536:
ABCDEFGHIJKLMNOPQRSTUVWXYZ ......... is impossible (G enciphers to D, yet B enciphers to G) G--B-H---X-Q ABCDEFGHIJKLMNOPQRSTUVWXYZ ......... is impossible (B enciphers to H, yet H enciphers to J) G--B-H---X-Q ABCDEFGHIJKLMNOPQRSTUVWXYZ ......... is impossible (Q apparently enciphers to Q) G--B-H---X-Q ABCDEFGHIJKLMNOPQRSTUVWXYZ ......... is impossible (G apparently enciphers to G) G--B-H---X-Q ABCDEFGHIJKLMNOPQRSTUVWXYZ ......... is impossible (G enciphers to H, yet H enciphers to M) G--B-H---X-Q ABCDEFGHIJKLMNOPQRSTUVWXYZ ......... is possible G--B-H---X-Q ABCDEFGHIJKLMNOPQRSTUVWXYZ ......... is possible G--B-H---X-Q ABCDEFGHIJKLMNOPQRSTUVWXYZ ......... is possible G--B-H---X-Q ABCDEFGHIJKLMNOPQRSTUVWXYZ ......... is impossible (H enciphers to Q, yet Q enciphers to W) G--B-H---X-Q ABCDEFGHIJKLMNOPQRSTUVWXYZ ......... is impossible (X enciphers to V, yet Q enciphers to X) G--B-H---X-Q ABCDEFGHIJKLMNOPQRSTUVWXYZ ......... is impossible (B enciphers to Q, yet Q enciphers to Y) G--B-H---X-Q ABCDEFGHIJKLMNOPQRSTUVWXYZ ......... is impossible (X enciphers to X) Q G--B-H---X-> ABCDEFGHIJKLMNOPQRSTUVWXYZ ......... is possible -Q G--B-H---X-> ABCDEFGHIJKLMNOPQRSTUVWXYZ ......... is impossible (Q enciphers to B, yet B enciphers to T) X-Q G--B-H---> ABCDEFGHIJKLMNOPQRSTUVWXYZ ......... is possible -X-Q G--B-H--> ABCDEFGHIJKLMNOPQRSTUVWXYZ ......... is impossible (X enciphers to B, yet B enciphers to V) --X-Q G--B-H-> ABCDEFGHIJKLMNOPQRSTUVWXYZ ......... is possible ---X-Q G--B-H-> ABCDEFGHIJKLMNOPQRSTUVWXYZ ......... is impossible (X enciphers to D, yet B enciphers to X) H---X-Q G--B-> ABCDEFGHIJKLMNOPQRSTUVWXYZ ......... is impossible (Q enciphers to G, yet G enciphers to V) -H---X-Q G--B-> ABCDEFGHIJKLMNOPQRSTUVWXYZ ......... is impossible (H enciphers to B, yet Q enciphers to H) B-H---X-Q G--> ABCDEFGHIJKLMNOPQRSTUVWXYZ ......... is possible (note the G enciphers to X, X enciphers to G property) -B-H---X-Q G-> ABCDEFGHIJKLMNOPQRSTUVWXYZ ......... is impossible (B enciphers to B) --B-H---X-Q G-> ABCDEFGHIJKLMNOPQRSTUVWXYZ ......... is possible
448:). If two sentences in English or German are written down one above the other, and a count is made of how often a letter in one message is the same as the corresponding letter in the other message; there will be more matches than would occur if the sentences were random strings of letters. For a random sequence, the repeat rate for single letters is expected to be 1 in 26 (around 3.8%), and for the German Navy messages it was shown to be 1 in 17 (5.9%). If the two messages were in depth, then the matches occur just as they did in the plaintexts. However, if the messages were not in depth, then the two ciphertexts will compare as if they were random, giving a repeat rate of about 1 in 26. This allows an attacker to take two messages whose indicators differ only in the third character, and slide them against each other looking for the giveaway repeat pattern that shows where they align in depth.
429:
31:
369:
The bigram tables themselves were not part of the capture, but Hut 8 were able to use the settings-lists to read, retrospectively, all the
Kriegsmarine traffic that had been intercepted from 22 to 27 April. This allowed them do a partial reconstruction of the bigram tables and start the first attempt
385:
was very out of date, but it did show that
Banburismus could work. It also allowed much more of the bigram tables to be reconstructed, which in turn allowed 14 April and 26 June to be broken. However, the Kriegsmarine had changed the bigram tables on 1 July. By the end of 1940, much of the theory of
509:
Turing calculated the scores for the number of single repeats in overlaps of so many letters, and the number of bigrams and trigrams. Tetragrams often represented German words in the plaintext and their scores were calculated according to the type of message (from traffic analysis), and even their
573:
Armed with a set of probable mid-wheel overlaps, Hut 8 could compose letter-chains for the middle wheel much in the same way as was illustrated above for the end wheel. That in turn (after
Scritchmus) would give at least a partial middle wheel alphabet, and hopefully at least some of the possible
562:
Once the end wheel is identified, these same principles can be extended to handle the middle rotor, though with the added complexity that the search is for overlaps in message-pairs sharing just the first indicator letter, and that the overlaps could therefore occur at up to 650 characters apart.
549:
Not only this, but such an end-wheel alphabet forces the conclusion that the end wheel is in fact "Rotor I". This is because "Rotor II" would have caused a mid-wheel turnover as it stepped from "E" to "F", yet that's in the middle of the span of the letter-chain "F----A--D---O". Likewise, all the
521:
Scritchmus was the part of the
Banburismus procedure that could lead to the identification of the right-hand (fast) wheel. The Banburist might have evidence from various message-pairs (with only the third indicator letter differing) showing that "X = Q−2", "H = X−4" and "B = G+3". He or she would
451:
The comparison of two messages to look for repeats was made easier by punching the messages onto thin cards about 250 millimetres (9.8 in) high by several metres (yards) wide, depending on the length of message. A hole at the top of a column on the card represented an 'A' at that position, a
539:
The so-called "end-wheel alphabet" is already limited to just nine possibilities, merely by establishing a letter-chain of five letters derived from a mere four message-pairs. Hut 8 would now try fitting other letter-chains — ones with no letters in common with the first chain — into these nine
424:
with each other. Normally, the indicators for two messages were never the same, but it could happen that, part-way through a message, the rotor positions became the same as the starting position of the rotors for another message, the parts of the two messages that overlapped in this way were in
535:
G--B-H---X-Q ABCDEFGHIJKLMNOPQRSTUVWXYZ ......... is possible G--B-H---X-Q ABCDEFGHIJKLMNOPQRSTUVWXYZ ......... is impossible (G enciphers to B, yet B enciphers to E) G--B-H---X-Q ABCDEFGHIJKLMNOPQRSTUVWXYZ ......... is impossible (H apparently enciphers to H) G--B-H---X-Q
505:
Turing's method of accumulating a score of a number of decibans allows the calculation of which of these situations is most likely to represent messages in depth. As might be expected, the former is the winner with odds of 5:1 on, the latter is only 2:1 on.
531:
If this is then compared at progressive offsets with the known letter-sequence of an Enigma rotor, quite a few possibilities are discounted due to violating either the "reciprocal" property or the "no-self-ciphering" property of the Enigma machine:
452:
hole at the bottom represented a 'Z'. The two message-cards were laid on top of each other on a light-box and where the light shone through, there was a repeat. This made it much simpler to detect and count the repeats. The cards were printed in
553:
That the different Enigma wheels had different turnover points was, presumably, a measure by the designers of the machine to improve its security. However, this very complication allowed
Bletchley Park to deduce the identity of the end wheel.
522:
search the deciban sheets for all distances with odds of better than 1:1 (i.e. with scores ≥ +34). An attempt was then made to construct the 'end wheel alphabet' by forming 'chains' of end-wheel letters out of these repeats.
417:
provided by key lists, and so it was the same for all messages on a particular day (or pair of days). This meant that the three-letter indicators were all enciphered with the same rotor settings so that they were all
361:
on 26 April 1940. The
Germans did not have time to destroy all their cryptographic documents, and the captured material revealed the precise form of the indicating system, supplied the plugboard connections and
370:
to use
Banburismus to attack Kriegsmarine traffic, from 30 April onwards. Eligible days were those where at least 200 messages were received, and for which the partial bigram-tables deciphered the
315:
In the first few months after arriving at
Bletchley Park in September 1939, Alan Turing correctly deduced that the message-settings of Kriegsmarine Enigma signals were enciphered on a common
490:
XCYBGDSLVWBDJLKWIPEHVYGQZWDTHRQXIKEESQSSPZXARIXEABQIRUCKHGWUEBPF YNSCFCCPVIPEMSGIZWFLHESCIYSPVRXMCFQAXVXDVUQILBJUABNLKMKDJMENUNQ - -- - - - - --
513:
Bletchley Park used the convention that the indicator plaintext of "VFX", being eight characters ahead of "VFG", or (in terms of just the third, differing, letter) that "X = G+8".
401:. The consequent decrypts allowed the statistical scoring system to be refined so that Banburismus could become the standard procedure against Kriegsmarine Enigma until mid-1943.
289:. Hut 8 performed the procedure continuously for two years, stopping only in 1943 when sufficient bombe time became readily available. Banburismus was a development of the "
510:
position within the message. These were tabulated and the relevant values summed by
Banburists in assessing pairs of messages to see which were likely to be in depth.
570:
to scan for tetragram repeats or better. That told them which banburies to set up on the light boxes (and with what overlap) to evaluate the whole repeat pattern.
550:
other possible mid-wheel turnovers are precluded. Rotor I does its turnover between "Q" and "R", and that's the only part of the alphabet not spanned by a chain.
577:
Taken together, the probable right hand and middle wheels would give a set of bombe runs for the day, that would be significantly reduced from the 336 possible.
409:
Banburismus utilised a weakness in the indicator procedure (the encrypted message settings) of
Kriegsmarine Enigma traffic. Unlike the German Army and Airforce
499:
XCYBGDSLVWBDJLKWIPEHVYGQZWDTHRQXIKEESQSSPZXARIXEABQIRUCKHGWUEBPF YNSCFCCPVIPEMSGIZWFLHESCIYSPVRXMCFQAXVXDVUQILBJUABNLKMKDJMENUNQ ---
307:
considered the process more an intellectual game than a job. It was "not easy enough to be trivial, but not difficult enough to cause a nervous breakdown".
487:
Hut 8 would punch these onto banburies and count the repeats for all valid offsets −25 letters to +25 letters. There are two promising positions:
1066:
366:
for 23 and 24 April and the operators' log, which gave a long stretch of paired plaintext and enciphered message for the 25th and 26th.
1185:
202:
574:
choices of rotor for the middle wheel could be eliminated from turnover knowledge (as was done in identifying the end wheel).
331:. However, without the bigram tables, Hut 8 were unable to start attacking the traffic. A breakthrough was achieved after the
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1240:
1164:
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848:
770:
699:
717:(1979). "Studies in the History of Probability and Statistics. XXXVII A. M. Turing's statistical work in World War II".
644:
1194:
979:
805:
794:
The Essential Turing: Seminal Writings in Computing, Logic, Philosophy, Artificial Intelligence, and Artificial Life
665:
Gilles, Donald (1990), "The Turing-Good Weight of Evidence Function and Popper's Measure of the Severity of a Test",
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in Oxfordshire. They became known as 'banburies' at Bletchley Park, and hence the procedure using them: Banburismus.
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300:
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to infer information about the likely settings of the Enigma machine. It gave rise to Turing's invention of the
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566:
The workload of doing this is beyond manual labour, so BP punched the messages onto 80-column cards and used
626:
493:
This offset of eight letters shows nine repeats, including two bigrams, in an overlap of 56 letters (16%).
459:
The application of the scritchmus procedure (see below) gives a clue as to the possible right-hand rotor.
195:
58:
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as a measure of the weight of evidence in favour of a hypothesis. This concept was later applied in
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1018:. The National Archives and Records Administration (NARA) 8601 Adelphi Road, College Park, Maryland
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1277:
690:
636:
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374:. The first day to be broken was 8 May 1940, thereafter celebrated as "Foss's Day" in honour of
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188:
120:
68:
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The principle behind Banburismus is relatively simple (and seems to be rather similar to the
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The left hand end of a "Banbury Sheet" from World War II found in 2014 in the roof space of
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Eventually they will hope to be left with just one candidate, maybe looking like this:
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on 3 March 1941 provided the complete keys for February – but no bigram tables or
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The aim of Banburismus was to reduce the time required of the electromechanical
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This offset of seven shows just a single trigram in an overlap of 57 letters.
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NUP F----A--D---O --X-Q G--B-H-> ABCDEFGHIJKLMNOPQRSTUVWXYZ
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1007:
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115:
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832:
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319:(starting position of the rotors), and were then super-enciphered with a
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includes a chapter discussing information theory aspects of Banburismus.
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714:
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1262:"All You Ever Wanted to Know About Banburismus but were Afraid to Ask"
1165:"All You Ever Wanted to Know About Banburismus but were Afraid to Ask"
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358:
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1264:— the whole procedure researched in detail, with a worked example.
16:
Cryptanalytic process developed by Alan Turing during World War II
1089:
843:, Oxford: Oxford University Press (published 1993), p. 118,
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324:
763:
Colossus: The Secrets of Bletchley Park's Codebreaking Computers
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XCYBGDSLVWBDJLKWIPEHVYGQZWDTHRQXIKEESQSSPZXARIXEABQIRUCKHGWUEBPF
483:
YNSCFCCPVIPEMSGIZWFLHESCIYSPVRXMCFQAXVXDVUQILBJUABNLKMKDJMENUNQ
347:
343:
320:
967:
Codes and ciphers: Julius Caesar, the Enigma, and the internet
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Although this method is frequently stated to be an example of
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machines by identifying the most likely right-hand and middle
30:
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327:
lookup table. These trigram tables were in a book called the
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239:
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970:. Cambridge: Publisher Cambridge University Press. p.
381:
This task took until November that year, by which time the
988:
892:
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913:"Service Histories of Royal Navy Warships in World War 2"
1130:
Cryptographic History of Work on the German Naval Enigma
671:), that the process is not really Bayesian, but rather
1186:
Information Theory, Inference, and Learning Algorithms
765:, Oxford: Oxford University Press, pp. 378–385,
386:
the Banburismus scoring system had been worked out.
930:
880:
868:
303:was regarded as the best of the Banburists. He and
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813:
1269:
1257:The 1944 Bletchley Park Cryptographic Dictionary
1147:Codebreakers: The inside story of Bletchley Park
1016:The 1944 Bletchley Park Cryptographic Dictionary
841:Codebreakers: The inside story of Bletchley Park
800:, Oxford: Oxford University Press, p. 261,
525:They could then construct a "chain" as follows:
269:and all the other methods used for breaking the
1133:, The National Archives, Kew, Reference HW 25/1
496:The other promising position looks like this:
1189:Cambridge: Cambridge University Press, 2003.
835:(1992), "Hut 8 and naval Enigma, Part I", in
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1123:
994:
898:
963:
947:
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1073:. University of St Andrews. Archived from
378:, the cryptanalyst who achieved the feat.
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189:
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427:
1235:. London: Cassell Military Paperbacks.
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1107:
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614:Banburismus revisited: depths and Bayes
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1207:"The History of Hut Eight 1939 – 1945"
1067:"Joan Elisabeth Lowther Clarke Murray"
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684:
664:
293:" invented by the Polish cryptanalyst
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1149:, Oxford: Oxford University Press,
13:
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238:. It was used by Bletchley Park's
14:
1309:
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1110:, 6.0 The Middle-Wheel alphabet.
1006:
694:, London: Vintage, p. 197,
669:, vol. 41, pp. 143–146
29:
1233:Enigma: The Battle for the Code
1117:
1054:
1039:, 2.3 Searching for "Evidence".
1000:
957:
917:HMS Griffin – G-class Destroyer
904:
557:
540:candidate end-wheel alphabets.
248:(naval) messages enciphered on
1141:(1993), "Enigma and Fish", in
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747:
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678:
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599:
1:
911:Mason, Geoffrey B (c. 2004).
592:
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404:
663:, Donald Gilles has argued (
7:
1051:, 4.2.2 Message Categories.
964:Churchhouse, R. F. (2002).
580:
276:
10:
1314:
462:
413:, the Kriegsmarine used a
342:, which was on its way to
310:
1065:Lord, Lynsey Ann (2008).
24:The Enigma cipher machine
1163:Hosgood, Steven (2008).
757:(2006), "Turingery", in
629:; Sprevak, Mark (2017).
477:Message with indicator "
467:Message with indicator "
329:Kenngruppenbuch (K book)
1145:; Stripp, Alan (eds.),
1125:Alexander, C. Hugh O'D.
1063:worked as a Banburist.
839:; Stripp, Alan (eds.),
733:10.1093/biomet/66.2.393
691:Alan Turing: The Enigma
637:Oxford University Press
610:Introducing Banburismus
335:in which the disguised
257:conditional probability
1293:Statistical algorithms
1229:Sebag-Montefiore, Hugh
441:
1298:Cryptographic attacks
1205:Mahon, A. P. (1945).
798:The Secrets of Enigma
788:(2004), "Enigma", in
431:
242:to help break German
222:process developed by
1183:MacKay, David J. C.
446:Index of Coincidence
287:wheels of the Enigma
69:Polish Cipher Bureau
995:Alexander (c. 1945)
899:Alexander (c. 1945)
587:Sequential analysis
391:First Lofoten pinch
252:. The process used
661:Bayesian inference
623:Bowen, Jonathan P.
568:Hollerith machines
442:
1242:978-1-4072-2129-8
1156:978-0-19-280132-6
1098:, 7.0 Scritchmus.
850:978-0-19-280132-6
790:Copeland, B. Jack
772:978-0-19-284055-4
759:Copeland, B. Jack
701:978-0-09-911641-7
667:Br. J. Phil. Sci.
619:Copeland, B. Jack
411:Enigma procedures
393:from the trawler
213:
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1199:on-line textbook
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1167:. Archived from
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632:The Turing Guide
612:and Chapter 38,
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350:, was seized by
236:Second World War
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1171:on 9 March 2016
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250:Enigma machines
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99:Zygalski sheets
59:Breaking Enigma
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1278:Bletchley Park
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1108:Hosgood (2008)
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1096:Hosgood (2008)
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1077:on 7 June 2011
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1049:Hosgood (2008)
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1037:Hosgood (2008)
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889:, p. 157.
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877:, p. 155.
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786:Copeland, Jack
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755:Copeland, Jack
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727:(2): 393–396.
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686:Hodges, Andrew
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646:978-0198747826
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1012:"Banburismus"
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954:, p. 26.
953:
948:
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939:, p. 22.
938:
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901:, p. 94.
900:
895:
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865:, p. 95.
864:
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838:
837:Hinsley, F.H.
834:
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822:, p. 17.
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295:Jerzy Różycki
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1210:. Retrieved
1184:
1173:. Retrieved
1169:the original
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1118:Bibliography
1103:
1091:
1079:. Retrieved
1075:the original
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1044:
1032:
1020:. Retrieved
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952:Mahon (1945)
937:Mahon (1945)
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920:. Retrieved
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1283:Alan Turing
1127:(c. 1945),
1081:16 November
1061:Joan Clarke
1022:15 November
887:Good (1993)
875:Good (1993)
234:during the
224:Alan Turing
216:Banburismus
116:Herivel tip
111:Banburismus
1272:Categories
1212:21 October
1139:Good, Jack
1008:Sale, Tony
922:28 October
720:Biometrika
715:Good, I.J.
593:References
517:Scritchmus
405:Principles
372:indicators
305:I. J. Good
254:sequential
89:Cyclometer
673:Popperian
376:Hugh Foss
359:North Sea
352:HMS
267:Turingery
1231:(2004).
688:(1992),
581:See also
421:in depth
277:Overview
153:PC Bruno
1288:Banbury
1197:. This
1175:9 March
792:(ed.),
761:(ed.),
741:0548210
463:Example
454:Banbury
425:depth.
357:in the
354:Griffin
340:Polares
325:trigram
311:History
232:Britain
167:Related
74:Doubles
1239:
1193:
1153:
978:
847:
804:
769:
739:
698:
643:
399:K book
348:Norway
344:Narvik
323:and a
321:bigram
218:was a
616:. In
434:Hut 6
395:Krebs
283:Bombe
240:Hut 8
176:Ultra
158:Cadix
146:Hut 8
141:Hut 6
136:Hut 4
131:Hut 3
126:Bombe
94:Bomba
84:Clock
79:Grill
1237:ISBN
1214:2009
1191:ISBN
1177:2016
1151:ISBN
1083:2009
1024:2009
976:ISBN
924:2009
845:ISBN
802:ISBN
796:plus
767:ISBN
696:ISBN
641:ISBN
389:The
121:Crib
729:doi
481:":
479:VFX
471:":
469:VFG
436:at
346:in
262:ban
230:in
226:at
1274::
1069:.
1014:.
1010:.
974:.
972:34
944:^
915:.
737:MR
735:.
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204:e
197:t
190:v
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