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尽管图灵的证明是在阿隆索 · 丘奇用他的 λ 微积分得到等价证明后不久发表的，但是图灵的方法比丘奇的方法更容易理解和直观。它还包含了一个通用机器的概念(现在被称为通用图灵机) ，其理念是这样一个机器可以执行任何其他计算机器的任务(实际上就像 Church 的 lambda 演算一样)。根据丘奇-图灵论文，图灵机和 lambda 微积分能够计算任何可计算的东西。约翰·冯·诺伊曼承认现代计算机的核心概念应归功于图灵的论文。“冯 · 诺依曼... ... 坚定地向我和其他人强调，基本概念应归功于图灵ーー这是巴贝奇、洛夫莱斯和其他人所没有预料到的。”斯坦利 · 弗兰克尔给布莱恩 · 兰德尔的信，1972年，引自杰克 · 科普兰(2004)《本质图灵》 ，第22页。直到今天，图灵机仍然是计算理论的中心研究对象。

尽管图灵的证明是在阿隆索 · 丘奇用他的 λ 微积分得到等价证明后不久发表的，但是图灵的方法比丘奇的方法更容易理解和直观。它还包含了一个通用机器的概念(现在被称为通用图灵机) ，其理念是这样一个机器可以执行任何其他计算机器的任务(实际上就像 Church 的 lambda 演算一样)。根据丘奇-图灵论文，图灵机和 lambda 微积分能够计算任何可计算的东西。约翰·冯·诺伊曼承认现代计算机的核心概念应归功于图灵的论文。“冯 · 诺依曼... ... 坚定地向我和其他人强调，基本概念应归功于图灵ーー这是巴贝奇、洛夫莱斯和其他人所没有预料到的。”斯坦利 · 弗兰克尔给布莱恩 · 兰德尔的信，1972年，引自杰克 · 科普兰(2004)《本质图灵》 ，第22页。直到今天，图灵机仍然是计算理论的中心研究对象。

From September 1936 to July 1938, Turing spent most of his time studying under Church at [[Princeton University]],<ref name="bowen19" /> in the second year as a [[Jane Eliza Procter Fellowship|Jane Eliza Procter Visiting Fellow]]. In addition to his purely mathematical work, he studied cryptology and also built three of four stages of an electro-mechanical [[binary multiplier]].<ref>{{Harvnb|Hodges|1983|p=138}}</ref> In June 1938, he obtained his PhD from the [[Princeton University Department of Mathematics|Department of Mathematics]] at Princeton;<ref>{{Cite journal

From September 1936 to July 1938, Turing spent most of his time studying under Church at [[Princeton University]],<ref name="bowen19" /> in the second year as a [[Jane Eliza Procter Fellowship|Jane Eliza Procter Visiting Fellow]]. In addition to his purely mathematical work, he studied cryptology and also built three of four stages of an electro-mechanical [[binary multiplier]].<ref>{{Harvnb|Hodges|1983|p=138}}</ref> In June 1938, he obtained his PhD from the [[Princeton University Department of Mathematics|Department of Mathematics]] at Princeton;<ref>{{Cite journal

从1936年9月到1938年7月，图灵大部分时间在普林斯顿大学的教堂学习，第二年成为 Jane Eliza Procter 访问学者。除了纯粹的数学工作外，他还研究了密码学，并建造了一台电子机械乘法器的四个阶段中的三个阶段。1938年6月，他获得了普林斯顿大学数学系的博士学位; 他的博士论文，基于序数的逻辑系统，引入了序数逻辑的概念和相对计算的概念，在这个概念中，图灵机被所谓的神谕加强，使得对图灵机无法解决的问题的研究成为可能。约翰·冯·诺伊曼想聘请他做博士后助理，但他回到了英国。约翰·冯·诺伊曼: 现代计算机、博弈论、核威慑及其他科学天才，

从1936年9月到1938年7月，图灵大部分时间在普林斯顿大学的教堂学习，第二年成为 Jane Eliza Procter 访问学者。除了纯粹的数学工作外，他还研究了密码学，并建造了一台电子机械乘法器的四个阶段中的三个阶段。1938年6月，他获得了普林斯顿大学数学系的博士学位; 他的博士论文，基于序数的逻辑系统，引入了序数逻辑的概念和相对计算的概念，在这个概念中，图灵机被所谓的神谕加强，使得对图灵机无法解决的问题的研究成为可能。约翰·冯·诺伊曼想聘请他做博士后助理，但他回到了英国。约翰·冯·诺伊曼: 现代计算机、博弈论、核威慑及其他科学天才，

==Career and research==

==Career and research==

When Turing returned to Cambridge, he attended lectures given in 1939 by [[Ludwig Wittgenstein]] about the [[foundations of mathematics]].<ref>{{Harvnb|Hodges|1983|p=152}}</ref> The lectures have been reconstructed verbatim, including interjections from Turing and other students, from students' notes.<ref>[[Cora Diamond]] (ed.), ''Wittgenstein's Lectures on the Foundations of Mathematics'', University of Chicago Press, 1976</ref> Cora Diamond (ed.), Wittgenstein's Lectures on the Foundations of Mathematics, University of Chicago Press, 1976Turing and Wittgenstein argued and disagreed, with Turing defending [[Philosophy of mathematics#Formalism|formalism]] and Wittgenstein propounding his view that mathematics does not discover any absolute truths, but rather invents them.<ref>{{Harvnb|Hodges|1983|pp=153–154}}</ref>

When Turing returned to Cambridge, he attended lectures given in 1939 by [[Ludwig Wittgenstein]] about the [[foundations of mathematics]].<ref>{{Harvnb|Hodges|1983|p=152}}</ref> The lectures have been reconstructed verbatim, including interjections from Turing and other students, from students' notes.<ref>[[Cora Diamond]] (ed.), ''Wittgenstein's Lectures on the Foundations of Mathematics'', University of Chicago Press, 1976</ref> Cora Diamond (ed.), Wittgenstein's Lectures on the Foundations of Mathematics, University of Chicago Press, 1976Turing and Wittgenstein argued and disagreed, with Turing defending [[Philosophy of mathematics#Formalism|formalism]] and Wittgenstein propounding his view that mathematics does not discover any absolute truths, but rather invents them.<ref>{{Harvnb|Hodges|1983|pp=153–154}}</ref>

当图灵回到剑桥后，他参加了路德维希·维特根斯坦1939年关于数学基础的讲座。这些讲座都是逐字逐句重建的，包括图灵和其他学生的感叹词，以及学生的笔记。图灵和维特根斯坦在1976年发表的《维特根斯坦关于数学基础的演讲》一书中对此提出了异议，图灵为形式主义辩护，维特根斯坦则提出了他的观点，即数学不是发现任何绝对真理，而是发明它们。

当图灵回到剑桥后，他参加了路德维希·维特根斯坦1939年关于数学基础的讲座。这些讲座都是逐字逐句重建的，包括图灵和其他学生的感叹词，以及学生的笔记。图灵和维特根斯坦在1976年发表的《维特根斯坦关于数学基础的演讲》一书中对此提出了异议，图灵为形式主义辩护，维特根斯坦则提出了他的观点，即数学不是发现任何绝对真理，而是发明它们。

===Cryptanalysis===

===Cryptanalysis===

During the Second World War, Turing was a leading participant in the breaking of German ciphers at [[Bletchley Park]]. The historian and wartime codebreaker [[Asa Briggs]] has said, "You needed exceptional talent, you needed genius at Bletchley and Turing's was that genius."<ref>{{Cite AV media | last = Briggs | first = Asa | author-link = Asa Briggs | title = Britain's Greatest Codebreaker | type = TV broadcast | publisher = [[Channel 4|UK Channel 4]] | date = 21 November 2011}}</ref>

During the Second World War, Turing was a leading participant in the breaking of German ciphers at [[Bletchley Park]]. The historian and wartime codebreaker [[Asa Briggs]] has said, "You needed exceptional talent, you needed genius at Bletchley and Turing's was that genius."<ref>{{Cite AV media | last = Briggs | first = Asa | author-link = Asa Briggs | title = Britain's Greatest Codebreaker | type = TV broadcast | publisher = [[Channel 4|UK Channel 4]] | date = 21 November 2011}}</ref>

第二次世界大战期间，图灵是在 Bletchley Park 破解德国密码的主要参与者。历史学家、战时密码破译者阿萨•布里格斯(Asa Briggs)曾说过: “你需要杰出的人才，你需要布莱奇利的天才，而图灵就是这种天才。”

第二次世界大战期间，图灵是在 Bletchley Park 破解德国密码的主要参与者。历史学家、战时密码破译者阿萨•布里格斯(Asa Briggs)曾说过: “你需要杰出的人才，你需要布莱奇利的天才，而图灵就是这种天才。”

From September 1938, Turing worked part-time with the [[Government Code and Cypher School]] (GC&CS), the British codebreaking organisation. He concentrated on [[cryptanalysis of the Enigma|cryptanalysis of the Enigma cipher machine]] used by [[Nazi Germany]], together with [[Dilly Knox]], a senior GC&CS codebreaker.<ref>{{Cite book | author-link = Jack Copeland | last = Copeland | first = Jack | chapter = Colossus and the Dawning of the Computer Age | page = 352 | title = Action This Day | publisher = Bantam | date = 2001 | isbn = 978-0-593-04910-5  | editor-first1 = Michael | editor-last1 = Smith  | editor-first2 = Ralph | editor-last2 = Erskine }}</ref> Soon after the July 1939 meeting near [[Warsaw]] at which the [[Polish Cipher Bureau]] gave the British and French details of the wiring of [[Enigma rotor details|Enigma machine's rotors]] and their method of decrypting [[Enigma machine]]'s messages, Turing and Knox developed a broader solution.<ref>{{Harvnb|Copeland|2004a|p=217}}</ref> The Polish method relied on an insecure [[Cryptanalysis#Indicator|indicator]] procedure that the Germans were likely to change, which they in fact did in May 1940. Turing's approach was more general, using [[Cryptanalysis of the Enigma#Crib-based decryption|crib-based decryption]] for which he produced the functional specification of the [[bombe]] (an improvement on the Polish [[Bomba (cryptography)|Bomba]]).<ref>{{cite news |last=Clark |first=Liat |url=https://www.wired.co.uk/news/archive/2012-06/18/turing-contributions?page=all |title=Turing's achievements: codebreaking, AI and the birth of computer science (Wired UK) |work=Wired |date=18 June 2012 |access-date=31 October 2013 |archive-url=https://web.archive.org/web/20131102122933/http://www.wired.co.uk/news/archive/2012-06/18/turing-contributions?page=all |archive-date=2 November 2013 |url-status=live }}</ref>

From September 1938, Turing worked part-time with the [[Government Code and Cypher School]] (GC&CS), the British codebreaking organisation. He concentrated on [[cryptanalysis of the Enigma|cryptanalysis of the Enigma cipher machine]] used by [[Nazi Germany]], together with [[Dilly Knox]], a senior GC&CS codebreaker.<ref>{{Cite book | author-link = Jack Copeland | last = Copeland | first = Jack | chapter = Colossus and the Dawning of the Computer Age | page = 352 | title = Action This Day | publisher = Bantam | date = 2001 | isbn = 978-0-593-04910-5  | editor-first1 = Michael | editor-last1 = Smith  | editor-first2 = Ralph | editor-last2 = Erskine }}</ref> Soon after the July 1939 meeting near [[Warsaw]] at which the [[Polish Cipher Bureau]] gave the British and French details of the wiring of [[Enigma rotor details|Enigma machine's rotors]] and their method of decrypting [[Enigma machine]]'s messages, Turing and Knox developed a broader solution.<ref>{{Harvnb|Copeland|2004a|p=217}}</ref> The Polish method relied on an insecure [[Cryptanalysis#Indicator|indicator]] procedure that the Germans were likely to change, which they in fact did in May 1940. Turing's approach was more general, using [[Cryptanalysis of the Enigma#Crib-based decryption|crib-based decryption]] for which he produced the functional specification of the [[bombe]] (an improvement on the Polish [[Bomba (cryptography)|Bomba]]).<ref>{{cite news |last=Clark |first=Liat |url=https://www.wired.co.uk/news/archive/2012-06/18/turing-contributions?page=all |title=Turing's achievements: codebreaking, AI and the birth of computer science (Wired UK) |work=Wired |date=18 June 2012 |access-date=31 October 2013 |archive-url=https://web.archive.org/web/20131102122933/http://www.wired.co.uk/news/archive/2012-06/18/turing-contributions?page=all |archive-date=2 November 2013 |url-status=live }}</ref>

从1938年9月开始，图灵在英国破译组织---- 英国政府通讯总部破译协会做兼职工作。他和资深 gc & cs 密码破译员 Dilly Knox 一起，专注于对纳粹德国使用的恩尼格玛密码机的密码分析。1939年7月在华沙附近的会议上，波兰密码局向英国和法国提供了关于恩尼格玛密码机转子的线路和解密密码机信息的方法的细节，不久之后，图灵和诺克斯开发了一个更广泛的解决方案。波兰的方法依赖于一种不安全的指示程序，德国人可能会改变这种程序，事实上他们在1940年5月就已经改变了。图灵的方法更为普遍，他使用了基于抄袭的解密方法，从而产生了炸弹的规格化(相对于波兰炸弹的改进)。[[File:Turing flat.jpg|thumb|right|Two cottages in the stable yard at [[Bletchley Park]]. Turing worked here in 1939 and 1940, before moving to [[Hut 8]].|链接=Special:FilePath/Turing_flat.jpg]]On 4 September 1939, the day after the UK declared war on Germany, Turing reported to Bletchley Park, the wartime station of GC&CS.<ref name="Copeland2006p378">Copeland, 2006 p.&nbsp;378.</ref> Like all others who came to Bletchley, he was required to sign the [[Official Secrets Act 1939|Official Secrets Act]], in which he agreed not to disclose anything about his work at Bletchley, with severe legal penalties for violating the Act.<ref name="Collins">{{cite web |last=Collins |first=Jeremy |title=Alan Turing and the Hidden Heroes of Bletchley Park: A Conversation with Sir John Dermot Turing |date=24 June 2020 |website=nationalww2museum.org |location=New Orleans |publisher=The National WWII Museum |url=https://www.nationalww2museum.org/war/articles/alan-turing-betchley-park |access-date=24 August 2021}}</ref>

 

 

从1938年9月开始，图灵在英国破译组织---- 英国政府通讯总部破译协会做兼职工作。他和资深 gc & cs 密码破译员 Dilly Knox 一起，专注于对纳粹德国使用的恩尼格玛密码机的密码分析。1939年7月在华沙附近的会议上，波兰密码局向英国和法国提供了关于恩尼格玛密码机转子的线路和解密密码机信息的方法的细节，不久之后，图灵和诺克斯开发了一个更广泛的解决方案。波兰的方法依赖于一种不安全的指示程序，德国人可能会改变这种程序，事实上他们在1940年5月就已经改变了。图灵的方法更为普遍，他使用了基于抄袭的解密方法，从而产生了炸弹的规格化(相对于波兰炸弹的改进)。

 

【最终版】从1938年9月开始，图灵在英国密码破译组织政府Code and Cypher School (GC&CS)兼职工作。他和GC&CS高级密码破解员迪利·诺克斯一起，专注于对纳粹德国使用的谜机进行密码分析。1939年7月，在华沙附近的一次会议上，波兰密码局向英国和法国提供了英格玛密码机转子连接的细节，以及他们对英格玛密码机信息进行解密的方法。此后不久，图灵和诺克斯提出了一个更广泛的解决方案。波兰的方法依赖于一个不安全的指示程序，德国人很可能会改变，他们实际上在1940年5月就改变了。图灵的方法更一般化，他使用了基于婴儿床的解密方法，为此他制作了炸弹机的功能规格(对波兰炸弹机的改进)。[[File:Turing flat.jpg|thumb|right|Two cottages in the stable yard at [[Bletchley Park]]. Turing worked here in 1939 and 1940, before moving to [[Hut 8]].|链接=Special:FilePath/Turing_flat.jpg]]On 4 September 1939, the day after the UK declared war on Germany, Turing reported to Bletchley Park, the wartime station of GC&CS.<ref name="Copeland2006p378">Copeland, 2006 p.&nbsp;378.</ref> Like all others who came to Bletchley, he was required to sign the [[Official Secrets Act 1939|Official Secrets Act]], in which he agreed not to disclose anything about his work at Bletchley, with severe legal penalties for violating the Act.<ref name="Collins">{{cite web |last=Collins |first=Jeremy |title=Alan Turing and the Hidden Heroes of Bletchley Park: A Conversation with Sir John Dermot Turing |date=24 June 2020 |website=nationalww2museum.org |location=New Orleans |publisher=The National WWII Museum |url=https://www.nationalww2museum.org/war/articles/alan-turing-betchley-park |access-date=24 August 2021}}</ref>

1939年9月4日，也就是英国对德国宣战的第二天，图灵向 Bletchley Park 报告了战时的 gc & cs 基地。2006 p. 378.像所有其他来到布莱奇利的人一样，他被要求签署《官方保密法案》(Official Secrets Act) ，在该法案中，他同意不披露任何有关他在布莱奇利工作的信息，违反该法案将受到严厉的法律制裁。

1939年9月4日，也就是英国对德国宣战的第二天，图灵向 Bletchley Park 报告了战时的 gc & cs 基地。2006 p. 378.像所有其他来到布莱奇利的人一样，他被要求签署《官方保密法案》(Official Secrets Act) ，在该法案中，他同意不披露任何有关他在布莱奇利工作的信息，违反该法案将受到严厉的法律制裁。

Specifying the bombe was the first of five major cryptanalytical advances that Turing made during the war. The others were: deducing the indicator procedure used by the German navy; developing a statistical procedure dubbed ''[[Banburismus]]'' for making much more efficient use of the bombes; developing a procedure dubbed ''[[Turingery]]'' for working out the cam settings of the wheels of the [[Lorenz SZ 40/42]] (''Tunny'') cipher machine and, towards the end of the war, the development of a portable [[secure voice]] scrambler at [[Her Majesty's Government Communications Centre|Hanslope Park]] that was codenamed ''Delilah''.

Specifying the bombe was the first of five major cryptanalytical advances that Turing made during the war. The others were: deducing the indicator procedure used by the German navy; developing a statistical procedure dubbed ''[[Banburismus]]'' for making much more efficient use of the bombes; developing a procedure dubbed ''[[Turingery]]'' for working out the cam settings of the wheels of the [[Lorenz SZ 40/42]] (''Tunny'') cipher machine and, towards the end of the war, the development of a portable [[secure voice]] scrambler at [[Her Majesty's Government Communications Centre|Hanslope Park]] that was codenamed ''Delilah''.

指定炸弹是图灵在战争期间取得的五大密码分析进展中的第一个。其他的是: 推导德国海军使用的指示程序; 开发一个统计程序称为 Banburismus，以便更有效地利用炸弹; 开发一个程序称为 Turingery，用于计算出 Lorenz SZ 40/42(Tunny)密码机车轮的凸轮设置; 战争快结束时，在 Hanslope Park 开发了一个代号为 Delilah 的便携式安全语音扰码器。

指定炸弹是图灵在战争期间取得的五大密码分析进展中的第一个。其他的是: 推导德国海军使用的指示程序; 开发一个统计程序称为 Banburismus，以便更有效地利用炸弹; 开发一个程序称为 Turingery，用于计算出 Lorenz SZ 40/42(Tunny)密码机车轮的凸轮设置; 战争快结束时，在 Hanslope Park 开发了一个代号为 Delilah 的便携式安全语音扰码器。

By using statistical techniques to optimise the trial of different possibilities in the code breaking process, Turing made an innovative contribution to the subject. He wrote two papers discussing mathematical approaches, titled ''The Applications of Probability to Cryptography''<ref>{{cite web | last = Turing | first = Alan | year = c. 1941 | title = The Applications of Probability to Cryptography | id = The National Archives (United Kingdom): HW 25/37 | url = http://discovery.nationalarchives.gov.uk/details/r/C11510465  | access-date = 25 March 2015 | archive-url = https://web.archive.org/web/20150407234050/http://discovery.nationalarchives.gov.uk/details/r/C11510465 | archive-date = 7 April 2015 | url-status = live }}</ref> and ''Paper on Statistics of Repetitions'',<ref>{{cite web | last = Turing | first = Alan | year = c. 1941 | title = Paper on Statistics of Repetitions | id = The National Archives (United Kingdom): HW 25/38 | url = http://discovery.nationalarchives.gov.uk/details/r/C11510466  | access-date = 25 March 2015 | archive-url = https://web.archive.org/web/20150408013845/http://discovery.nationalarchives.gov.uk/details/r/C11510466 | archive-date = 8 April 2015 | url-status = live }}</ref> which were of such value to GC&CS and its successor [[Government Communications Headquarters|GCHQ]] that they were not released to the [[The National Archives (United Kingdom)|UK National Archives]] until April 2012, shortly before the centenary of his birth. A GCHQ mathematician, "who identified himself only as Richard," said at the time that the fact that the contents had been restricted under the Official Secrets Act for some 70 years demonstrated their importance, and their relevance to post-war cryptanalysis:<ref name="bbcrichard">{{cite news |last=Vallance |first=Chris |title=Alan Turing papers on code breaking released by GCHQ |url=https://www.bbc.co.uk/news/technology-17771962 |access-date=20 April 2012 |work=BBC News |date=19 April 2012 |archive-url=https://web.archive.org/web/20121004192554/http://www.bbc.co.uk/news/technology-17771962 |archive-date=4 October 2012 |url-status=live }}</ref>{{blockquote|text=[He] said the fact that the contents had been restricted "shows what a tremendous importance it has in the foundations of our subject". ... The papers detailed using "mathematical analysis to try and determine which are the more likely settings so that they can be tried as quickly as possible." ... Richard said that GCHQ had now "squeezed the juice" out of the two papers and was "happy for them to be released into the public domain".}}通过使用统计技术来优化破译过程中不同可能性的试验，图灵对这个课题做出了创新性的贡献。他写了两篇讨论数学方法的论文，题目是《概率在密码学中的应用》和《重复次数的统计学论文》 ，这两篇论文对 gc & cs 和它的继任者 GCHQ 如此重要，以至于直到2012年4月，也就是他诞辰100周年之前不久，才向英国国家档案馆发表。一位“自称理查德”的英国国家通信总局(GCHQ)数学家当时表示，《官方机密法案》(Official Secrets Act)约70年来一直对其内容加以限制，这一事实证明了它们的重要性，以及它们与战后密码分析的关联性:

By using statistical techniques to optimise the trial of different possibilities in the code breaking process, Turing made an innovative contribution to the subject. He wrote two papers discussing mathematical approaches, titled ''The Applications of Probability to Cryptography''<ref>{{cite web | last = Turing | first = Alan | year = c. 1941 | title = The Applications of Probability to Cryptography | id = The National Archives (United Kingdom): HW 25/37 | url = http://discovery.nationalarchives.gov.uk/details/r/C11510465  | access-date = 25 March 2015 | archive-url = https://web.archive.org/web/20150407234050/http://discovery.nationalarchives.gov.uk/details/r/C11510465 | archive-date = 7 April 2015 | url-status = live }}</ref> and ''Paper on Statistics of Repetitions'',<ref>{{cite web | last = Turing | first = Alan | year = c. 1941 | title = Paper on Statistics of Repetitions | id = The National Archives (United Kingdom): HW 25/38 | url = http://discovery.nationalarchives.gov.uk/details/r/C11510466  | access-date = 25 March 2015 | archive-url = https://web.archive.org/web/20150408013845/http://discovery.nationalarchives.gov.uk/details/r/C11510466 | archive-date = 8 April 2015 | url-status = live }}</ref> which were of such value to GC&CS and its successor [[Government Communications Headquarters|GCHQ]] that they were not released to the [[The National Archives (United Kingdom)|UK National Archives]] until April 2012, shortly before the centenary of his birth. A GCHQ mathematician, "who identified himself only as Richard," said at the time that the fact that the contents had been restricted under the Official Secrets Act for some 70 years demonstrated their importance, and their relevance to post-war cryptanalysis:<ref name="bbcrichard">{{cite news |last=Vallance |first=Chris |title=Alan Turing papers on code breaking released by GCHQ |url=https://www.bbc.co.uk/news/technology-17771962 |access-date=20 April 2012 |work=BBC News |date=19 April 2012 |archive-url=https://web.archive.org/web/20121004192554/http://www.bbc.co.uk/news/technology-17771962 |archive-date=4 October 2012 |url-status=live }}</ref>{{blockquote|text=[He] said the fact that the contents had been restricted "shows what a tremendous importance it has in the foundations of our subject". ... The papers detailed using "mathematical analysis to try and determine which are the more likely settings so that they can be tried as quickly as possible." ... Richard said that GCHQ had now "squeezed the juice" out of the two papers and was "happy for them to be released into the public domain".}}通过使用统计技术来优化破译过程中不同可能性的试验，图灵对这个课题做出了创新性的贡献。他写了两篇讨论数学方法的论文，题目是《概率在密码学中的应用》和《重复次数的统计学论文》 ，这两篇论文对 gc & cs 和它的继任者 GCHQ 如此重要，以至于直到2012年4月，也就是他诞辰100周年之前不久，才向英国国家档案馆发表。一位“自称理查德”的英国国家通信总局(GCHQ)数学家当时表示，《官方机密法案》(Official Secrets Act)约70年来一直对其内容加以限制，这一事实证明了它们的重要性，以及它们与战后密码分析的关联性: