The Special Fish Report
Albert W. Small (December 1944)
Codes and Ciphers
TOP SECRET Special Fish Report Page 23
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
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
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
This rectangle is usually constructed on the Colossus, or on Garbo. The
rectangle is converged crudely by hand, by methods already familiar to
"Accurate convergence" is almost never used.