Enigma and Bletchley Park
The German cipher machine considered unbreakable, and the Polish and British effort that broke it anyway and read the enemy's mail for most of the war.
The Enigma was an electromechanical cipher machine, patented commercially just after the First World War and adopted in modified form by the German military. Pressing a key sent current through a plugboard and a set of rotating wheels to light a different letter; the rotors advanced with every keystroke, so a letter never enciphered the same way twice. With the wheel order, ring settings, starting positions, and plugboard pairings all changed daily, the number of possible configurations runs to something like 150 million million million.
German cryptographers concluded from that number that the system was unbreakable. They were wrong, and the reason they were wrong is the most useful thing in the story: a cipher is broken not by exhausting its key space but by exploiting how people use it.
The Poles got there first
This part is routinely left out of British accounts. The Polish Cipher Bureau, working from the 1930s under the mathematician Marian Rejewski with Jerzy Rozycki and Henryk Zygalski, reconstructed the internal wiring of the military Enigma without ever having seen one, using permutation theory and material obtained by French intelligence. They built machines called bomby to search for daily settings and were reading German traffic years before the war.
When German procedures changed in 1938 and the Polish methods began to fail, and with invasion clearly coming, the Poles handed everything they had, methods, machines, and reconstructed wiring, to British and French intelligence in July 1939, five weeks before the war began. Britain did not start from zero. It started from a Polish solution.
Bletchley Park
The British operation at Bletchley Park, a Victorian house in Buckinghamshire, grew from a few dozen people to around ten thousand, roughly three quarters of them women, and industrialized the attack.
Alan Turing's contribution was to design a machine, the Bombe, that did not search for the key directly but tested assumptions for contradictions. It worked from cribs, guessed fragments of plaintext, which were available because German operators were predictable: weather reports at the same hour, formulaic sign-offs, a station that sent nothing but the word nothing to report. A design feature of the Enigma also helped enormously, since no letter could ever encipher as itself, which let cryptanalysts eliminate crib positions instantly.
The resulting intelligence was codenamed Ultra, and protecting the fact of it was as demanding as producing it. Decrypts could only be acted on if a plausible innocent source, a reconnaissance flight, an agent, could be arranged to explain how the information was obtained.
What it was worth
The clearest case is the Atlantic. Knowing where the U-boat wolfpacks were let convoys be routed around them, which was the difference between Britain being supplied and Britain being starved. When the German navy introduced a fourth rotor in February 1942, Bletchley went blind for most of that year and shipping losses rose sharply; the blackout ended largely because material was recovered from a sinking submarine.
Historians commonly estimate Ultra shortened the war by around two years. That figure is a guess, but the direction is not in dispute.
The echo
Two long shadows.
The first is technological. Bletchley also built Colossus to attack a different German cipher, and Colossus was the first programmable electronic digital computer. The people and ideas went directly into postwar computing, and Turing's wartime work is inseparable from the birth of the field.
The second is institutional. The habit of large-scale signals intelligence, and the wartime sharing arrangement between Britain and the United States, hardened into the Five Eyes alliance that still governs how the English-speaking world collects intelligence. Modern debates about encryption, government access, and mass interception are conducted in an institutional structure that was built around a wooden box with three wheels in it.
It was all kept secret until the 1970s. Turing, prosecuted in 1952 for homosexuality and dead by his own hand in 1954, was never publicly credited in his lifetime for any of it.