The Special Fish Report
Albert W. Small (December 1944)
Codes and Ciphers
TOP SECRET Special Fish Report Page 3
Characterisation of D
To solve (1) for pseudo plain text D, it is necessary to know
the characteristics of D, as resulting from equation (2).
Individual letters of D are random in appearance, since D
results from the enciphering of plain text P by the addition of PSI' key,
the letters of which are fairly random when distributed singly.
However the successive letters of PSI' text are non-random in their pairings,
since PSI' is generated from PSI wheels which either all move with probability
of "a," or all don't move with a probability of "l-a." A letter in the
PSI' key will be repeated when the Mtotal= . ; or when the, Mtotal = x and every
one of the five signs of PSI, remains the same. Thus a double letter
in the PSI' text has a probability of occurrence of (1-a) + a(l-b)^5.
Similarly, a letter which is the exact inverse will follow a PSI' character
with a probability of ab^5.
In teletype modulo-two arithmetic, if a letter is added to
itself the result to always the teletype character "/" and if it is
added to its inverse the result is always an "8". Thus by taking each
letter of the PSI' text and adding it to a letter equal to the letter
following it, this resultant text, called Delta-PSI' text, shows a slant for
every double letter. Thus, / has a probability of occurrence of
(1 - a) + a(l - b)^5. Similarly, 8 has a probability of occurrence in
Delta-PSI' of ab^5. The probabilities of Delta-PSI' text characters are as shown
in the following table: