# The Special Fish Report

## Albert W. Small (December 1944)

### Page 29

Tony Sale's
Codes and Ciphers

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CX/MSS
TOP SECRET Special Fish Report Page 29

For a wheel to be accepted as final: 1. It must integrate;

2. The lowest two pips whose signs may be changed and still give a

legal wheel, must sum to more than 50 decibans:; 3. Individual characters

of the wheels must be at least 20 decibans each, or if lower in spots

they must be so marked and Testery notified for its information in solving

delta psi prime; 4. The 32-letter count of the final Delta-D text

must be satisfactory.

There is also available a brief significance test to use

on a single wheel-breaking run:

Let R = total number of positions actually looked at in the run.

W = wheel length.

X = sum of the moduli resulting from the run (= Sum|xi|.)

The expected value of X at a wrong setting can be shown to

be = .8(R*W)^1/2 from the normal distribution. And the variance of X at random

can be shown to equal R(l - 2/Pi) or approximately .36R. So the

standard deviation is approximately equal to .6(R)^1/2, and two standard

deviations above expected random = .8(R*W)^1/2 + 1.2(R)^1/2. This is graphed for

usable values of R and W. X of an run must exceed this if the run

is to be called significant.

With these miscellaneous facts gathered together, we are prepared

to study wheel-breaking.

Pages 30-34 inclusive are the actual work-sheets used during the

runs shown in Pages 35 to 54.

The first line of the Delta-X1 worksheet (Page 30) and the first line of

the Delta-X2 worksheet (Page 31) shows the pips obtained from converging a

rectangle made on Garbo. Message involved, SB 3037 part l.(Original rectangle

not available.) Other messages used during the run: SB 3037 pt 2; SG 917 Roll I

pts 2 and 3.

A "part" is usually 10,168 characters (8x41x31); the last part is what is left.

(Note by Tony Sale: pages 30-54 are at present omitted (some are A3 size) until a
way can be found to put these on the web at a reasonable Kbyte size)

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