Re: Good enough for crypto?

Mok-Kong Shen <[EMAIL PROTECTED]> wrote in message news:<[EMAIL PROTECTED]>...
> Scott Wilber wrote:
> > 
>  [snip]
> > The theorem relates specifically and only to non-deterministic
> > sequences as is clearly stated.  No inference may be made from this
> > concerning deterministic generators.
> 
> While my math knowledge is poor, I rather doubt that,
> starting from the concept of 'non-deterministic' and
> independent of any assumptions from the side of physics,
> you could establish a (purely) mathematical theorem on 
> the behaviour of ACF in the way you said. Isn't it that
> a strictly theoretical white noise has zero ACF
> excepting at the origin? So a practical one should
> have some tiny values that are non-zero as it should be
> due to statistical 'fluctuations'.

Please keep in mind that we are discussing "real," not theoretical
noise sources.  In a theoretical white noise source, infinite
bandwidth and zero ACF are possible, while in any real device with
physical components, this is not possible.  By the way, the actual
PDF, mean and standard deviation (approximate for large N only) of the
statistical 'fluctuations' are known.

> I just don't yet see
> in this (admittedly heuristic) sense why ACF 'has' to 
> vanish at sufficiently far distance from the origin.
> Thanks.
> 
> M. K. Shen
> -------------------------------
> http://home.t-online.de/home/mok-kong.shen


I did say I wouldn't present this derivation in the context of this
thread, but here is an outline.

Every non-deterministic, truly random sequence (excluding chaotic
generators) is composed of an entropy source, say, shot noise in a
diode, and associated amplifiers and other electronic components.  The
noise source, amplifiers and components have a certain knowable, i.e.,
theoretically modelable or accurately measurable transfer function. 
To determine the mathematical description of the ACF:
1) Determine the Laplace transform, G(s), of the entropy source and
measurement system.
2) Determine the impulse response, g(t), by the inverse Laplace
transform.
3) Calculate the step response of the system, h(T), by integrating
g(t) from 0 to T.
4) The autocorrelation, AC(T)=1.0-h(T).  (This works for continuous
data.)
5) For discrete data, calculate each order by setting T=sampling
interval times the order number.  By inspection, AC(0)=1.0 as
expected.  Also by inspection, every step response function will
converge to zero as time increases without bound.

Given a complete mathematical representation of the ACF, it is a
simple exercise to show the behavior of that function over all time.

Again, this is to my knowledge original and previously unpublished.
This may not be complete since I had no intention to publish this
information at this time in this forum, and it was thrown together
rather hurriedly.

Scott Wilber
ComScire - Quantum World Corporation