24X Page 136
accurate convergence one of the numbers x,y is generally far too large for the approximation to be valid. In this case the formula
| log cosh(x+y) δ - log cosh(x-y) δ |
There is another type of flag, called the Jacobs flag (see R2 p 101) in which the function xy is replaced by
| sign(xy) min(|x|,|y|). |
This type of flag was used for one of the methods of starting the convergence of a rectangle, because it is quicker than multiplication, though much less accurate. It would be a good approximation for large values of δ. If all the entries in the rectangle are ±1 or 0 then Jacobs flag and the ordinary (scalar product) flag are the same thing. This remark applies in the case of most key rectangles.
For mechanical flag-making for cipher tapes see R3 pp 63, 78, 82, 106, R2 p 101, and ch. 91.
24X SIGNIFICANCE TESTS
(a) Introductory remarks.
We are about to discuss a number of significance tests for rectangles. The first one, 'significance test 0' is designed for rectangles not converged. Tests I to IV are for converged rectangles. The standard one is significance test IV, and is the most difficult to understand.
(b) Tests for unconverged rectangles (historical).
No rectangle was made with mechanical aid of any sort until after the autoclave had been generally introduced (January 1944). It was then suggested (R1 p 32) that if the rectangles were made on a Robinson, with a set total, the number of readings that came up would be an indication of how good the rectangle was likely to be. Such a test was particularly important at a time when it was troublesome to make rectangles. It was thought at first that such a test would be quite powerful and that it might even be possible to stop Robinson in the middle of the run. However some figures were then produced (R1 pp 34, 38) depending on a single message that had been rectangled by hand a long time before, and these figures tended to show
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