Editing Cryptanalysis (section)

===Ciphers from World War I and World War II===
{{See also|Cryptanalysis of the Enigma|Cryptanalysis of the Lorenz cipher}}
[[Image:Zimmermann-telegramm-offen.jpg|thumb|right|The decrypted [[Zimmermann Telegram]].]]

In [[World War I]], the breaking of the [[Zimmermann Telegram]] was instrumental in bringing the United States into the war. In [[World War II]], the [[Allies of World War II|Allies]] benefitted enormously from their joint success cryptanalysis of the German ciphers – including the [[Enigma machine]] and the [[Lorenz cipher]] – and Japanese ciphers, particularly [[Purple (cipher machine)|'Purple']] and [[JN-25]]. [[Ultra (cryptography)|'Ultra']] intelligence has been credited with everything between shortening the end of the European war by up to two years, to determining the eventual result. The war in the Pacific was similarly helped by [[Magic (cryptography)|'Magic']] intelligence.<ref>{{Harvnb|Smith|2000|p=4}}</ref>

Cryptanalysis of enemy messages played a significant part in the [[Allies of World War II|Allied]] victory in World War II. [[F. W. Winterbotham]], quoted the western Supreme Allied Commander, [[Dwight D. Eisenhower]], at the war's end as describing [[Ultra (cryptography)|Ultra]] intelligence as having been "decisive" to Allied victory.{{sfn|Winterbotham|2000|p=229}} [[Harry Hinsley|Sir Harry Hinsley]], official historian of British Intelligence in World War II, made a similar assessment about Ultra, saying that it shortened the war "by not less than two years and probably by four years"; moreover, he said that in the absence of Ultra, it is uncertain how the war would have ended.{{sfn|Hinsley|1993}}

In practice, frequency analysis relies as much on [[linguistics|linguistic]] knowledge as it does on statistics, but as ciphers became more complex, [[mathematics]] became more important in cryptanalysis. This change was particularly evident before and during [[World War II]], where efforts to crack [[Axis Powers|Axis]] ciphers required new levels of mathematical sophistication. Moreover, automation was first applied to cryptanalysis in that era with the Polish [[Bomba (cryptography)|Bomba]] device, the British [[Bombe]], the use of [[punched card]] equipment, and in the [[Colossus computers]] – the first electronic digital computers to be controlled by a program.<ref>{{Harvnb|Copeland|2006|p=1}}</ref><ref>{{Harvnb|Singh|1999|p=244}}</ref>

====Indicator====
With reciprocal machine ciphers such as the [[Lorenz cipher]] and the [[Enigma machine]] used by [[Nazi Germany]] during [[World War II]], each message had its own key. Usually, the transmitting operator informed the receiving operator of this message key by transmitting some plaintext and/or ciphertext before the enciphered message. This is termed the ''indicator'', as it indicates to the receiving operator how to set his machine to decipher the message.<ref>{{Harvnb|Churchhouse|2002|pp=33, 34}}</ref>

Poorly designed and implemented indicator systems allowed first [[Biuro Szyfrów|Polish cryptographers]]<ref>{{Harvnb|Budiansky|2000|pp=97–99}}</ref> and then the British cryptographers at [[Bletchley Park]]<ref>{{Harvnb|Calvocoressi|2001|p=66}}</ref> to break the Enigma cipher system. Similar poor indicator systems allowed the British to identify ''depths'' that led to the diagnosis of the [[Lorenz cipher|Lorenz SZ40/42]] cipher system, and the comprehensive breaking of its messages without the cryptanalysts seeing the cipher machine.<ref name="Tutte 1998">{{Harvnb|Tutte|1998}}</ref>

====Depth====
Sending two or more messages with the same key is an insecure process. To a cryptanalyst the messages are then said to be ''"in depth."''<ref>{{Harvnb|Churchhouse|2002|p=34}}</ref><ref>The [[Bletchley Park]] 1944 Cryptographic Dictionary defined a depth as <br /> 1. A series of code messages reciphered with the same, or the same part of a, reciphering key especially when written under one another so that all the groups (usually one in each message) that are reciphered with the same group of the subtractor lie under each other and form a 'column'.<br /> (b) two or more messages in a transposition cipher that are of the same length and have been enciphered on the same key;<br /> (c) two or more messages in a machine or similar cipher that have been enciphered on the same machine-setting or on the same key.<br /> 2. be in depth: (of messages). Stand to each other in any of the relationships described above.<br />{{Citation |title=The Bletchley Park 1944 Cryptographic Dictionary formatted by Tony Sale (c) 2001 |page=27 |url=https://www.codesandciphers.org.uk/documents/cryptdict/cryptxtt.pdf}}</ref> This may be detected by the messages having the same ''[[Enigma machine#Indicator|indicator]]'' by which the sending operator informs the receiving operator about the [[Key (cryptography)|key generator initial settings]] for the message.<ref>{{Harvnb|Churchhouse|2002|pp= 33, 86}}</ref>

Generally, the cryptanalyst may benefit from lining up identical enciphering operations among a set of messages. For example, the [[Gilbert Vernam|Vernam cipher]] enciphers by bit-for-bit combining plaintext with a long key using the "[[exclusive or]]" operator, which is also known as "[[Modular arithmetic|modulo-2 addition]]" (symbolized by ⊕ ):
::::Plaintext ⊕ Key = Ciphertext
Deciphering combines the same key bits with the ciphertext to reconstruct the plaintext:
::::Ciphertext ⊕ Key = Plaintext
(In modulo-2 arithmetic, addition is the same as subtraction.) When two such ciphertexts are aligned in depth, combining them eliminates the common key, leaving just a combination of the two plaintexts:
::::Ciphertext1 ⊕ Ciphertext2 = Plaintext1 ⊕ Plaintext2
The individual plaintexts can then be worked out linguistically by trying ''probable words'' (or phrases), also known as ''"cribs,"'' at various locations; a correct guess, when combined with the merged plaintext stream, produces intelligible text from the other plaintext component:
::::Cyphertext1 ⊕ Cyphertext2 ⊕ Plaintext1 = Plaintext2
The recovered fragment of the second plaintext can often be extended in one or both directions, and the extra characters can be combined with the merged plaintext stream to extend the first plaintext. Working back and forth between the two plaintexts, using the intelligibility criterion to check guesses, the analyst may recover much or all of the original plaintexts. (With only two plaintexts in depth, the analyst may not know which one corresponds to which ciphertext, but in practice this is not a large problem.) When a recovered plaintext is then combined with its ciphertext, the key is revealed:
::::Plaintext1 ⊕ Ciphertext1 = Key
Knowledge of a key then allows the analyst to read other messages encrypted with the same key, and knowledge of a set of related keys may allow cryptanalysts to diagnose the system used for constructing them.<ref name="Tutte 1998"/>