Turing's Treatise on the Enigma

Each of the 263 trigramme must occur equally often on cards in this urn, and must of course have with it the number of previous occurrences of this trigramme. Now imagine that we have worked out a certain number of V.K.G. using a given bigramme table, and that we have found out how many times each of them had occurred before. This can be compared with being given one of the urns, and told "It is Q:1 on this being the random urn", and then drawing a certain number of cards from the urn. After the draw we have a new idea of the odds that the urn is the random urn, and we should have a correspondingly modified idea of the odds that the bigramme list is the right one. Let us suppose that the trigrammes, in the order as they were worked out, had the numbers r1, r2, ... r5 of previous occurrences, and that correspondingly the cards drawn from the urn bore the numbers r1, r2, ... r5. The proportion of cases of draws of s cards from the urn, giving these results with the same order, is ur1, ur2, ... ur5 where ur is the proportion of r-cards in the urn.

Likewise the proportion of cases where this happpens with the other urn is ur1', ... ur5' with a corresponding meaning for ur'. Then the odds on the urn not being the random one after the experiment are

In other words the drawing of a card with the number r m improves the odds by a factor of

which is equal to

except in the case m=0 when it is

The same method may be applied for the identification of some unknown bigrammes By taking into account a number of days traffic all using the same bigramme table we may find a number of indicators whose V.K.G. would be completely known if we knew the value of a certain bigramme. If we make the right hypothesis as to the value, we should get trigrammes agreeing with the ststistics as before. In this sort of case, as the data is liable to be very scanty, it is essential to use the accurate theory as described above.


< previous