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. -- ____________________________________________________________________ The law is applied philosophy and a philosphical system is only as valid as its first principles. James Patrick Kelly - "Wildlife" [EMAIL PROTECTED] www.ssz.com [EMAIL PROTECTED] www.open-forge.org --------------------------------------------------------------------