General Report on Tunny

---

24W Page 130

24W     THEORY OF CONVERGENCE

(a) Elementary properties of the convergence of a rectangle.

Let the length of message be N.

Let the entry in the cell which is the i^th row and j^th column be θ_ij.

If some value of δ is assumed then the odds that ΔΧ₁₂ = dot in all (i,j) are where .

This is a trivial consequence of Bayes' theorem, if the message is assumed to contain no slide (see R3, p 130). Another way of stating the result is that the hypothesis ΔΧ₁₂ = dot is Θ_ij pips up, where a pip is 10log₁₀ζ decibans. This enables one to regard the rectangle as an array of 1271 pieces of probability information, arranged in a convenient form for attempting to find the ΔΧ₁ and ΔΧ₂ patterns. We now give a description of methods used for doing this.

A partial wheel pattern can be regarded as a sequence of numbers, ε₁, ε₂ ,... each equal to ±1 or 0, where +1 stands for a dot and -1 for a cross and 0 for a doubt. The process of taking the pattern through the rectangle consists in forming scalar products

The numbers y_i are then called the scores in pips of the characters of the other wheel.

In the original form of convergence one would take ε_j' = sign y_j,

i.e.	εj' = +1 if yj > 0
	εj' = -1 if yj < 0
	εj' =  0 if yj = 0

and take this wheel pattern back through the rectangles giving pippages

The resulting pattern is then taken back in a similar way, giving new values for y_i and so on until the pattern on one side of the rectangle is the same twice running. The rectangle is then said to be converged (crudely). The result of the convergence may depend on the particular pattern with which the convergence is started. The sum of the moduli of the scores of one wheel is called X,X = Σ|x_i|. This score is independent of which of the two wheels is used, when the convergence is completed, for

---

< previous

next >

Back to General Report on Tunny. Contents.

Back to The Enigma and the Bombe main page

Home