[ExI] Deterministic Randomness (was Probability)

The Avantguardian avantguardian2020 at yahoo.com
Wed Jul 16 23:45:00 UTC 2008


Ok so now that people have had time to try wrap their head around probability
being "subjectively objective", I will take the concept one step further by
stating quite confidently that randomness and determinism are not opposites
because they are not mutually exlusive. Mathematics is filled with things that
seem random but are deterministic. Eliezer's example of pi is one example. Pi
is an example of an irrational number.

Irrational numbers are nature's joke on mathematicians because all are
constants that have an infinite sequence of digits past the decimal point that
never repeat. This caused great consternation amongst the early mathematicians.
In fact, according to legend, the poor Greek that first discovered that the
square root of 2 cannot be rendered as a fraction was thrown into the ocean and
drowned by his fellow Pythagoreans.

Most irrational numbers can be derived from formulae of varying degrees of
complexity but many are also *inherent* in nature. Pi is an example. It is
*literally* the ratio of a circle's circumference to a circle's diameter in
flat space. Other examples are the base of the natural logorithm e and the
copeland-erdos constant 0.235711131719232931374143... which one gets by
concatenating the prime numbers consecutively past the decimal point in base
ten.

There are algorthms and formulas for computing pi out to as many digits as
desired. There are no published closed-form formulas for computing the sequence
of prime numbers but there are primality tests out there that can determine if
a given number is prime. My point however is that both sequences are
simultaneously *absolutely determined* AND *randomly distributed*.

They are deterministic because they are constants and cannot be changed no
matter what goes on in your head. They are random because, you can't predict
what the next digit of the sequence will be unless you both *know* and *do* the
math. Furthermore it has been proven for some irrationals, and suspected for
many others, that the digits follow a uniform distribution. That is to say that
each digit occurs with equal probability like rolling a ten-sided die.

Now the truly astounding thing is this. In mathematics almost all complex and
real numbers are irrational. The rational numbers are countable. That is to say
that, even though there are infinitely more rational numbers than there are
protons, neutrons, electrons, neutrons, and photons in the universe, if you had
forever, you could in theory count all rational numbers. The irrationals are
uncountable. God himself cannot count the irrationals, not even in theory.

Now where am I going with all this? Well I'll leave you your philosophizing
about Copenhagen, MWI, Einstein, and Bell. Suffice to say that all this was
just an introduction to a mathematical toy I have been playing around with for
a while now. I call it the Coin function. It is a deterministic formula with an
associated irrational number that I hope will someday be called LaForge's
Constant or something similar. What this function does is take an input number,
usually an integer, and use that number to generate to generate either a 1 or a
0, heads or tails, if you would.

I have been studying this function for a while now, both analytically and
empirically, and it's output passes all statistical tests for randomness that I
have performed on it. I have yet to rigorously prove it to be random, but that
is difficult because there is no rigorous definition of random in math. For
example, every possible real sequence of coin tosses is guaranteed to be a
subsequence of the Coin sequence.

Here is the first 64 digits (bits actually) of the irrational Constant. the
Constant is also a transcendental number for those who grock:

0.1010000100001000010111010001110010100111000101111001101011011111...

When I used the Ent random number generator testing program to analyze a 1 MB
bitstream my function generated, it gave me these results:

Entropy = 1.000000 bits per bit.

Optimum compression would reduce the size
of this 8257536 bit file by 0 percent.

Chi square distribution for 8257536 samples is 0.12, and randomly
would exceed this value 72.99 percent of the times.

Arithmetic mean value of data bits is 0.4999 (0.5 = random).
Monte Carlo value for Pi is 3.112374442 (error 0.93 percent).
Serial correlation coefficient is -0.000092 (totally uncorrelated = 0.0).

Considering it is it is incompressible, yet the formula is only a few
characters long, Kolmogorov complexity/randomness takes on a whole new meaning.
Also notice that a truly random sequence of numbers can be used to derive pi by
the Monte Carlo method, although it takes *many* such random numbers.

In any case, I found that makes a really neat encryption algorithm that I call
Flipstream. Unlike 128 bit encryption that is the gold standard of the
encryption industry, Flipstream is infinite bit encryption and you don't need
any gigantic prime numbers, just the Coin Function and a private key that can
be any integer. Other amazing things about it is that the Flipstream algorithm
is less than 2KB long, and encrypted files are the same size as the original.
Furthermore, the same Coin Function both encodes and decodes the message,
flipping the text back and forth from plain text to encrypted text.

How uncrackable is Flipstream without the Coin function and the key? Well here
is a favorite text of mine. You all have have seen this text before but you
won't recognize it:

----begin encrypted text--------

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--------end encrypted text---------

I dare anybody to crack it. Hint: don't forget my constant, it will help with
the first 64 characters. Thereafter, I know you'll be able to figure out the
rest of the text because you will have seen it before. Also I made it easy on
you, the private key I used was the number 0.

I am also open to suggestions as to what to do with this technology. I am
sympathetic to the open-source software movement. But open source encryption
kind of defeats the point of trying to keep secrets in the first place.
Especially since in this case, all one needs is a "Magic Coin" and an integer
key. If you have the Coin, you can brute force the key. Perhaps an alternative
to the GNU public license would be an Extropian Public License or something
similar. Where I let *known* Extropes have it for free provided that they don't
give away the Coin to anyone that cannot be trusted to uphold the principles of
extropy. 

My implementation of Flipstream is in Python but if someone would like to a
write a compiled version where the Coin is invisible, I would welcome a
business partner. The Coin itself can be studied with a pencil, paper, and a
calculator for those of you aren't into computer programming.

Suggestions? Comments? Complaints?


Stuart LaForge
alt email: stuart"AT"ucla.edu

"In ancient times they had no statistics so that they had to fall back on lies."- Stephen Leacock


      



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