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Codes, chess and Kubrick: the life of Jack Good<br>A GCHQ senior mathematician celebrates the talent of Bletchley Park codebreaker Jack Good in the centenary of his birth
Jack Good was always willing to work things out in his own way. His personal resume describes how he independently rediscovered the irrationality of the square root of two at the age of nine, the principle of mathematical induction at the age of 13, and Fermat's theorem on sums of two squares as a sixth former. His capacity for independent, original thought saw him recognised as a mathematical prodigy and he studied at Cambridge University, completing a doctorate on Fourier analysis under the great pure mathematician G. H. Hardy in 1941.<br>It was both his aptitude for mathematics and for chess1 that brought him to the attention of Hugh Alexander and Bletchley Park. He was initially posted to Hut 8 which provided a whole new outlet for his ingenuity, working German naval Enigma traffic. Good's cryptanalytic career got off to a rocky start when he annoyed the head of Hut 8 (one Alan Turing) by napping on a night shift2. For a while Turing refused to acknowledge Good, but soon Good was able to prove his worth and one of Bletchley's most successful partnerships took off.<br>Turing and Good seem to be kindred spirits and would discuss many topics of mutual interest such as mathematics, computation, artificial intelligence, and philosophy; they would play Go3 at Turing's billet in The Crown Inn. Most importantly, they would discuss statistics; a topic which neither of the two pure mathematicians had devoted much time to at Cambridge, but which they now realised was a critical tool in cryptanalysis. Between them they set about developing their own statistical ideas, some of which were rediscoveries and some of which were new. This could be viewed as the dawn of a new age in statistics as they promoted a viewpoint now known as Bayesian statistics4. It was a viewpoint that Good championed to great effect for the rest of his life.<br>Statistics begins by thinking about a fixed underlying truth and reasons about what events one might observe, but the reasoning can also be reversed and given an observed set of events one can try to reason about the underlying truth. Classical statistics treats the underlying truth as fixed and immutable, but unknown; Bayesian statistics treats the underlying truth as a competing set of explanations with associated level of belief. Good and Turing found that cryptanalysis acquires a much stronger expression in Bayesian statistics. They were particularly interested in cases where observations arrived sequentially over time and how the Bayesian beliefs could be updated continually. This led to results such as Good-Turing frequency estimation which allows measurement of the probability of observing an event which had not occurred in previous observations. Ideas such as these had a great impact on the analysis of the naval Enigma via processes such as Banburismus scoring and also on Good's later work under Max Newman attacking the Fish ciphers of the German High Command.<br>Equally important to Bletchley was Good's intuition for computation and how to use the new machines such as the Newmanry's COLOSSUS to best effect. In one throwaway anecdote, Good tells how he was able to double the efficiency of the COLOSSUS simply by reducing the precision to which one of the statistical scores were computed (using his statistical expertise to show that the error introduced was not great enough to change the conclusions).<br>After the war, Max Newman quickly recruited Good as a lecturer at Manchester University with the intent of also using him to help develop the Manchester Mark I computer. A dislike of lecturing and frustrations at finding a publisher for his (now classic) treatise on Bayesian statistics "Probability and the Weighing of Evidence" drew Good away from academia and returned him to classified work at GCHQ. The Cold War and the growth of computational power meant Good's talents were incredibly valuable to GCHQ as he continued to contribute to the revolution in cryptanalysis brought about by efficient computers. It was during this time that Good developed a powerful computational technique which is the basis of modern methods known as Fast Fourier Transforms (FFTs). FFTs have found an immense range of applications from signal processing, to solving differential equations, to providing the best known methods for a computer to multiply large numbers. They massively reduce the computational power required to achieve these tasks and the idea of the FFT has been described as the most important algorithm in modern...