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During the second world war the British were doing cryptanalysis of the German Enigma cipher machine and subsequent signals intelligence.

One of the issue was about the safeguarding of the source of the intelligence, see for example http://en.m.wikipedia.org/wiki/Ultra#Safeguarding_of_sources

In the recent movie "The imitation game" about the Enigma code breaking, the character of Alan Turing refers to the use of "Statistical analysis" to protect the sources. What analysis is talking about?

Warning: a movie spoiler is following.

STEWART MENZIES [Agent of MI6]

... Why are you telling me this?

ALAN TURING

We need your help to keep this from the Admiralty. Army. RAF. No one can know we broke Enigma, not even Denniston [Royal Navy Officer in charge of the cryptoanalysis].

STEWART MENZIES

Who is in the process of having you fired.

JOAN CLARKE [Cryptoanalyst]

You’ll take care of that.

ALAN TURING

While we develop a system for determining how much intelligence to act on. Which attacks to stop, which to let through. Statistical analysis. The minimum number of actions it’ll take to win the war, but the maximum number we’re able to take before the Germans get suspicious.

STEWART MENZIES

You’re going to trust this all to statistics?

To maths?

ALAN TURING

Correct.

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What analysis is talking about?

The script was written by Graham Moore based on a book by Andrew Hodges.

We have a dramatic (and possibly fictional) conversation that Turing may have had, or may have had one vaguely like, or one he may never have actually had, but which serves a dramatic purpose in conveying something that would be otherwise less easy or less watchable to convey. Guessing what was in the scriptwriter's mind may be difficult without asking him.

We might make guesses, but they may in the end be little more than that.

The following is clear enough:

  • If they act on every signal they intercept, it will very quickly become clear that the explanation for this is that the code can be broken, at which point all future value from being able to do it is lost.

  • if they never do anything with the knowledge, they will of course learn a lot, and will be unlikely to be discovered, but they can't get operational benefit from it.

  • if they make careful use of the information, acting only when it matters most, they may be able to reap a great deal of reward.

  • the more they use the information, the more "signal" they send that they can decode the messages.

In a very clear sense, then, there's a statistical problem for the people using Enigma (yet another mission has been lost - perhaps because the enemy was well-prepared; has our unbreakable cipher been broken?).

If it hasn't been broken, changing the code will be a difficult, expensive (and even somewhat risky, since a completely new, hastily constructed system may be vulnerable in unexpected ways compared to a tried and true secure) process -- you really want to avoid changing it if you don't need to. On the other hand, if it has been broken, you really need to use a new code.

So then the Enigma-user will be trying to pick up a weak and noisy signal from the Allied actions -- we might even frame it as an hypothesis test -- do they "know" more than can be explained by chance? (or as a decision-theory problem, by minimizing the loss function across the two different actions of change-the-code and keep-the-code in the presence of the data on allied actions)

In modern terms, game theory* would suggest there's an optimal mixed strategy for teh Allies - for any given level of benefit from using a particular piece of information, there's a corresponding probability with which you should use that information (realistically, there's some situations for which you may be able to fake a plausible mechanism by which you knew that didn't involve code breaking -- but if you start doing that, you also risk that deception being discovered -- even a few more people knowing about such a thing runs a great risk; this sort of 'the more people that know' risk is mentioned in the above exchange)

* (which Turing might or might not have had any formal background in but some of the ideas may well have been clear to him anyway)

So Turing is right - given that they can break the messages (at least pretty frequently), they'll have an important flow of very specific information, which will be of enormous benefit. But how should that information best be used in such a way as to avoid* conveying that the code can be broken? A probabilistic (/statistical) approach is actually a very smart strategy, because it allows to you make the signal as hard as possible to detect for a given level of use of the information.

Indeed, the more you "pick and choose" instead, the easier you make it to detect that you know, since whatever basis you use to choose might be exploited by someone occasionally feeding just the right false information into the communications system in the hopes of catching you (if it seems particular kinds of information tend to be acted on, you can focus on those ... and quickly reveal that they are). [By contrast a carefully tailored probabilistic approach makes every kind of response to such a tactic fairly noisy. So you would need to observe for a lot longer to really be confident that the code was broken.]

* 'avoid' in the sense that they can get as much total benefit from it as possible

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  • $\begingroup$ this is a helpful overview of some of the issues involved, but the original question is about the type of analysis that was actually used, and not what could have been used. By the way, game theory as such (especially mixed strategies) are a later or at best contemporaneous invention, although the strategic issues involved would have been apparent without the game-theoretical formalism. And you seem to equate a statistical approach to the problem with a randomization strategy and the connection is not implied. $\endgroup$ – Dimiter Jan 18 '15 at 14:15
  • $\begingroup$ @Dimiter I don't believe there was any formal statistical approach used as to what information would be acted on. In this (possibly fictional) conversation, Turing was giving advice about how information ought to be acted on, not setting policy. None of what was done with the information was in his hands. In effect, you're left with nothing but hypotheticals and what might be done in such circumstances. If you're able to provide a better answer, I'll gladly upvote it. $\endgroup$ – Glen_b -Reinstate Monica Jan 18 '15 at 22:38
  • $\begingroup$ One way or another, there must have been a policy/strategy about which intelligence to act upon and which to pass. To me, it is not obvious what the optimal strategy is, and it is not obvious what a feasible strategy would have been given the circumstances. So I just want to emphasize the relevance of the original question about what policy/strategy was in fact used. $\endgroup$ – Dimiter Jan 19 '15 at 21:32
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Since in this kind of problems you lack of data, you have to include external knowledge in statistical model. Including prior information in a model is a key feature of Bayesian model. Numerous sources (e.g. here or here) quote that some kind of Bayesian analysis was employed during World War II for different aims such as cracking Enigma, hunting U-boats etc. Actually, Bayesian estimation is used for similar purposes like this even today (e.g. for search and rescue, see here or here).

Note that for cracking Enigma computers were used for the first time and also nowadays computers are a key part of simulation-based Bayesian estimation methods.

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