Comments inline...
On Wed, 24 Apr 2002, R. A. Hettinga wrote:
> I seem to be channeling mathematicians this morning...
>
> Cheers,
> RAH
>
> --- begin forwarded text
>
>
> Status: U
> From: Somebody with a sheepskin...
> To: "R. A. Hettinga" <[EMAIL PROTECTED]>
> Subject: Re: Two ideas for random number generation
> Date: Wed, 24 Apr 2002 08:44:41 -0600
>
> Bob,
>
> Tim's examples are unnecessarily complicated.
>
> The logistic function f(x) = Ax(1-x) maps the interval [0,1] into itself for
> A in the range [0,4]. Hence, for any such A, it can be iterated.
>
> That is, one may start with an x|0 get x|1= f(x|0) -- where x|j means x sub
> j -- and repeat, thus: x|(n+1) = f(x|n).
>
> For small enough values of A, the iteration provably converges to a single
> value. For slightly larger values, it converges to a pair of values that
> alternate every other time -- known as a period 2 sequence. For a slightly
> larger value of A it converges to 4 values that come up over and over
> again -- a period 4 sequence. Some of this is provable, too.
This is called 'bifurcation'...a mathematician should know that.
> This increase in multiple period states continues briefly for smaller and
> smaller changes in the parameter A. At some point the period becomes
> infinite, and the sequence becomes not detectably different from random.
> This is an empirical fact, not yet proven so far as I know.
This person should start with Mandelbrots book....x=(x^2)+1 is
'deterministic' as well...but their is a catch to this...
> Note that the function is completely deterministic. If you know x exactly,
> you know x|n -- exactly. But if you know x to only finite precision, you
> know very little about x|n. Specifically, you know only that it is in the
> range [0,1].
You -can't, even in -principle-, know the precision to finite levels. This
is the entire basis for chaos theory.
> So.... Pick A large enough. Pick an arbitrary double precision floating
> point number (about 14 digits for 64 bit arithmetic) on a given machine.
The selection of a special analysis point to prove a general behaviour of
the function is a logical fallicy.
> sequence of 7 least significant digits. They're probably uniformly
> distributed in the 7 digit integers.
Probably not, in general (there is a special name for this distribution
but I don't remember and I've got four days mail to catch up on) but the
numbers, [0...9], do -not- appear in nature in equal distribution. If you
are basing anything on this then you're in for a world of hurt.
> If you don't know the seed, you don't know the sequence, so I guess you can
> encrypt with the thing, too.
Actually that's overly general. Other keys could generate that sequence
(PRNG's repeat, and RNG's aren't predictable) so this test is really
worthless.
> But you can't prove squat about it!
Hardly.
--
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