts agree far more often than chance, roughly one time in 17, which is about 5.9 per cent.
That difference, between 3.85 and 5.9 per cent, is the entire lever.
Now bring in the machine. Two Enigma messages sent on the same day with rotor start positions close together will, over their overlapping stretch, be encrypted by nearly the same sequence of machine states. Where the underlying plaintexts happen to share a letter, the ciphertexts will share a letter too. So if two intercepted messages are aligned at the correct offset, the count of matching letters between the ciphertexts jumps above the random rate. Align them wrongly and it sits at chance.
You do not need to decrypt anything. You need to count.
The Sheets
Doing it by eye across thousands of positions is impossible, so Turing designed a physical apparatus for it.
Each message was punched onto a long sheet of card, about ten inches deep and up to several feet long, with a grid of 26 rows for the alphabet and one column per letter of the message. A hole was punched in the row corresponding to each letter. The sheets were printed in Banbury, hence the name of the method, and the staff called them Banburies.
To compare two messages, you laid one sheet over the other on a table lit from below and slid it along. Wherever both sheets had a hole in the same place, light came through. The number of points of light was the number of coincidences at that offset.
Slide, count the lights, write down the number, slide again. When a particular offset produced a conspicuous excess, you had the distance between the two messages’ starting positions.
It was done mostly by the women of Hut 8, and it was tedious, exacting work that required precision for hours at a stretch. Enough of these offsets, from enough pairs of messages, constrained the position of the right hand rotor and then the identity and order of the wheels. Only then, with the field cut from 336 wheel orders to a small number of candidates, was the Bombe asked to do the rest.
The Invention Inside the Invention
Here is the part that outlived the war.
Each coincidence count is weak evidence. A slight excess at one offset might be luck. Turing needed a way to combine many weak pieces of evidence into a decision, and to know when he had enough to act, which meant putting a number on how much a piece of evidence was worth.
His answer was to work with the logarithm of the odds. For each hypothesis, start with the prior odds, and for every new observation add a quantity representing how much more likely that observation is if the hypothesis is true than if it is false. Because the quantities are logarithms, evidence adds instead of multiplying, which is the only way a room full of people with pencils can accumulate it.
He needed a unit for that quantity, and he named it after the sheets. One ban is a factor of ten in the odds. A deciban is a tenth of a ban, which Turing took to be roughly the smallest change in plausibility a human being can sensibly notice. Scores were tabulated in decibans and added up.
Three things about that are remarkable.
It is Bayesian inference, applied operationally, at a time when Bayesian methods were out of fashion and the dominant school of statistics was hostile to them.
It is sequential analysis: evidence accumulates until a threshold is crossed and a decision is taken, rather than a fixed sized test performed once. Abraham Wald published the sequential probability ratio test in the United States in 1943 and 1945, and it was regarded as a major advance. Turing had been using the idea in Hut 8 since 1940, and could not publish a word of it.
And it is a logarithmic measure of information, arrived at independently of, and slightly before, Shannon’s definition of the bit in 1948. The ban is the same kind of object in base ten that the bit is in base two. Turing and Shannon met at Bell Labs in 1943 and talked, guardedly, each forbidden to say what he was actually working on.
Turing’s assistant I. J. Good spent much of his postwar career developing the weight of evidence framework properly and putting it into the statistical literature, where it has become standard. Log likelihood ratios are now routine in signal detection, medical testing, machine learning and modern cryptanalysis, and the deciban is the ancestor of all of it.
What It Bought
Banburismus ran from early 1940 until the summer of 1943. It did not break Enigma by itself, and it was never meant to: it was a filter that turned an impossible amount of machine work into a feasible amount.
Good, who was there, estimated that it cut the Bombe time needed for naval Enigma by a large factor. During the period when there were only a handful of Bombes in existence and U boats were sinking Atlantic shipping faster than it could be built, that was the difference between reading the traffic and not reading it.
It was retired in 1943, and for a good reason: enough American built Bombes had come into service that brute force was cheaper than the labour. When you have a hundred machines you do not need a hundred people sliding sheets over a light table. Turing’s method was killed by the arrival of enough computing power, which is a fate that has since befallen a great many clever algorithms.
Where to Read Turing on His Own Method
Banburismus is documented inside Turing’s own wartime treatise on the Enigma, the manuscript known at Bletchley Park as the Prof’s Book. He sets out the properties of the machine, the statistics of coincidences, the scoring in decibans and the reasoning behind the whole procedure, in his own words, with his handwritten corrections still in the margins.
It is an unusual document because it is not a paper written for publication. It is a working explanation, written by the person who invented the methods, for the people who had to use them. Our remastered edition of Turing’s treatise on the Enigma reproduces it with the annotations traced and the typography reset so that it can actually be read.
If you want the machine side of the story first, our posts on how Enigma actually worked and on the Bombe cover the hardware that Banburismus was feeding.
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