Re: Random number generation using radioactivity

From: Michael Jørgensen (ingen_at_ukendt.dk)
Date: 08/09/04 

Date: Mon, 9 Aug 2004 08:27:33 +0200

<juuitchan@hotmail.com> wrote in message
news:9159d95e.0408082112.428be67b@posting.google.com...
> I wonder if this would be a good method for random number generation
> using radioactive decay:
>
> You have a radioactive source and a Geiger counter. This setup is
> connected to a computer. The computer's internal timer is good enough
> to split the average time between two consecutive decays into several
> hundred parts.
>
> You take the intervals between successive decays modulo P (P is a
> smallish prime number, like 41 or 59). When you get about 30 or so of
> these numbers, you concatenate them, read it as a single base-P
> number, convert it to decimal, throw away the first 10 or so digits,
> and keep the rest as random digits.

Well, it won't be *exactly* uniform.

Each measured interval follows an known distribution (geometric?) with an
average value of around 200 (you wrote "several hundred parts"). Taken
modulo 41, we have a random integer in the range [0, 40]. The distribution
is still not uniform, there will be a substantial bias towards low numbers.

What happens after that I'm not quite sure about. However, if we look at
information content, then each measurement gives you approximately 5 bits of
information. After collecting 30 values and converting to decimal and
throwing away 10 digits we get (5*30 - 10*3) = 120 bits of information. That
should give you approximately 40 random digits, but keeping the
non-uniformity in mind, I would not trust them all to be independent.

Here's an alternate approach, that tries to achieve uniformity:

[disclaimer: This is something I just thought of, while replying to this
post. Use at your own risk!]

Take the difference between pairwise measurements. This gives an integer in
the range [-40, 40], with a peak at the value 0. Now add together 8 of such
values (using a total of 16 measurements). This gives an integer in the
range [-15*40, 15*40] which almost follows a normal distribution. You must
now normalize, so that the standard deviation becomes 1.

Now if my memory serves me well; if X and Y are normal distributed
independent variables, then exp(-(X^2 + Y^2)) is uniformly distributed in
the interval ]0,1].

-Michael.

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