# The Special Fish Report

## Albert W. Small (December 1944)

### Page 23 Tony Sale's
Codes and Ciphers

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CX/MSS
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Wheel Breaking

The most interesting solution of Delta-Z + Delta-X = Delta-D occurs when

Delta-X is known and Delta-D are unknown (except for general characteristics

which depend upon the type of traffic.)

This is called wheel-breaking. It is an attempt, through the study

of at least 5,000 characters of Delta-Z, to formulate Delta-X and Delta-D
simultaneously,
thereby obtaining the X wheels as well as the Delta-D text.

We need mathematics in wheel-breaking. Wheel-breaking procedure

involves establishing hypotheses of varying probabilities, and building

further hypotheses on them. Odds in favour or the final outcome are most
important
to know. By the theory of probability, these final odds are the product of

the prior odds and all the factors derived from the individual pieces of

evidence. Because a product is involved, statements of odds and of factors

are usually made in logarithms (to base 10) called "bans;" the actual

working unit is for convenience a "deciban."

Note the following equations in probability:

P(E+A) = P(E) P(A/E) = P(A) P(E/A)
P(E+notA) = P(E) P(notA/E) = P(notA) P(E/notA)

If the center and right hand-member of the upper are divided by the center and

right hand member of the lower, we obtain:

Odds of A, given E = (Prior odds of A)P(E/A) / P(E/notA)

and the fraction on the right is the "factor" referred to above.

A "pip" is the factor in favour of a hypothesis, resulting from the

occurrence of some standard event.

It is the custom in wheel-breaking to begin with the Delta-Z1 Delta-Z2

rectangle.

This rectangle is usually constructed on the Colossus, or on Garbo. The

rectangle is converged crudely by hand, by methods already familiar to

Arlington.

"Accurate convergence" is almost never used.

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