By Laurence Peter
The secluded mansion north of London had 10-12,000 staff in WWII
A silk scarf bearing the image of a horse race was a suitably cryptic gift for a Polish mathematician to receive from a British code-breaker.
The Poles had got there first - that seemed to be the message.
Dillwyn "Dilly" Knox was delighted with the Polish copy of an Enigma - a top secret German military cipher machine.
But his meeting with code breakers in Poland in July 1939 - just weeks before Hitler invaded their country - had initially put him in a sour mood. He had been struggling to figure out the machine's wiring - a key part of the complex jigsaw puzzle called Enigma.
Marian Rejewski, a talented Polish mathematician, had guessed correctly that the wiring connections between the machine's keyboard and encoding mechanism were simply in alphabetical order.
Of course, there were numerous other problems to solve, but Rejewski had made a major breakthrough, by devising equations to match permutations in the machine's settings.
For decades after the war the contributions of Rejewski and other Polish cipher experts to the Allied victory over Nazi Germany went unrecognised.
Dutch invention, first used by German military in 1926
Typewriter-style keypad used to input plain text
Encryption done by three or more rotors and electrical plugboard
Daily instructions for settings, known as "key for the day"
Each message also had "message setting" chosen by sender
Receiving operator used message setting to recover signal on his Enigma
Morse code signals intercepted by British
But Bletchley Park, the nerve centre of Britain's wartime code breaking operations, has just held its annual Polish Day - a celebration of the Polish achievements that laid the foundations for British success in cracking German codes.
The fictional film Enigma, made in 2000, had dismayed Poles by neglecting these achievements and portraying a Pole as a traitor.
It has taken a long time to establish the historical facts, but the picture is much clearer now, in the run-up to the 70th anniversary of the outbreak of World War II.
"This event is tremendous - we're very pleased that the British remember the Poles," said Derek Celinski, a Polish army veteran who survived the Nazi destruction of Warsaw.
One of the lessons the British learned from the Polish experience was the importance of engaging the country's best mathematicians in the code-breaking project.
While British code-breakers were undoubtedly bright - Knox was a translator of ancient Greek poetry - they were not necessarily mathematicians.
Dancing in the park: Many Poles were among the Polish Day visitors
Polish historian Eugenia Maresch says that Alastair Denniston, the first director of Bletchley Park, was inspired by his meeting with the cryptologists at Pyry, the small Polish decoding centre in woods outside Warsaw. There the Poles divulged their methods and Enigma secrets to British and French intelligence.
The Poles were already deciphering Enigma messages in 1933, Mrs Maresch explained, whereas the British did not seriously turn their attention to Enigma until the Spanish Civil War in 1936, when the Axis powers' aggression started threatening British interests in the Mediterranean.
Rejewski was the brightest of three top Polish mathematicians who were recruited for code-breaking, Bletchley Park historian Frank Carter says. The other two were Henryk Zygalski and Jerzy Rozycki.
They had graduated from a University of Poznan cryptology course, set up by Polish officer Maksymilian Ciezki, who had been trained by the Germans before Poland became independent in 1918.
Bletchley Park has a replica of Turing's "Bombe" - the originals were destroyed
Although Zygalski and Rejewski were smuggled out of fascist Spain by British agents during the war the veil of secrecy meant they were not allowed to join the Bletchley Park team, Mr Carter explained.
German changes to the Enigma machines during the war meant much greater resources were required to crack them, and that was where the inventiveness of Alan Turing and the other British code-breakers was key.
The Enigma configurations changed daily - and the "key for the day" could be any one of about 364,000 million possible settings.
"Many Enigma keys were never found," Mr Carter told the BBC.
"Probably less than 25% of the naval codes were broken, but it was still a significant success.
"The easiest was the German air force - they weren't as security-minded and made blunders. They were broken daily."
Turing created the "Bombe" at Bletchley Park - a more sophisticated decoding machine than an earlier Polish machine called the "Bomba".
The Polish machine exploited a weakness in the German "indicators" - the starting positions for sending Enigma messages. But when the Germans changed the indicator system in May 1940 the Polish method became redundant.
The British "Bombes" however did work, based on "cribs" - recurring patterns in German secret messages, such as the words "special arrangements for".
The German naval codes were the hardest to crack - and that mattered hugely while U-boats were wreaking havoc, torpedoing Allied ships in the North Atlantic.
But Bletchley Park's work is reckoned to have shortened the war by as much as two years.