A proper law

[Clive D. W. Feather]

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In article <3E6774CB.6346.1D7DC2F@localhost>, David Hansen 
<davidh@spidacom.co.uk> writes
>The one that still fascinates me is how the plugboard was made
>essentially irrelevant, by the diagonal board IIRC (I must read up
>again some time). Even Alan Turing was stunned by that idea.

 From the memories of a talk at BP that I attended, the diagonal board 
did *not* make the plugboard irrelevant.

The purpose of the bombes was to find a position of the rotors that was 
consistent with a crib or guessed decrypt. For this you needed to 
construct loops (e.g. N in position p decrypts to T, T in position p+1 
decrypts to I, I in position n+4 decrypts to N, and there's your loop). 
With no loops, any of the 26^3 rotor positions was consistent. With one 
loop, on average 26^2 would be consistent by random chance. With two it 
was 26^1, with three 26^0, and with four 26^-1. So you needed four 
separate loops to make it worth running the bombe, and even then you had 
a 4% chance of a false drop.

The diagonal board had the same effect as one loop for each symmetry it 
was able to exploit. On average it reduced the number of loops needed 
from four to three, which was a major benefit.

But it was the bombes that overcame the plugboard; the diagonal board 
was just an optimisation.

- -- 
Clive D.W. Feather, writing for himself  | Home: <clive@davros.org>
Tel: +44 20 8371 1138 (work)             | Web:  <http://www.davros.org>
Fax: +44 870 051 9937                    | Work: <clive@demon.net>
Written on my laptop; please observe the Reply-To address

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