# The Special Fish Report

## Albert W. Small (December 1944)

### Page 8

Tony Sale's
Codes and Ciphers

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CX/MSS
TOP SECRET Special Fish Report page - 8

In the preceding table, "Expected sigmage" is the excess

of the average observed causal score over random score, divided by

random sigma. The variance of such sigmage is assumed to be the

sum of the variances of the sigmages of the letters involved. (These

individual variances are in turn the mean square deviation-from-average-sigmage

in the 16 messages.) The "estimated standard deviation" is the square root

of the variance so computed. "Actual standard deviation of sigmage" is

computed from sigmages recorded in runs.

It might almost be said that the whole story of the Newmanry is

told in the preceding table, and for this reason such a table deserves

close scrutiny. Different traffic networks would or course have different

tables, and while such have not been computed for all cases where they

are needed, they exist in the minds and experience of the Colossus operators.

A thorough knowledge of Delta-D characteristics is needed either for

setting known X wheels or for breaking unknown wheels.

Setting known X's is of course easier. In an ideal world the basis

for setting them would be to "run" all positions of Delta(X1, X2, X3, X4, X5)

against Delta-Z text; and to match the resultant Delta-D text against the

theoretical distribution. Where the match would be greatest, there would be

the most probable answer. Since 41x31x29x26x23 settings would require a

run of the Z tape 22,041,684 times through Colossus, this is obviously impossible

on today's machinery. Compromises must be found. Accordingly the X's are

set in smaller combinations rather than in toto; and this is the reason the

table of Delta-D impulse combinations is so important. Any run involving the

setting of two X wheels at once is called a "long run;" any involving the setting

of only one X wheel is called a "short run". The average long run takes 8

minutes* as against 2 minutes for a short run.

*Utilising five counters in "multiple testing" to cut the time in one fifth.

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