The publication of F. W. Winterbotham's The Ultra Secret in 1974 revealed to the English-speaking world the now-familiar Enigma story. During most of the second world war, the Allies were reading much of the German armed forces' radio communications enciphered with their Enigma machine, and the intelligence derived from reading German orders and dispatches was of inestimable value to the Allied cause.

Winterbotham, whose job was to supervise the security of the intelligence flowing from the Government Code and Cypher School (GCCS) at Bletchley Park, knew nothing of the technicalities of this feat of cryptanalysis and had no access to contemporary official documents. In the following years, a host of other books on the subject have corrected Winterbotham's occasional errors and have supplied a wealth of new detail. Gordon Welchman's Hut Six Story (1982), for instance, provided a personal account by one of the designers of the Enigma-cracking bombe machine, and Harry Hinsley and Alan Stripp's Codebreakers (1993) gave an anthology of accounts by Bletchley Park veterans. With a completely different tone, the official multi-volume history, British Intelligence during the Second World War , edited by Hinsley, appeared between 1979 and 1988.

In all this literature several kinds of exposition appear again and again. There is the recruitment narrative, in which the young undergraduate (say) is told by his tutor to attend a mysterious interview where he is asked if he likes crossword puzzles. Then he is whisked off to Bletchley Park where the work is intense and at times frustrating. The brilliant but eccentric colleagues make the wartime austerities bearable, and eventually great progress is made. Possibly, the narrator is then shipped out to the Middle East or Asia, where a new set of austerities prevail, but again the work prospers. Another kind of exposition gives details of how the codebreaking work influenced the course of the war. For instance, there was no cryptographic forewarning of the Bismarck 's outward course during its 1941 raid into the North Atlantic, but there was for its return course, allowing Admiral Tovey to concentrate his forces on the approaches to France rather than to Norway and thus to sink the Bismarck . Finally, there are explications of codebreaking techniques. These always involve chains of deductions made from a diagram of some sort, rather like explanations of chess positions or bridge plays. The reasoning is often hard to follow, although enlivened by the Bletchley Park jargon: "cillies", "lobsters", "crashing" and "Yoxallismus". These three kinds of exposition - with their strengths and weaknesses - correspond to well-known categories of military historiography: war as seen by privates from the foxholes, as seen by generals from the hilltops and as seen by military technologists from the back rooms.

Although television documentaries or stage plays understandably ignore the point, it is important to realise that there was no one unitary Enigma secret to be solved once and for all. At any given time there were dozens of different Enigma keys, each used by a different branch or command of the Wehrmacht for some specialised purpose. So too for other German and Japanese codes and ciphers. During the war, most keys were changed every day, and each key change posed a separate puzzle for the overworked staff at Bletchley Park and their American counterparts. The German navy was particularly aggressive about introducing improvements to the Enigma, and its keys -especially those used by the U-boat service -proved especially difficult to solve.

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In the last five years, GCHQ has declassified the secret documents used by Hinsley et al in his official History and placed them in the Public Records Office; in America, the wartime records inherited by the National Security Agency have gone to the National Archives. These papers have given a fresh impetus to the history of wartime cryptography and all three books make extensive use of them.

Hugh Sebag-Montefiore's Enigma: The Battle for the Code is the most satisfying read, in large part because he focuses on a relatively narrow aspect of the subject -the road to the solution of the German naval Enigma -and organises a long series of third-person accounts into a clear storyline, with a leavening of technical exposition (readers who are not interested in the fine points can ignore the appendices).

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He gives much new information about H. T. Schmidt, the spy who supplied the French (and through them, the Poles and British) with Enigma instruction manuals and key lists in the early 1930s. He gives ample space to the group of Polish mathematicians who used Schmidt's information to make the first break into Enigma, figure out the Enigma rotor wirings and so read German Enigma traffic during 1933-38. Sebag-Montefiore follows them through the hair-raising story of their escape to Britain in 1943.

Things became more complicated when the war began. Building on the Polish work, Alan Turing devised a method ("Banburismus" in Bletchley Park codespeak) for attacking German naval Enigma messages. The method, however, required precise information about the wiring of the extra rotors in the navy's Enigma machines (the other services used only five, the navy used eight) and about the tables it used to disguise message headers. This data was obtained from captured German ships and occasionally from sinking U-boats. At various times in 1941-43 there were not enough data at hand to allow Banburismus; the resulting fluctuations in naval intelligence led to corresponding sea victories or convoy losses. Sebag-Montefiore gives many gripping accounts of such naval actions. (In a few places he shows some unfamiliarity with wartime terminology, using the tautology "Asdic sonar", or glossing "Hedgehog" as "depth charge" and B-Dienst as "German secret service" instead of "German naval signals intelligence service", for example, but these are minor lapses.) Stephen Budiansky's Battle of Wits: The Complete Story of Codebreaking in World War II is, as its title suggests, more sprawling and less tightly organised. The Enigma machine was not the only cryptographic system used by the Germans: the foreign office used code books with additive tables, and the army used a different cipher machine, the ³§³¦³ó±ô¨¹²õ²õ±ð±ô³ú³Ü²õ²¹³Ù³ú (known as Tunny to the Allies) for its higher-level communications. And the Japanese used a vast multiplicity of code books and several different cipher machines working on different principles from either Enigma or Tunny.

On the codebreaking side, both the American army and navy had their own codebreaking establishments (SIS and Op-20-G, respectively) -in contrast to the all-services GCCS of the British. So the narratives of how, between them, the British and Americans broke essentially all these systems, ramify and criss-cross, and it is easy for the reader to get confused.

Budiansky has a drier, more scholarly tone than Sebag-Montefiore. He relies somewhat less on personal reminiscences and somewhat more on official archives; his explanations of technicalities are more confident than Sebag-Montefiore's. In addition to the original Polish work on Enigma, the high-speed Enigma crib-checking bombes of Turing and Welchman, Banburismus and, to a certain extent, the attack on Tunny, Budiansky describes the techniques used to break the Japanese Purple cipher machine and to break super-enciphered codes such as the Japanese navy's JN-25 and the German foreign office's codes.

Budiansky is especially attracted by the technology of the various pieces of special-purpose "rapid analytic machinery" (Ram) invented in America and Britain, of which the bombes were just the first examples. It is unfortunate that Budiansky's book went to press before the declassification of the 1945 General Report on Tunny , which describes in great detail the use of the war's most technologically advanced piece of Ram, the electronic Tunny-breaking Colossus.

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Michael Smith's The Emperor's Codes: Bletchley Park and the Breaking of Japan's Secret Ciphers aims to celebrate the achievements of the GCCS against Japanese cryptographic systems. He relies primarily on personal accounts (lots of recruitment narratives here) and the newly available British documents. Smith's introduction makes clear that, like Alan Stripp's Codebreaker in the Far East (1989), his book is meant as a corrective to the widely accepted oversimplification that America achieved all of the considerable Allied cryptographic success against the Japanese.

Smith's account is often confusing because of frequent changes of scenery and of subject, as he follows the various British codebreaking groups working on different parts of the Japanese problem in roughly chronological order. (Meanwhile, in Mombasa...) These groups were at Bletchley Park itself, at various GCCS outstations and attached to various American and Australian units. Moreover, his decision to concentrate on British work on Japanese codes is artificially limiting. In fact, the Allies cooperated in the effort, assigning some tasks to Britain, some to America and sharing still others. By concentrating on the British parts of the story, Smith occasionally slides from saying "we did it too" to "we did it alone".

The chief example of this intermittent chauvinism concerns work on the Japanese JN-25 naval code. In his introduction, Smith claims that it was broken not by the Americans but instead by the GCCS's John Tiltman in 1939, soon after the Japanese started using it. But when Japan entered the war, they were using a different edition, JN-25B, with a larger additive table and more secure code book, against which the GCCS and Op-20-G had made comparable meagre progress.

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The GCCS outstation specialising in JN-25 - whose work was disrupted by successive evacuations from Hong Kong to Singapore to Colombo to Kenya, and by painfully slow communications with Bletchley Park - also laboured under the distrust and even enmity of the Op-20-G and its Melbourne branch. It is indeed shameful that this group was so treated, but what emerges from Smith is not an account of the attack on JN-25 (which took place primarily at the Op-20-G and its outstations) but a grievance narrative. In other places, such as in his treatment of the Water Transport Code, which was solved by the SIS and the Delhi branch of the GCCS, Smith is more even-handed.

Balancing the demands of readability, historical significance, and accurate explication of cryptographic technique is no easy task. None of these books gets the mix entirely right. But all of them convey the intense intellectual engagement of the wartime codebreakers, the feeling described by the author of the wartime History of Hut Eight : "In finishing this account of Hut Eight's activities, I think that it should be said that while we broke German Naval Cyphers because it was our job to do so and because we believed it to be worthwhile, we also broke them because the problem was an interesting and amusing one. The work of Hut Eight combined, to a remarkable extent, a sense of urgency and importance and the pleasure of playing an intellectual game."

James A. Reeds is a mathematician in the mathematics and cryptography research department, AT&T Labs, Florham Park, New Jersey, United States.