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There is no such thing like a Gaussian distribution over [0,1]; this doesn't make any sense. So it is not clear what you have to begin with. However, if you have independent random values, you can generate a random bit by taking two values A and B and comparing them. E.g., if $A<B$ you set the bit to 0, otherwise you set the bit to 1. A sequence of 8 ...


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Knowledgeable crypto practitioners do not calculate $\pi$ using a Monte Carlo method to determine if a series of numbers are random. The test alluded to in the question is a general-purpose randomness test for random number generators with output a real number expected to be uniform of the range $[0\dots1[$. The test consists of drawing pairs $(x,y)$, and ...


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I think that I've realised why you might calculate pi. It's easy. x^2 + y^2 = 1 is a simple calculation, thus fast. If you picked another 2 dimensional shape with a non trivial perimeter, you'd have a harder time estimating which side of the perimeter a point had fallen.


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Depending on the parameters of the Gaussian, every $X_i$ byte will have some entropy < 8 bits. So you cannot produce cryptographically random bytes from each of them, unless you add some entropy from another source. You can, however, turn them into smaller values. For example, if they have at least 1 bit of entropy, you can turn them into bits. Like if ...



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