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In 1951, two policemen, Nock and Staehl, investigate an apparent break-in at the home of mathematician Alan Turing. Turing's suspicious behaviour and lack of war records triggers Nock's suspicion that he might be a Soviet spy. During his interrogation by Nock, Turing tells of his time working at Bletchley Park.
When Britain declares war on Germany in 1939, Turing travels to Bletchley Park, where, under the direction of Commander Alastair Denniston, he joins the cryptography team ofHugh Alexander, John Cairncross, Peter Hilton, Keith Furman, and Charles Richards. The team are trying to break the ciphers created by the Enigma machine, which the Nazis use to provide security for their radio messages.
Turing is difficult to work with, and considers his colleagues inferior. There is a flashback to 1927, when Turing was unhappy and bullied at boarding school. He developed a friendship with Christopher Morcom, who sparks an interest in cryptography, and eventually Turing developed romantic feelings for him. Before Turing could confess his love, Morcom died from bovine tuberculosis.
Turing works alone to design a machine to decipher Enigma. After Denniston refuses to fund construction of the machine, Turing writes to Prime Minister Winston Churchill, who puts Turing in charge of the team and funds the machine. Turing fires Furman and Richards and places a difficult crossword in newspapers to find replacements. Joan Clarke, a Cambridge graduate, and Jack Good pass Turing's test, but Clarke's parents will not allow her to work as the only female among male cryptographers. Turing arranges for her to live and work with the female clerks who intercept Enigma messages, and shares his plans with her.
Turing's machine, which he names Christopher, is constructed, but Turing cannot determine the Enigma settings fast enough before the Germans reset the encryption each day. Denniston orders it destroyed and Turing fired, but the other cryptographers threaten to leave if Turing is dismissed. After Clarke plans to leave on the wishes of her parents, Turing proposes marriage, which she accepts. During their reception, Turing confirms his homosexuality to Cairncross, who warns him to keep it secret. After overhearing a conversation with a clerk about messages she receives, Turing has an epiphany, realising he can program the machine to decode words he already knows exist in certain messages—such as the word "weather" (as the morning messages always include a weather forecast) and the phrase "Heil Hitler." After he recalibrates the machine, it quickly decodes a message and the cryptographers celebrate; however, Turing realises they cannot act on every decoded message or the Germans will realise Enigma has been broken.
Turing discovers that Cairncross is a Soviet spy. When Turing confronts him, Cairncross argues that the Soviets are allies working for the same goals and threatens to disclose Turing's homosexuality if Turing talks. When MI6 agent Stewart Menzies tells Turing that Clarke has been taken into custody, Turing identifies Cairncross as the real spy. Menzies reveals that he knew this already and that he planted Cairncross among them in order to leak the information he wants the Soviets to have. Fearing for her safety, Turing tells Clarke to leave Bletchley Park, revealing that he is homosexual and lying about never having cared for her. They break up, but she remains at Bletchley. After the war, Menzies tells the cryptographers to destroy their work and not reveal what they have done.
In the 1950s, Turing is convicted of indecency and, given the choice of a jail sentence or chemical castration, he selects the latter so he can continue his work. Clarke visits him in his home and witnesses his physical and mental deterioration. They reconcile, as she reminds him that his work saved millions of lives.
In the end, the group is seen burning the documents, and a caption reveals Alan Turing committed suicide when he was 41 years old.
