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124-406: The bombe ( UK : / b ɒ m b / ) was an electro-mechanical device used by British cryptologists to help decipher German Enigma-machine -encrypted secret messages during World War II . The US Navy and US Army later produced their own machines to the same functional specification, albeit engineered differently both from each other and from Polish and British bombes. The British bombe

248-576: A West Germanic language that originated from the Anglo-Frisian dialects brought to Britain by Germanic settlers from various parts of what is now northwest Germany and the northern Netherlands. The resident population at this time was generally speaking Common Brittonic —the insular variety of Continental Celtic , which was influenced by the Roman occupation. This group of languages ( Welsh , Cornish , Cumbric ) cohabited alongside English into

372-417: A contradiction , since, by hypothesis, we assumed that P ( A ) = Y at the outset. This means that the initial assumption must have been incorrect, and so that (for this rotor setting) P ( A ) ≠ Y (this type of argument is termed reductio ad absurdum or "proof by contradiction"). The cryptanalyst hypothesised one plugboard interconnection for the bombe to test. The other stecker values and

496-414: A polyalphabetic substitution cipher, which turns plaintext into ciphertext and back again. The Enigma's scrambler contains rotors with 26 electrical contacts on each side, whose wiring diverts the current to a different position on the two sides. When a key is pressed on the keyboard, an electric current flows through an entry drum at the right-hand end of the scrambler, then through the set of rotors to

620-456: A reflecting drum (or reflector) which turns it back through the rotors and entry drum, and out to illuminate one of the lamps on the lampboard. At each key depression, the right-hand or "fast" rotor advances one position, which causes the encipherment to change. In addition, once per rotation, the right-hand rotor causes the middle rotor to advance; the middle rotor similarly causes the left-hand (or "slow") rotor to advance. Each rotor's position

744-559: A century as Received Pronunciation (RP). However, due to language evolution and changing social trends, some linguists argue that RP is losing prestige or has been replaced by another accent, one that the linguist Geoff Lindsey for instance calls Standard Southern British English. Others suggest that more regionally-oriented standard accents are emerging in England. Even in Scotland and Northern Ireland, RP exerts little influence in

868-413: A certain stretch of ciphertext, say, WSNPNLKLSTCS . The letters of the crib and the ciphertext were compared to establish pairings between the ciphertext and the crib plaintext. These were then graphed as in the diagram. It should be borne in mind that the relationships are reciprocal so that A in the plaintext associated with W in the ciphertext is the same as W in the plaintext associated with A in

992-473: A chance of drawing two of a kind. This probability can then be normalized by multiplying it by some coefficient, typically 26 in English. where c is the normalizing coefficient (26 for English), n a is the number of times the letter "a" appears in the text, and N is the length of the text. We can express the index of coincidence IC for a given letter-frequency distribution as a summation: where N

1116-458: A coincidence in this situation is 62.5% (6.25% for AA + 56.25% for BB), exactly the same as for the unencrypted "plaintext" case. In effect, the new alphabet produced by the substitution is just a uniform renaming of the original character identities, which does not affect whether they match. Now suppose that only one message (say, the second) is encrypted using the same substitution cipher (A,B)→(B,A). The following pairs can now be expected: Now

1240-448: A compression bar, the set of three rotors on their spindle can be removed from the machine and their sequence (called the "wheel order" at Bletchley Park) altered. Multiplying 17,576 by the six possible wheel orders gives 105,456 different ways that the scrambler can be set up. Although 105,456 is a large number, it does not guarantee security. A brute-force attack is possible: one could imagine using 100 code clerks who each tried to decode

1364-408: A cryptanalyst might suppose that P ( A ) = Y . Looking at position 10 of the crib:ciphertext comparison, we observe that A encrypts to T , or, expressed as a formula: Due to the function P being its own inverse, we can apply it to both sides of the equation and obtain the following: This gives us a relationship between P ( A ) and P ( T ) . If P ( A ) = Y , and for

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1488-613: A diagonal board. On 26 April 1940, HMS  Griffin captured a German trawler ( Schiff 26 , the Polares ) flying a Dutch flag; included in the capture were some Enigma keys for 23 to 26 April. Bletchley retrospectively attacked some messages sent during this period using the captured material and an ingenious Bombe menu where the Enigma fast rotors were all in the same position. In May and June 1940, Bletchley succeeded in breaking six days of naval traffic, 22–27 April 1940. Those messages were

1612-423: A given text. The chance of drawing a given letter in the text is (number of times that letter appears / length of the text). The chance of drawing that same letter again (without replacement) is (appearances − 1 / text length − 1). The product of these two values gives you the chance of drawing that letter twice in a row. One can find this product for each letter that appears in the text, then sum these products to get

1736-508: A greater movement, normally [əʊ], [əʉ] or [əɨ]. Dropping a morphological grammatical number , in collective nouns , is stronger in British English than North American English. This is to treat them as plural when once grammatically singular, a perceived natural number prevails, especially when applying to institutional nouns and groups of people. The noun 'police', for example, undergoes this treatment: Police are investigating

1860-406: A lesser class or social status and often discounted or considered of a low intelligence. Another contribution to the standardisation of British English was the introduction of the printing press to England in the mid-15th century. In doing so, William Caxton enabled a common language and spelling to be dispersed among the entirety of England at a much faster rate. Samuel Johnson's A Dictionary of

1984-409: A menu, derived from a crib. Because each Enigma machine had 26 inputs and outputs, the replica Enigma stacks are connected to each other using 26-way cables. In addition, each Enigma stack rotor setting is offset a number of places as determined by its position in the crib; for example, an Enigma stack corresponding to the fifth letter in the crib would be four places further on than that corresponding to

2108-575: A message could be found using the index of coincidence . Many major powers (including the Germans) could break Enigma traffic if they knew the rotor wiring. The German military knew the Enigma was weak. In 1930, the German army introduced an additional security feature, a plugboard ( Steckerbrett in German; each plug is a Stecker , and the British cryptologists also used the word) that further scrambled

2232-468: A message using 1000 distinct rotor settings. The Poles developed card catalogs so they could easily find rotor positions; Britain built " EINS " (the German word for one) catalogs. Less intensive methods were also possible. If all message traffic for a day used the same rotor starting position, then frequency analysis for each position could recover the polyalphabetic substitutions. If different rotor starting positions were used, then overlapping portions of

2356-453: A plugboard ( Steckerbrett in German) which swapped letters (indicated here by P ) before and after the main scrambler's change (indicated by S ). The plugboard connections were known to the cryptanalysts as Stecker values. If there had been no plugboard, it would have been relatively straightforward to test a rotor setting; a Typex machine modified to replicate Enigma could be set up and

2480-463: A practical illustration of the use of I.C., suppose that we have intercepted the following ciphertext message: (The grouping into five characters is just a telegraphic convention and has nothing to do with actual word lengths.) Suspecting this to be an English plaintext encrypted using a Vigenère cipher with normal A–Z components and a short repeating keyword, we can consider the ciphertext "stacked" into some number of columns, for example seven: If

2604-659: A process called T-glottalisation . National media, being based in London, have seen the glottal stop spreading more widely than it once was in word endings, not being heard as "no [ʔ] " and bottle of water being heard as "bo [ʔ] le of wa [ʔ] er". It is still stigmatised when used at the beginning and central positions, such as later , while often has all but regained /t/ . Other consonants subject to this usage in Cockney English are p , as in pa [ʔ] er and k as in ba [ʔ] er. In most areas of England and Wales, outside

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2728-520: A regional accent or dialect. However, about 2% of Britons speak with an accent called Received Pronunciation (also called "the King's English", "Oxford English" and " BBC English" ), that is essentially region-less. It derives from a mixture of the Midlands and Southern dialects spoken in London in the early modern period. It is frequently used as a model for teaching English to foreign learners. In

2852-668: A rotatable reflector (the M4 or Four-rotor Enigma) for communicating with its U-boats . This could be set up in 1.8×10 different ways. By late 1941 a change in German Navy fortunes in the Battle of the Atlantic , combined with intelligence reports, convinced Admiral Karl Dönitz that the Allies were able to read the German Navy's coded communications, and a fourth rotor with unknown wiring

2976-406: A set of five (hence there were now 60 possible wheel orders) for the army and air force Enigmas, and three out of eight (making 336 possible wheel orders) for the navy machines. In addition, ten leads were used on the plugboard, leaving only six letters unsteckered. This meant that the air force and army Enigmas could be set up in 1.5×10 ways. In 1941 the German navy introduced a version of Enigma with

3100-725: Is also due to London-centric influences. Examples of R-dropping are car and sugar , where the R is not pronounced. British dialects differ on the extent of diphthongisation of long vowels, with southern varieties extensively turning them into diphthongs, and with northern dialects normally preserving many of them. As a comparison, North American varieties could be said to be in-between. Long vowels /iː/ and /uː/ are usually preserved, and in several areas also /oː/ and /eː/, as in go and say (unlike other varieties of English, that change them to [oʊ] and [eɪ] respectively). Some areas go as far as not diphthongising medieval /iː/ and /uː/, that give rise to modern /aɪ/ and /aʊ/; that is, for example, in

3224-408: Is around 1.73, reflecting the unevenness of natural-language letter distributions. Sometimes values are reported without the normalizing denominator, for example 0.067 = 1.73/26 for English; such values may be called κ p ("kappa-plaintext") rather than IC, with κ r ("kappa-random") used to denote the denominator 1/ c (which is the expected coincidence rate for a uniform distribution of

3348-401: Is based on British English, but has more influence from American English , often grouped together due to their close proximity. British English, for example, is the closest English to Indian English, but Indian English has extra vocabulary and some English words are assigned different meanings. Index of coincidence Because letters in a natural language are not distributed evenly ,

3472-402: Is considerable value in truly indexing each I.C. against the value to be expected for the null hypothesis (usually: no match and a uniform random symbol distribution), so that in every situation the expected value for no correlation is 1.0. Thus, any form of I.C. can be expressed as the ratio of the number of coincidences actually observed to the number of coincidences expected (according to

3596-740: Is included in style guides issued by various publishers including The Times newspaper, the Oxford University Press and the Cambridge University Press . The Oxford University Press guidelines were originally drafted as a single broadsheet page by Horace Henry Hart, and were at the time (1893) the first guide of their type in English; they were gradually expanded and eventually published, first as Hart's Rules , and in 2002 as part of The Oxford Manual of Style . Comparable in authority and stature to The Chicago Manual of Style for published American English ,

3720-418: Is indicated by a letter of the alphabet showing through a window. The Enigma operator rotates the wheels by hand to set the start position for enciphering or deciphering a message. The three-letter sequence indicating the start position of the rotors is the "message key". There are 26 = 17,576 different message keys and different positions of the set of three rotors. By opening the lid of the machine and releasing

3844-496: Is not hard to generate random text with a frequency distribution similar to real text, artificially raising the coincidence count. Nevertheless, this technique can be used effectively to identify when two texts are likely to contain meaningful information in the same language using the same alphabet, to discover periods for repeating keys, and to uncover many other kinds of nonrandom phenomena within or among ciphertexts. Expected values for various languages are: The above description

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3968-553: Is only an introduction to use of the index of coincidence, which is related to the general concept of correlation . Various forms of Index of Coincidence have been devised; the "delta" I.C. (given by the formula above) in effect measures the autocorrelation of a single distribution, whereas a "kappa" I.C. is used when matching two text strings. Although in some applications constant factors such as c {\displaystyle c} and N {\displaystyle N} can be ignored, in more general situations there

4092-644: Is termed a known plaintext attack and had been used to a limited extent by the Poles, e.g., the Germans' use of "ANX" — "AN", German for "To", followed by "X" as a spacer. A £100,000 budget for the construction of Turing's machine was acquired and the contract to build the bombes was awarded to the British Tabulating Machine Company (BTM) at Letchworth . BTM placed the project under the direction of Harold 'Doc' Keen . Each machine

4216-414: Is the length of the text and n 1 through n c are the frequencies (as integers) of the c letters of the alphabet ( c = 26 for monocase English ). The sum of the n i is necessarily N . The products n ( n − 1) count the number of combinations of n elements taken two at a time. (Actually this counts each pair twice; the extra factors of 2 occur in both numerator and denominator of

4340-409: Is used in frequency analysis of substitution ciphers . Coincidences involving the letter E, for example, are relatively likely. So when any two English texts are compared, the coincidence count will be higher than when an English text and a foreign-language text are used. This effect can be subtle. For example, similar languages will have a higher coincidence count than dissimilar languages. Also, it

4464-547: The Chambers Dictionary , and the Collins Dictionary record actual usage rather than attempting to prescribe it. In addition, vocabulary and usage change with time; words are freely borrowed from other languages and other varieties of English, and neologisms are frequent. For historical reasons dating back to the rise of London in the ninth century, the form of language spoken in London and

4588-658: The East Midlands became standard English within the Court, and ultimately became the basis for generally accepted use in the law, government, literature and education in Britain. The standardisation of British English is thought to be from both dialect levelling and a thought of social superiority. Speaking in the Standard dialect created class distinctions; those who did not speak the standard English would be considered of

4712-576: The English language in England , or, more broadly, to the collective dialects of English throughout the British Isles taken as a single umbrella variety, for instance additionally incorporating Scottish English , Welsh English , and Northern Irish English . Tom McArthur in the Oxford Guide to World English acknowledges that British English shares "all the ambiguities and tensions [with]

4836-493: The Royal Spanish Academy with Spanish. Standard British English differs notably in certain vocabulary, grammar, and pronunciation features from standard American English and certain other standard English varieties around the world. British and American spelling also differ in minor ways. The accent, or pronunciation system, of standard British English, based in southeastern England, has been known for over

4960-490: The Scots language or Scottish Gaelic ). Each group includes a range of dialects, some markedly different from others. The various British dialects also differ in the words that they have borrowed from other languages. Around the middle of the 15th century, there were points where within the 5 major dialects there were almost 500 ways to spell the word though . Following its last major survey of English Dialects (1949–1950),

5084-573: The University of Leeds has started work on a new project. In May 2007 the Arts and Humanities Research Council awarded a grant to Leeds to study British regional dialects. The team are sifting through a large collection of examples of regional slang words and phrases turned up by the "Voices project" run by the BBC , in which they invited the public to send in examples of English still spoken throughout

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5208-610: The West Country and other near-by counties of the UK, the consonant R is not pronounced if not followed by a vowel, lengthening the preceding vowel instead. This phenomenon is known as non-rhoticity . In these same areas, a tendency exists to insert an R between a word ending in a vowel and a next word beginning with a vowel. This is called the intrusive R . It could be understood as a merger, in that words that once ended in an R and words that did not are no longer treated differently. This

5332-403: The ciphertext . Finding cribs was not at all straightforward; it required considerable familiarity with German military jargon and the communication habits of the operators. However, the codebreakers were aided by the fact that the Enigma would never encrypt a letter to itself. This helped in testing a possible crib against the ciphertext, as it could rule out a number of cribs and positions, where

5456-404: The "bulge" of a distribution, measures the discrepancy between the observed I.C. and the null value of 1.0. The number of cipher alphabets used in a polyalphabetic cipher may be estimated by dividing the expected bulge of the delta I.C. for a single alphabet by the observed bulge for the message, although in many cases (such as when a repeating key was used) better techniques are available. As

5580-629: The 21st century. RP, while long established as the standard English accent around the globe due to the spread of the British Empire , is distinct from the standard English pronunciation in some parts of the world; most prominently, RP notably contrasts with standard North American accents. In the 21st century, dictionaries like the Oxford English Dictionary , the Longman Dictionary of Contemporary English ,

5704-836: The English Language (1755) was a large step in the English-language spelling reform , where the purification of language focused on standardising both speech and spelling. By the early 20th century, British authors had produced numerous books intended as guides to English grammar and usage, a few of which achieved sufficient acclaim to have remained in print for long periods and to have been reissued in new editions after some decades. These include, most notably of all, Fowler's Modern English Usage and The Complete Plain Words by Sir Ernest Gowers . Detailed guidance on many aspects of writing British English for publication

5828-666: The Germanic schwein ) is the animal in the field bred by the occupied Anglo-Saxons and pork (like the French porc ) is the animal at the table eaten by the occupying Normans. Another example is the Anglo-Saxon cu meaning cow, and the French bœuf meaning beef. Cohabitation with the Scandinavians resulted in a significant grammatical simplification and lexical enrichment of the Anglo-Frisian core of English;

5952-422: The Germans that their messages were secure. Conversely, the Allies learned that the Germans had broken the naval cipher almost immediately from Enigma decrypts, but lost many ships due to the delay in changing the cipher. The following settings of the Enigma machine must be discovered to decipher German military Enigma messages. Once these are known, all the messages for that network for that day (or pair of days in

6076-490: The IC is higher for such texts than it would be for uniformly random text strings. What makes the IC especially useful is the fact that its value does not change if both texts are scrambled by the same single-alphabet substitution cipher , allowing a cryptanalyst to quickly detect that form of encryption. The index of coincidence provides a measure of how likely it is to draw two matching letters by randomly selecting two letters from

6200-922: The Oxford Manual is a fairly exhaustive standard for published British English that writers can turn to in the absence of specific guidance from their publishing house. British English is the basis of, and very similar to, Commonwealth English . Commonwealth English is English as spoken and written in the Commonwealth countries , though often with some local variation. This includes English spoken in Australia , Malta , New Zealand , Nigeria , and South Africa . It also includes South Asian English used in South Asia, in English varieties in Southeast Asia , and in parts of Africa. Canadian English

6324-712: The South East, there are significantly different accents; the Cockney accent spoken by some East Londoners is strikingly different from Received Pronunciation (RP). Cockney rhyming slang can be (and was initially intended to be) difficult for outsiders to understand, although the extent of its use is often somewhat exaggerated. Londoners speak with a mixture of accents, depending on ethnicity, neighbourhood, class, age, upbringing, and sundry other factors. Estuary English has been gaining prominence in recent decades: it has some features of RP and some of Cockney. Immigrants to

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6448-550: The UK in recent decades have brought many more languages to the country and particularly to London. Surveys started in 1979 by the Inner London Education Authority discovered over 125 languages being spoken domestically by the families of the inner city's schoolchildren. Notably Multicultural London English , a sociolect that emerged in the late 20th century spoken mainly by young, working-class people in multicultural parts of London . Since

6572-506: The US east coast. In May 1942 the US began using the convoy system and requiring a blackout of coastal cities so that ships would not be silhouetted against their lights, but this yielded only slightly improved security for Allied shipping. The Allies' failure to change their cipher for three months, together with the fact that Allied messages never contained any raw Enigma decrypts (or even mentioned that they were decrypting messages), helped convince

6696-640: The United Kingdom , as well as within the countries themselves. The major divisions are normally classified as English English (or English as spoken in England (which is itself broadly grouped into Southern English , West Country , East and West Midlands English and Northern English ), Northern Irish English (in Northern Ireland), Welsh English (not to be confused with the Welsh language ), and Scottish English (not to be confused with

6820-465: The West Scottish accent. Phonological features characteristic of British English revolve around the pronunciation of the letter R, as well as the dental plosive T and some diphthongs specific to this dialect. Once regarded as a Cockney feature, in a number of forms of spoken British English, /t/ has become commonly realised as a glottal stop [ʔ] when it is in the intervocalic position, in

6944-410: The adjective little is predominant elsewhere. Nevertheless, there is a meaningful degree of uniformity in written English within the United Kingdom, and this could be described by the term British English . The forms of spoken English, however, vary considerably more than in most other areas of the world where English is spoken and so a uniform concept of British English is more difficult to apply to

7068-488: The award of the grant in 2007, Leeds University stated: that they were "very pleased"—and indeed, "well chuffed"—at receiving their generous grant. He could, of course, have been "bostin" if he had come from the Black Country , or if he was a Scouser he would have been well "made up" over so many spondoolicks, because as a Geordie might say, £460,000 is a "canny load of chink". Most people in Britain speak with

7192-413: The bombe to be wired up according to the menu. The 'fast' drum rotated at a speed of 50.4 rpm in the first models and 120 rpm in later ones, when the time to set up and run through all 17,576 possible positions for one rotor order was about 20 minutes. The first bombe was named "Victory". It was installed in "Hut 1" at Bletchley Park on 18 March 1940. It was based on Turing's original design and so lacked

7316-399: The case of the German navy) could be decrypted. Internal settings (that required the lid of the Enigma machine to be opened) External settings (that could be changed without opening the Enigma machine) The bombe identified possible initial positions of the rotor cores and the stecker partner of a specified letter for a set of wheel orders. Manual techniques were then used to complete

7440-409: The change in the number of rotors used became official, and the Allies' ability to read German submarines' messages ceased until a snatch from a captured U-boat revealed not only the four-rotor machine's ability to emulate a three-rotor machine, but also that the fourth rotor did not move during a message. This along with the aforementioned retransmission eventually allowed the code breakers to figure out

7564-498: The ciphertext. At position 1 of the plaintext-ciphertext comparison, the letter A is associated with W , but A is also associated with P at position 4, K at position 7 and T at position 10. Building up these relationships into such a diagram provided the menu from which the bombe connections and drum start positions would be set up. In the illustration, there are three sequences of letters which form loops (or cycles or closures ), ATLK , TNS and TAWCN . The more loops in

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7688-419: The circuit continued to a plugboard located on the left-hand end panel, which was wired to imitate an Enigma reflector and then back through the outer pair of contacts. At each end of the "double-ended Enigma", there were sockets on the back of the machine, into which 26-way cables could be plugged. The bombe drums were arranged with the top one of the three simulating the left-hand rotor of the Enigma scrambler,

7812-567: The country. The BBC Voices project also collected hundreds of news articles about how the British speak English from swearing through to items on language schools. This information will also be collated and analysed by Johnson's team both for content and for where it was reported. "Perhaps the most remarkable finding in the Voices study is that the English language is as diverse as ever, despite our increased mobility and constant exposure to other accents and dialects through TV and radio". When discussing

7936-456: The crib letter A encrypted on it, and compared with the ciphertext, W . If they matched, the next letter would be tried, checking that T encrypted to S and so on for the entire length of the crib. If at any point the letters failed to match, the initial rotor setting would be rejected; most incorrect settings would be ruled out after testing just two letters. This test could be readily mechanised and applied to all 17 576 settings of

8060-413: The current in an Enigma passing in one direction through the scrambler, and the inner pair equivalent to the current flowing in the opposite direction. The interconnections within the drums between the two sets of input and output contacts were both identical to those of the relevant Enigma rotor. There was permanent wiring between the inner two sets of contacts of the three input/output plates. From there,

8184-492: The danger of bombes at Bletchley Park being lost if there were to be a bombing raid, bombe outstations were established, at Adstock , Gayhurst and Wavendon , all in Buckinghamshire . In June–August 1941 there were 4 to 6 bombes at Bletchley Park, and when Wavendon was completed, Bletchley, Adstock and Wavenden had a total of 24 to 30 bombes. When Gayhurst became operational there were a total of 40 to 46 bombes, and it

8308-564: The decryption process. In the words of Gordon Welchman , "... the task of the bombe was simply to reduce the assumptions of wheel order and scrambler positions that required 'further analysis' to a manageable number". The bombe was an electro-mechanical device that replicated the action of several Enigma machines wired together. A standard German Enigma employed, at any one time, a set of three rotors , each of which could be set in any of 26 positions. The standard British bombe contained 36 Enigma equivalents, each with three drums wired to produce

8432-481: The drums had to make reliable contact with the terminals on the template. There were 104 brushes per drum, 720 drums per bombe, and ultimately around 200 bombes. British English British English (abbreviations: BrE , en-GB , and BE ) is the set of varieties of the English language native to the United Kingdom of Great Britain and Northern Ireland . More narrowly, it can refer specifically to

8556-413: The encryption. This regularity was exploited by Welchman's "diagonal board" enhancement to the bombe, which vastly increased its efficiency. With six plug leads in use (leaving 14 letters "unsteckered"), there were 100,391,791,500 possible ways of setting up the plugboard. An important feature of the machine from a cryptanalyst's point of view, and indeed Enigma's Achilles' heel , was that the reflector in

8680-420: The entire column for each of the 26 possibilities A–Z for the key letter, and choosing the key letter that produces the highest correlation between the decrypted column letter frequencies and the relative letter frequencies for normal English text. That correlation, which we don't need to worry about normalizing, can be readily computed as where n i {\displaystyle n_{i}} are

8804-411: The false stops, built up the set of plugboard connections and established the positions of the rotor alphabet rings. Eventually, the result would be tested on a Typex machine that had been modified to replicate an Enigma, to see whether that decryption produced German language . A bombe run involved a cryptanalyst first obtaining a crib — a section of plaintext that was thought to correspond to

8928-485: The first breaks of Kriegsmarine messages of the war, "[b]ut though this success expanded Naval Section's knowledge of the Kriegsmarines's signals organization, it neither affected naval operations nor made further naval Enigma solutions possible." The second bombe, named " Agnus dei ", later shortened to "Agnes", or "Aggie", was equipped with Welchman's diagonal board, and was installed on 8 August 1940; "Victory"

9052-426: The first letter. Practical bombes used several stacks of rotors spinning together to test multiple hypotheses about possible setups of the Enigma machine, such as the order of the rotors in the stack. While Turing's bombe worked in theory, it required impractically long cribs to rule out sufficiently large numbers of settings. Gordon Welchman came up with a way of using the symmetry of the Enigma stecker to increase

9176-407: The formula and thus cancel out.) Each of the n i occurrences of the i -th letter matches each of the remaining n i − 1 occurrences of the same letter. There are a total of N ( N − 1) letter pairs in the entire text, and 1/ c is the probability of a match for each pair, assuming a uniform random distribution of the characters (the "null model"; see below). Thus, this formula gives

9300-458: The idea of two different morphemes, one that causes the double negation, and one that is used for the point or the verb. Standard English in the United Kingdom, as in other English-speaking nations, is widely enforced in schools and by social norms for formal contexts but not by any singular authority; for instance, there is no institution equivalent to the Académie française with French or

9424-403: The key size happens to have been the same as the assumed number of columns, then all the letters within a single column will have been enciphered using the same key letter, in effect a simple Caesar cipher applied to a random selection of English plaintext characters. The corresponding set of ciphertext letters should have a roughness of frequency distribution similar to that of English, although

9548-408: The key size is most likely five. If the actual size is five, we would expect a width of ten to also report a high I.C., since each of its columns also corresponds to a simple Caesar encipherment, and we confirm this. So we should stack the ciphertext into five columns: We can now try to determine the most likely key letter for each column considered separately, by performing trial Caesar decryption of

9672-523: The last southern Midlands accent to use the broad "a" in words like bath or grass (i.e. barth or grarss ). Conversely crass or plastic use a slender "a". A few miles northwest in Leicestershire the slender "a" becomes more widespread generally. In the town of Corby , five miles (8 km) north, one can find Corbyite which, unlike the Kettering accent, is largely influenced by

9796-518: The later Norman occupation led to the grafting onto that Germanic core of a more elaborate layer of words from the Romance branch of the European languages. This Norman influence entered English largely through the courts and government. Thus, English developed into a "borrowing" language of great flexibility and with a huge vocabulary . Dialects and accents vary amongst the four countries of

9920-443: The letter A is used 75% of the time, and the letter B is used 25% of the time. If two texts in this language are laid side by side, then the following pairs can be expected: Overall, the probability of a "coincidence" is 62.5% (56.25% for AA + 6.25% for BB). Now consider the case when both messages are encrypted using the simple monoalphabetic substitution cipher which replaces A with B and vice versa: The overall probability of

10044-428: The letter identities have been permuted (shifted by a constant amount corresponding to the key letter). Therefore, if we compute the aggregate delta I.C. for all columns ("delta bar"), it should be around 1.73. On the other hand, if we have incorrectly guessed the key size (number of columns), the aggregate delta I.C. should be around 1.00. So we compute the delta I.C. for assumed key sizes from one to ten: We see that

10168-409: The letters, both before and after they passed through the rotor-reflector system. The Enigma encryption is a self-inverse function , meaning that it substitutes letters reciprocally: if A is transformed into R , then R is transformed into A . The plugboard transformation maintained the self-inverse quality, but the plugboard wiring, unlike the rotor positions, does not change during

10292-493: The machine); and third, the number of plug-board leads had to remain relatively small so that the majority of letters were unsteckered . Six machines were built, one for each possible rotor order. The bomby were delivered in November 1938, but barely a month later the Germans introduced two additional rotors for loading into the Enigma scrambler, increasing the number of wheel orders by a factor of ten. Building another 54 bomby

10416-457: The mass internal migration to Northamptonshire in the 1940s and given its position between several major accent regions, it has become a source of various accent developments. In Northampton the older accent has been influenced by overspill Londoners. There is an accent known locally as the Kettering accent, which is a transitional accent between the East Midlands and East Anglian . It is

10540-450: The menu, the more candidate rotor settings the bombe could reject, and hence the fewer false stops. Alan Turing conducted a very substantial analysis (without any electronic aids) to estimate how many bombe stops would be expected according to the number of letters in the menu and the number of loops. Some of his results are given in the following table. Recent bombe simulations have shown similar results. The German military Enigma included

10664-560: The middle one the middle rotor, and the bottom one the right-hand rotor. The top drums were all driven in synchrony by an electric motor. For each full rotation of the top drums, the middle drums were incremented by one position, and likewise for the middle and bottom drums, giving the total of 26 × 26 × 26 = 17 576 positions of the 3-rotor Enigma scrambler. The drums were colour-coded according to which Enigma rotor they emulated: I red; II maroon; III green; IV yellow; V brown; VI cobalt (blue); VII jet (black); VIII silver. At each position of

10788-463: The modern period, but due to their remoteness from the Germanic languages , influence on English was notably limited . However, the degree of influence remains debated, and it has recently been argued that its grammatical influence accounts for the substantial innovations noted between English and the other West Germanic languages. Initially, Old English was a diverse group of dialects, reflecting

10912-466: The null model), using the particular test setup. From the foregoing, it is easy to see that the formula for kappa I.C. is where N {\displaystyle N} is the common aligned length of the two texts A and B , and the bracketed term is defined as 1 if the j {\displaystyle j} -th letter of text A matches the j {\displaystyle j} -th letter of text B , otherwise 0. A related concept,

11036-403: The observed column letter frequencies and f i {\displaystyle f_{i}} are the relative letter frequencies for English. When we try this, the best-fit key letters are reported to be " EVERY ," which we recognize as an actual word, and using that for Vigenère decryption produces the plaintext: from which one obtains: after word divisions have been restored at

11160-527: The positions of the indicator drums and the indicator unit on the Bombe's right-hand end panel. The operator then restarted the run. The candidate solutions, stops as they were called, were processed further to eliminate as many false stops as possible. Typically, there were many false bombe stops before the correct one was found. The candidate solutions for the set of wheel orders were subject to extensive further cryptanalytical work. This progressively eliminated

11284-503: The power of the bombe. His suggestion was an attachment called the diagonal board that further improved the bombe's effectiveness. The Polish cryptologic bomba (Polish: bomba kryptologiczna ; plural bomby ) had been useful only as long as three conditions were met. First, the form of the indicator had to include the repetition of the message key; second, the number of rotors available had to be limited to three, giving six different "wheel orders" (the three rotors and their order within

11408-427: The probability of a coincidence is only 37.5% (18.75% for AA + 18.75% for BB). This is noticeably lower than the probability when same-language, same-alphabet texts were used. Evidently, coincidences are more likely when the most frequent letters in each text are the same. The same principle applies to real languages like English, because certain letters, like E, occur much more frequently than other letters—a fact which

11532-411: The ratio of the total number of coincidences observed to the total number of coincidences that one would expect from the null model. The expected average value for the IC can be computed from the relative letter frequencies f i of the source language: If all c letters of an alphabet were equally probable, the expected index would be 1.0. The actual monographic IC for telegraphic English text

11656-409: The reflector and the other for output from the reflector, so that the reflected signal could pass back through a separate set of contacts. Each drum had 104 wire brushes, which made contact with the plate onto which they were loaded. The brushes and the corresponding set of contacts on the plate were arranged in four concentric circles of 26. The outer pair of circles (input and output) were equivalent to

11780-416: The ring settings were worked out by hand methods. To automate these logical deductions, the bombe took the form of an electrical circuit. Current flowed around the circuit near-instantaneously, and represented all the possible logical deductions which could be made at that position. To form this circuit, the bombe used several sets of Enigma rotor stacks wired up together according to the instructions given on

11904-421: The rotor setting under consideration S 10 ( Y ) = Q (say), we can deduce that While the crib does not allow us to determine what the values after the plugboard are, it does provide a constraint between them. In this case, it shows how P ( T ) is completely determined if P ( A ) is known. Likewise, we can also observe that T encrypts to L at position 8. Using S 8 , we can deduce

12028-422: The rotors, an electric current would or would not flow in each of the 26 wires, and this would be tested in the bombe's comparator unit. For a large number of positions, the test would lead to a logical contradiction , ruling out that setting. If the test did not lead to a contradiction, the machine would stop. The operator would then find the point at which the test passed, record the candidate solution by reading

12152-420: The rotors. However, with the plugboard, it was much harder to perform trial encryptions because it was unknown what the crib and ciphertext letters were transformed to by the plugboard. For example, in the first position, P ( A ) and P ( W ) were unknown because the plugboard settings were unknown. Turing's solution to working out the stecker values (plugboard connections) was to note that, even though

12276-453: The same alphabet, 0.0385=1/26 for English). English plaintext will generally fall somewhere in the range of 1.5 to 2.0 (normalized calculation). The index of coincidence is useful both in the analysis of natural-language plaintext and in the analysis of ciphertext ( cryptanalysis ). Even when only ciphertext is available for testing and plaintext letter identities are disguised, coincidences in ciphertext can be caused by coincidences in

12400-418: The same language using the same alphabet . (This technique has been used to examine the purported Bible code ). The causal coincidence count for such texts will be distinctly higher than the accidental coincidence count for texts in different languages, or texts using different alphabets, or gibberish texts. To see why, imagine an "alphabet" of only the two letters A and B. Suppose that in our "language",

12524-425: The same letter occurred in the same position in both the plaintext and the ciphertext. This was termed a crash at Bletchley Park. Once a suitable crib had been decided upon, the cryptanalyst would produce a menu for wiring up the bombe to test the crib against the ciphertext. The following is a simplified explanation of the process of constructing a menu. Suppose that the crib is ATTACKATDAWN to be tested against

12648-452: The same scrambling effect as the Enigma rotors. A bombe could run two or three jobs simultaneously. Each job would have a ' menu ' that had to be run against a number of different wheel orders. If the menu contained 12 or fewer letters, three different wheel orders could be run on one bombe; if more than 12 letters, only two. In order to simulate Enigma rotors, each rotor drum of the bombe had two complete sets of contacts, one for input towards

12772-413: The scrambler prevented a letter from being enciphered as itself. Any putative solution that gave, for any location, the same letter in the proposed plaintext and the ciphertext could therefore be eliminated. In the lead-up to World War II , the Germans made successive improvements to their military Enigma machines. By January 1939, additional rotors had been introduced so that three rotors were chosen from

12896-476: The set of rotors in use and their positions in the machine; the rotor core start positions for the message—the message key —and one of the wirings of the plugboard . The Enigma is an electro-mechanical rotor machine used for the encryption and decryption of secret messages. It was developed in Germany in the 1920s. The repeated changes of the electrical pathway from the keyboard to the lampboard implement

13020-401: The spoken language. Globally, countries that are former British colonies or members of the Commonwealth tend to follow British English, as is the case for English used by European Union institutions. In China, both British English and American English are taught. The UK government actively teaches and promotes English around the world and operates in over 200 countries . English is

13144-456: The steckered value for L as well using a similar argument, to get, say, Similarly, in position 6, K encrypts to L . As the Enigma machine is self-reciprocal, this means that at the same position L would also encrypt to K . Knowing this, we can apply the argument once more to deduce a value for P ( K ) , which might be: And again, the same sort of reasoning applies at position 7 to get: However, in this case, we have derived

13268-603: The theft of work tools worth £500 from a van at the Sprucefield park and ride car park in Lisburn. A football team can be treated likewise: Arsenal have lost just one of 20 home Premier League matches against Manchester City. This tendency can be observed in texts produced already in the 19th century. For example, Jane Austen , a British author, writes in Chapter 4 of Pride and Prejudice , published in 1813: All

13392-485: The thin 'B' reflector, and the rotor and ring were set to 'A', the pair acted as a 'B' reflector coupled with three rotors. Fortunately for the Allies, in December 1941, before the machine went into official service, a submarine accidentally sent a message with the fourth rotor in the wrong position, and then retransmitted the message with the rotor in the correct position to emulate the three-rotor machine. In February 1942

13516-403: The traditional accent of Newcastle upon Tyne , 'out' will sound as 'oot', and in parts of Scotland and North-West England, 'my' will be pronounced as 'me'. Long vowels /iː/ and /uː/ are diphthongised to [ɪi] and [ʊu] respectively (or, more technically, [ʏʉ], with a raised tongue), so that ee and oo in feed and food are pronounced with a movement. The diphthong [oʊ] is also pronounced with

13640-526: The underlying plaintext. This technique is used to cryptanalyze the Vigenère cipher , for example. For a repeating-key polyalphabetic cipher arranged into a matrix, the coincidence rate within each column will usually be highest when the width of the matrix is a multiple of the key length, and this fact can be used to determine the key length, which is the first step in cracking the system. Coincidence counting can help determine when two texts are written in

13764-441: The values for, say, P ( A ) or P ( W ) , were unknown, the crib still provided known relationships amongst these values; that is, the values after the plugboard transformation. Using these relationships, a cryptanalyst could reason from one to another and, potentially, derive a logical contradiction, in which case the rotor setting under consideration could be ruled out. A worked example of such reasoning might go as follows:

13888-750: The varied origins of the Anglo-Saxon kingdoms of England. One of these dialects, Late West Saxon , eventually came to dominate. The original Old English was then influenced by two waves of invasion: the first was by speakers of the Scandinavian branch of the Germanic family, who settled in parts of Britain in the eighth and ninth centuries; the second was the Normans in the 11th century, who spoke Old Norman and ultimately developed an English variety of this called Anglo-Norman . These two invasions caused English to become "mixed" to some degree (though it

14012-690: The wiring of both the 'beta' and 'gamma' fourth rotors. The first half of 1942 was the " Second Happy Time " for the German U-boats, with renewed success in attacking Allied shipping, as the US had just entered war unprepared for the onslaught, lacking in anti-submarine warfare (ASW) aircraft, ships, personnel, doctrine and organization. Also, the security of the new Enigma and the Germans' ability to read Allied convoy messages sent in Naval Cipher No. 3 contributed to their success. Between January and March 1942, German submarines sank 216 ships off

14136-402: The word 'British' and as a result can be used and interpreted in two ways, more broadly or more narrowly, within a range of blurring and ambiguity". Variations exist in formal (both written and spoken) English in the United Kingdom. For example, the adjective wee is almost exclusively used in parts of Scotland, north-east England, Northern Ireland, Ireland, and occasionally Yorkshire , whereas

14260-568: The world are good and agreeable in your eyes. However, in Chapter 16, the grammatical number is used. The world is blinded by his fortune and consequence. Some dialects of British English use negative concords, also known as double negatives . Rather than changing a word or using a positive, words like nobody, not, nothing, and never would be used in the same sentence. While this does not occur in Standard English, it does occur in non-standard dialects. The double negation follows

14384-422: Was about 7 feet (2.1 m) wide, 6 feet 6 inches (1.98 m) tall, 2 feet (0.61 m) deep and weighed about a ton. On the front of each bombe were 108 places where drums could be mounted. The drums were in three groups of 12 triplets. Each triplet, arranged vertically, corresponded to the three rotors of an Enigma scrambler. The bombe drums' input and output contacts went to cable connectors, allowing

14508-465: Was added to German Navy Enigmas used for U-boat communications, producing the Triton system, known at Bletchley Park as Shark . This was coupled with a thinner reflector design to make room for the extra rotor. The Triton was designed in such a way that it remained compatible with three-rotor machines when necessary: one of the extra 'fourth' rotors, the 'beta', was designed so that when it was paired with

14632-479: Was beyond the Poles' resources. Also, on 1 January 1939, the number of plug-board leads was increased to ten. The Poles therefore had to return to manual methods, the Zygalski sheets . Alan Turing designed the British bombe on a more general principle, the assumption of the presence of text, called a crib , that cryptanalysts could predict was likely to be present at a defined point in the message. This technique

14756-752: Was developed from a device known as the " bomba " ( Polish : bomba kryptologiczna ), which had been designed in Poland at the Biuro Szyfrów (Cipher Bureau) by cryptologist Marian Rejewski , who had been breaking German Enigma messages for the previous seven years, using it and earlier machines. The initial design of the British bombe was produced in 1939 at the UK Government Code and Cypher School (GC&CS) at Bletchley Park by Alan Turing , with an important refinement devised in 1940 by Gordon Welchman . The engineering design and construction

14880-660: Was expected that the total would increase to about 70 bombes run by some 700 Wrens (Women's Royal Naval Service) . But in 1942 with the introduction of the naval four-rotor Enigma, "far more than seventy bombes" would be needed. New outstations were established at Stanmore and Eastcote , and the Wavendon and Adstock bombes were moved to them, though the Gayhurst site was retained. The few bombes left at Bletchley Park were used for demonstration and training purposes only. Production of bombes by BTM at Letchworth in wartime conditions

15004-479: Was later returned to Letchworth to have a diagonal board fitted. The bombes were later moved from "Hut 1" to "Hut 11". The bombe was referred to by Group Captain Winterbotham as a "Bronze Goddess" because of its colour. The devices were more prosaically described by operators as being "like great big metal bookcases". During 1940, 178 messages were broken on the two machines, nearly all successfully. Because of

15128-422: Was never a truly mixed language in the strictest sense of the word; mixed languages arise from the cohabitation of speakers of different languages, who develop a hybrid tongue for basic communication). The more idiomatic, concrete and descriptive English is, the more it is from Anglo-Saxon origins. The more intellectual and abstract English is, the more it contains Latin and French influences, e.g. swine (like

15252-661: Was nowhere near as rapid as the Americans later achieved at NCR in Dayton, Ohio. Sergeant Jones was given the overall responsibility for Bombe maintenance by Edward Travis . Later Squadron Leader and not to be confused with Eric Jones , he was one of the original bombe maintenance engineers, and experienced in BTM techniques. Welchman said that later in the war when other people tried to maintain them, they realised how lucky they were to have him. About 15 million delicate wire brushes on

15376-453: Was the work of Harold Keen of the British Tabulating Machine Company . The first bombe, code-named Victory , was installed in March 1940 while the second version, Agnus Dei or Agnes , incorporating Welchman's new design, was working by August 1940. The bombe was designed to discover some of the daily settings of the Enigma machines on the various German military networks : specifically,

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