Talk:Cryptanalysis of the Enigma
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[edit] Plugboard Combinations
One's first intuition is usually that there are 26! plugboard combinations - but this is wrong, because, for example, if A transforms to H, then H has to transform to A - whereas if all combinations were possible, then H should be able to transform to another letter.
- the number of ways to connect no wires in the plug board is 1
- the number of ways to connect 1 wire is sum(1..25)
This is the "triangle" function, which I defined thus:
tri(x) := sum(i, i, 1, x); (ref: http://maxima.sourceforge.net/docs/manual/en/maxima_6.html#IDX194 )
- the number of ways to connect 2 wires is tri(25) * tri(23)
- the number of ways to connect 3 wires is tri(25) * tri(23) * tri(21)
- I have defined a function for the number of ways to connect n wires thus:
tricons(y) := product(tri(25 - (2 * k)), k, 0, y - 1); (ref: http://maxima.sourceforge.net/docs/manual/en/maxima_6.html#IDX188 )
- therefore the total number of plug board combinations is:
sum(tricons(z) / z!, z, 0, 13)
The division by z! is because the actual order of the connections doesn't matter.
This gives a result of 532985208200576 - or 5.3 * 10^14
Could someone rewite this using the sum and product symbols and add it to the article, please? Thanks for this! New Thought 08:33, 17 April 2007 (UTC)
- A couple of quick comments (sorry, I'm a bit busy atm!). The formula I came up with is the following, derived a slightly different way:
. As a source, we can use this paper for related discussion. — Matt Crypto 18:12, 17 April 2007 (UTC)
-
- Turning this into a Maxima expression, sum( 26! / (2^n * n! * (26 - 2*n)!), n, 0, 13), yields the same result I calculated independently above (532,985,208,200,576) - excellent! :-) . That formula is sufficiently succinct to add to the article without excessively adding to its size. New Thought 19:34, 17 April 2007 (UTC)
[edit] Remark
Hello
I am very surprised after reading this article about Enigma. The authors cite the article of Gaj and Orlowski and the book of W. Kozaczuk, but there in the article is not written that polish mathematicians as first people in the world broke Enigma cipher ( I haven't found such sentence). There is only described their work, however without clear ascertaintion, that they broke this cipher. Only concerning Bletchley Park, the authors write about breaking the cipher of Enigma. The reader, who is not an expert in this matter may be desoriented.
I think it is not good.
Observer
- Thanks for taking the time to post your observations! It's certainly true that this article needs work, and that the nature and importance of the role of the Poles needs to be much more clearly stated — I'll try and get round to it soon, if noone else works on it (this being Wikipedia, you're very welcome to edit the page yourself, of course!).
- One note, though — the Polish were the first to break Enigma in the sense that they were the first to read live encrypted German military Enigma traffic, which they did after 1932. However, other codebreakers had "broken" some versions of Enigma earlier, in the sense of finding cryptanalytic weaknesses — GC&CS's Hugh Foss had found weaknesses in the commerical Enigma C as early as 1927 (and I suspect he wouldn't have been the only one). The Polish work was orders of magnitude more significant, but we just have to be careful and precise in how we phrase it. — Matt 06:55, 23 Sep 2004 (UTC)
sorry for direct language but the pole with wooden mockup enigam model is bs , no question. it comes from the time of around 1974 when Winterbothams "the ultra secret" book came out, archives were still secret and polish sources were ignored in the western world. that stuff should be deleted !
[edit] Key-size
Does anybody know how many different keys there are? It is hinted at in the Pre-World War II section:
- Finding the proper chains from the 105,456 possibilities was a tremendous task. The Poles, particularly Rejewski's classmates Jerzy Różycki and Henryk Zygalski, developed a number of methods. One technique used clear strips for each rotor showing which letters could be chained, with the letters that could not chain being blacked out. Users would pick up the strips and lay them over each other, looking for selections where the three letters were clear all the way through. The British had also developed such a technique when they succeeded in breaking the common commercial Enigma, though they failed to break the military versions of the Enigma.
But it isn't exactly clear whether 105,456 is, in fact, the right number. I think it would make sense to add, to the end of the Security section, something like "the XXXXXX Enigma had approximately 100000 possible keys" or whatever the correct number is. Ealex292 19:27, 10 Apr 2005 (UTC)
- I've added a reference which addresses this topic, but I haven't yet incorporated it into the text. The number of keys is a slightly ambiguous question, as you really need to first state how much you already known about the machine. For example, the number of possible keys is a lot less if it's assumed that the attacker already possesses the rotor wiring. — Matt Crypto 22:38, 6 November 2005 (UTC)
[edit] "Jeffreys sheets"
Not "Jefferson sheets." (And Zygalski invented them.) logologist 03:24, 11 Apr 2005 (UTC)
- Yes. Welchman referred to Zygalski's perforated sheets as Jeffreys Sheets, but it appears that he misremembered and Jeffreys Sheets were actually sheets used for a different purpose. — Matt Crypto 22:38, 6 November 2005 (UTC)
[edit] "Clear strips" technique
What is the source on the supposed Polish "technique [that] used clear strips for each rotor showing which letters could be chained..."? I don't recall seeing in the Polish documentation any mention of such a technique. I would be interested in reading about it. logologist 00:42, 12 Apr 2005 (UTC)
- My best guess it that it's a garbled description of perforated sheets. I've moved the text here pending verification/clarification
- One technique used clear strips for each rotor showing which letters could be chained, with the letters that could not chain being blacked out. Users would pick up the strips and lay them over each other, looking for selections where the three letters were clear all the way through.
- — Matt Crypto 18:27, 17 February 2006 (UTC)
[edit] Poles' use of "cribs"
It's a bit misleading to say that "British attacks usually required cribs, whereas the Polish attacks exploited the indicator system." The Poles also used cribs. Rejewski writes, for example (Kozaczuk, 1984 Enigma, pp. 243-44): "The last phase in reconstructing daily keys was finding the settings of the rings [on the rotors]. In that phase, we relied on the fact that the greater number of messages began with the letters ANX [German for "To," followed by "x" as a spacer]. [...] But the Germans [...] later, probably after 1940, [...] introduced a rule that, at the beginning of a message, one should place some word that has no meaning in connection with the context of the message, so that the message would not open with the letters ANX." logologist 23:29, 10 September 2005 (UTC)
- Ah, interesting, I didn't know about that. Finding the first six permutations (A,B,C,D,E,F from AD, BE and CF) also required some insight into the non-random choices the operators made for the message settings (e.g. "lll"). But as a general statement, it's fair to characterise the Polish attacks as not needing cribs, whereas the British methods wouldn't have worked without them. — Matt Crypto 09:04, 11 September 2005 (UTC)
[edit] At what point did Germany discover any of this?
It would be a good addition to the article to mention: Did the Germans add the extra rotors in 1939 because they had learned of the Polish decryption? Tempshill 17:55, 22 July 2006 (UTC)
[edit] 26 * 25 * 26
Is there any reason, why the calculation of rotorsettings is 26 * 25 * 26 and not 26 * 26 * 26? I don't get this...--JolleJ 14:06, 24 May 2007 (UTC)
- The number of ways you can set the three rotors is 26 * 26 * 26, but the period of the machine is slightly less: 26 * 25 * 26. The reason is that at certain points the middle rotor steps twice in two consecutive key strokes: see [1]. — Matt Crypto 18:06, 24 May 2007 (UTC)
[edit] Polish?
This article is fairly poorly written and has an obvious strong Polish bias. No I won't cite or otherwise, I know how wikibureaucracy works. But a rewrite would be nice if someone could be bothered. 86.41.67.200 23:16, 13 November 2007 (UTC)
- OK. How about if that someone was you? — Matt Crypto 17:16, 14 November 2007 (UTC)
