[Hardware] Notes... The case for an open client

Elektron elektron_rc5 at yahoo.ca
Tue Aug 17 00:29:37 EDT 2004

>> Which for 453 or so bits, is biased, but not horribly so.
> The next question a statistician will ask himself is wether this  
> observed bias is statistically significant. Generally the result is  
> significant if there is less than a 5% chance that the result was due  
> to random fluctuations.
> You don't need to be a math head to compute the significance. Just  
> generate a large number of random samples just like the set above and  
> count the number that have the same or greater bias.

I wouldn't sum the run lengths, since all the keys are of different  
lengths. Although it's not entirely correct, for simplicity's sake, we  
can string all the keys together:


232/456 bits are ones. That's a little on the high side, but only by  

As for run lengths, we can split it up into words (14.25 of them) and  
look at max run lengths (you can look at all of them, but then you have  
trouble measuring the bias). This is probably too small a sample to  
draw any conclusions, though.

There's also a small problem with measuring the bias of run lengths,  
but for simplicity's sake, if we have 456/(2^32*32) (approx. 3.318e-9)  
of the bits in the study, then we should have the same proportion of  
each run length (pretending they're all 32-bit, which they aren't, but  
anyway). E.g. with 35433480192 length-1 runs, in 456 bits, we should  
have 35433480192*3.318e-9 = 117.6. Continuing this,

   1     2     3     4     5    6    7    8    9
117.6  57.0  27.6  13.4   6.5  3.1  1.5  0.7  0.3    Predicted runs
  97    67    27    18    11    1    0    1    0      Actual runs

Then we have the problem of deciding just how unlikely this is (and  
then, though there are more 5-bit runs than usual, there are also more  
2-bit runs than usual).

- Purr

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