A Random essay on Random Numbers

A Random essay on Random Numbers

What do we mean by Random Number ?

Conventioanlly a set of numbers that are independent and identically distributed is called a random set of numbers. There are a
lot of hidden properties not very clearly defined here !!!. Huh!, what do you mean by a set of numbers, what is the cardinality
of the set, what do you mean by independent, and what is identically distributed ???.

A set of random numbers:: Usually in this context, it is any run of numbers. As an example, 1,1,1,1, 2, 901, 357, … Yes in this case
it is an infinite sequence, and might very well be a random sequence of numbers. Just by looking at first 4 or 5 terms, eyes tell
us that it is definitely not a random sequence, at least intuitively we can not say they are random, can we ! So in this infinte
sequence of number, how long do we need to wait and how long should the sequence be to be considered for us to determine if
it is a random sequence or not. Moreover, what do we do with very very large numbers that pops up once in a while in that rundom
sequence, for example 89123490812349809871098138973456710… upto 5000 digits, I can not possibly fit naturally inside my alorithm
for the generator of this sequence ??

Well, for the part of how long a sub sequence should we consider, sampling theory plays a major role, and all we need to do is
to have a fairly large sample, and many samples from the generator to statiscally justify that the gererators generate a
statistically random sequence. FAIRLY LARGE, AND FAIRLY MANY are not really interesting, since it has been well studied in the
area of statistics and DATA ANALYSIS. OKAY THEN, WE HAVE SOME REASONALBLY LARGE RUNS OF SEQUENCES, AND WE HAVE REASONABLY MANY
INSTANCES OF THOSE RUNS. But what about those single very large number !, how could we handle those in our algorithms?. THIS IS
ONE OF THE MANY REASONS THE TERM PSEUDO-RANDOM WAS COINED I THINK, but pls check.

Now we can see that an important aspect, cardinality of the run ( or set) is brushed away quite nicely for our practical purpose !.
But one thing to note is that generation of random events has been under rigorous studies for long time due to various curiosities
about the enviroment&nature around us … So they discoverd countable and un-countable infinites. When we look at most of the
literature we almost always see that the definitions and descriptions, almost always, encompasses (0, 1), an uncountably infite
set, as the domain of discourse.

AND THIS SET IS MOST PROBABLY CONSIDERED FROM TWO POINTS OF VIEW -

  1. Without loss of generality it captures the sense of time in an apparently small subsets
  2. And there is a one to one and onto map between (0,1) and (0, infinity), hence cardinality is preserved !!!

DO YOU STILL THINK YOU COULD BE A FAN OF CARDINALS !. Not me for sure.

What is Independent business ?

Given today is saturday, we know tomorrow must be sunday !. This is not independent. Essentially knowing any subsequence of any
length of the run, does not help in predicting what would be the next number of the sequence when the genrator produces the
next number. Any geometic sequence, or arithmatic sequence represents examples of dependent sequences… Most of the functions
we know of are not independent ( sin, cos, cosec, tan all are not independent …).

OKAY, WE CAN BELIVE THAT, THEN WHAT IS THE BUSINESS OF IDENTICALLY DISTRIBUTED ?

If we roll a dice, and say that we win when ever we get a face up that is not marked “Occured before” then the roles are not
identically distributed. First roll has 6 equally prossible event, 2nd one has 5 equally …, so on and so forth…

Instead, if we considered all faces on all trials, then each face is identically distributed from the uniform distribution in
[1 … 6]. Here the distribution happens to be uniform based on the trust we have on the maker of the dice ! No bias !

Given a random sequence of numbers, the underlying distribution ( that is the probability mass/density function ) could be
different. It could be exponential, it could be normal, poissson, and others. BUT WE ONLY CONSIDER UNIFORM DISTRIBUTION, SINCE
having a good analysis and a subsequent generator for a pseudo-random sequence of numbers within (0,1) is enough to have a
capability to generates other important distribution(s). Since the others are mere trasormation from uniform !!!

Today, most of the generator(s) of ordinary uses are based on :: Fast algorithm, large periods, indipendent and identically distributed
over (0, 1) following uniform distribution. AND THERE ARE SOME STANDARD TESTS TO FIND HOW GOOD IS A RANDOM GENRATOR. The tests
can be found in any good lituratures for Applied probability and Data Analysis.

Conclusion::

So giving a wide variety of algorithms to generate a pseudo-random sequence as well as different function(s) for different
use, one can easily pick one over the other by looking at the analysis made on those generators, or if very adventourous we
can always do those analysis taking samples from the generators… Alternatively, … trust someone or something.!!

Finally congruence is a favorite base to use in those generator, including symmetric crypto. Mainly due to its fastness, and already
proven capability to gernerate good enough rand(). Also some amount of mathematical simplicity is important to make it a
subject under analysis. CONGRUENCE IS WELL STUDIED UNDER NUMBER THEORY. CLASSES COMING OUT OF CONGRUNCES FORMS FIELDS
(AN ABSTRACT MATHEMATICAL OBJECT ) that obeys our common notion of algebra, and it is one of the method of coming up with lots
of finte fields. Then the immediate high in the hierarchy is the field of finite polinomials, and assymetric crypto graphy
depends on this by using finite polYnomials of large prime number. SO FACTORING POLYNOMIALS IS THE CRUX OF ITS BEAUTY.

Happy Halloween !

-pro

Hello,

Speaking of random numbers, randomness (a very
subtle and tricky topic of philosophy, I suggest
looking at Kolmogorov-Solomorov-Chaitin complexity:

http://www.cs.auckland.ac.nz/CDMTCS/chaitin/

http://homepages.cwi.nl/~tromp/cl/cl.html

http://homepages.cwi.nl/~tromp/cl/LC.pdf

Regards, Vasili

— Prokash Sinha wrote:

>
> A Random essay on Random Numbers
> ---------------------------------
>
> What do we mean by Random Number ?
>
> Conventioanlly a set of numbers that are independent
> and identically distributed is called a random set
> of numbers. There are a
> lot of hidden properties not very clearly defined
> here !!!. Huh!, what do you mean by a set of
> numbers, what is the cardinality
> of the set, what do you mean by independent, and
> what is identically distributed ???.
>
> A set of random numbers:: Usually in this context,
> it is any run of numbers. As an example, 1,1,1,1, 2,
> 901, 357, … Yes in this case
> it is an infinite sequence, and might very well be a
> random sequence of numbers. Just by looking at first
> 4 or 5 terms, eyes tell
> us that it is definitely not a random sequence, at
> least intuitively we can not say they are random,
> can we ! So in this infinte
> sequence of number, how long do we need to wait and
> how long should the sequence be to be considered
> for us to determine if
> it is a random sequence or not. Moreover, what do we
> do with very very large numbers that pops up once in
> a while in that rundom
> sequence, for example
> 89123490812349809871098138973456710… upto 5000
> digits, I can not possibly fit naturally inside my
> alorithm
> for the generator of this sequence ??
>
> Well, for the part of how long a sub sequence should
> we consider, sampling theory plays a major role, and
> all we need to do is
> to have a fairly large sample, and many samples from
> the generator to statiscally justify that the
> gererators generate a
> statistically random sequence. FAIRLY LARGE, AND
> FAIRLY MANY are not really interesting, since it has
> been well studied in the
> area of statistics and DATA ANALYSIS. OKAY THEN, WE
> HAVE SOME REASONALBLY LARGE RUNS OF SEQUENCES, AND
> WE HAVE REASONABLY MANY
> INSTANCES OF THOSE RUNS. But what about those single
> very large number !, how could we handle those in
> our algorithms?. THIS IS
> ONE OF THE MANY REASONS THE TERM PSEUDO-RANDOM WAS
> COINED I THINK, but pls check.
>
>
>
> Now we can see that an important aspect, cardinality
> of the run ( or set) is brushed away quite nicely
> for our practical purpose !.
> But one thing to note is that generation of random
> events has been under rigorous studies for long time
> due to various curiosities
> about the enviroment&nature around us … So they
> discoverd countable and un-countable infinites. When
> we look at most of the
> literature we almost always see that the definitions
> and descriptions, almost always, encompasses (0, 1),
> an uncountably infite
> set, as the domain of discourse.
>
> AND THIS SET IS MOST PROBABLY CONSIDERED FROM TWO
> POINTS OF VIEW -
>
> 1) Without loss of generality it captures the sense
> of time in an apparently small subsets
> 2) And there is a one to one and onto map between
> (0,1) and (0, infinity), hence cardinality is
> preserved !!!
>
> DO YOU STILL THINK YOU COULD BE A FAN OF CARDINALS
> !. Not me for sure.
>
>
> What is Independent business ?
>
> Given today is saturday, we know tomorrow must be
> sunday !. This is not independent. Essentially
> knowing any subsequence of any
> length of the run, does not help in predicting what
> would be the next number of the sequence when the
> genrator produces the
> next number. Any geometic sequence, or arithmatic
> sequence represents examples of dependent
> sequences… Most of the functions
> we know of are not independent ( sin, cos, cosec,
> tan all are not independent …).
>
> OKAY, WE CAN BELIVE THAT, THEN WHAT IS THE BUSINESS
> OF IDENTICALLY DISTRIBUTED ?
>
> If we roll a dice, and say that we win when ever we
> get a face up that is not marked “Occured before”
> then the roles are not
> identically distributed. First roll has 6 equally
> prossible event, 2nd one has 5 equally …, so on
> and so forth…
>
> Instead, if we considered all faces on all trials,
> then each face is identically distributed from the
> uniform distribution in
> [1 … 6]. Here the distribution happens to be
> uniform based on the trust we have on the maker of
> the dice ! No bias !
>
>
> Given a random sequence of numbers, the underlying
> distribution ( that is the probability mass/density
> function ) could be
> different. It could be exponential, it could be
> normal, poissson, and others. BUT WE ONLY CONSIDER
> UNIFORM DISTRIBUTION, SINCE
> having a good analysis and a subsequent generator
> for a pseudo-random sequence of numbers within (0,1)
> is enough to have a
> capability to generates other important
> distribution(s). Since the others are mere
> trasormation from uniform !!!
>
>
> Today, most of the generator(s) of ordinary uses are
> based on :: Fast algorithm, large periods,
> indipendent and identically distributed
> over (0, 1) following uniform distribution. AND
> THERE ARE SOME STANDARD TESTS TO FIND HOW GOOD IS A
> RANDOM GENRATOR. The tests
> can be found in any good lituratures for Applied
> probability and Data Analysis.
>
>
>
> Conclusion::
> -----------
>
> So giving a wide variety of algorithms to generate a
> pseudo-random sequence as well as different
> function(s) for different
> use, one can easily pick one over the other by
> looking at the analysis made on those generators, or
> if very adventourous we
> can always do those analysis taking samples from the
> generators… Alternatively, … trust someone or
> something.!!
>
> Finally congruence is a favorite base to use in
> those generator, including symmetric crypto. Mainly
> due to its fastness, and already
> proven capability to gernerate good enough rand().
> Also some amount of mathematical simplicity is
> important to make it a
> subject under analysis. CONGRUENCE IS WELL STUDIED
> UNDER NUMBER THEORY. CLASSES COMING OUT OF
> CONGRUNCES FORMS FIELDS
> (AN ABSTRACT MATHEMATICAL OBJECT ) that obeys our
> common notion of algebra, and it is one of the
> method of coming up with lots
> of finte fields. Then the immediate high in the
> hierarchy is the field of finite polinomials, and
> assymetric crypto graphy
> depends on this by using finite polYnomials of large
> prime number. SO FACTORING POLYNOMIALS IS THE CRUX
> OF ITS BEAUTY.
>
>
>
> Happy Halloween !
>
> -pro
>
>
>
>
>
>
>
>
> —
> Questions? First check the Kernel Driver FAQ at
> http://www.osronline.com/article.cfm?id=256
>
> You are currently subscribed to ntdev as: unknown
> lmsubst tag argument: ‘’
> To unsubscribe send a blank email to
xxxxx@lists.osr.com

_______________________________
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Declare Yourself - Register online to vote today!
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I see you have been reading Wolfram’s book. :slight_smile:

Jamey


From: xxxxx@lists.osr.com
[mailto:xxxxx@lists.osr.com] On Behalf Of Prokash Sinha
Sent: Saturday, October 16, 2004 11:23 AM
To: Windows System Software Devs Interest List
Subject: [ntdev] A Random essay on Random Numbers

A Random essay on Random Numbers

What do we mean by Random Number ?

Conventioanlly a set of numbers that are independent and identically
distributed is called a random set of numbers. There are a
lot of hidden properties not very clearly defined here !!!. Huh!, what do
you mean by a set of numbers, what is the cardinality
of the set, what do you mean by independent, and what is identically
distributed ???.

A set of random numbers:: Usually in this context, it is any run of numbers.
As an example, 1,1,1,1, 2, 901, 357, … Yes in this case
it is an infinite sequence, and might very well be a random sequence of
numbers. Just by looking at first 4 or 5 terms, eyes tell
us that it is definitely not a random sequence, at least intuitively we can
not say they are random, can we ! So in this infinte
sequence of number, how long do we need to wait and how long should the
sequence be to be considered for us to determine if
it is a random sequence or not. Moreover, what do we do with very very large
numbers that pops up once in a while in that rundom
sequence, for example 89123490812349809871098138973456710… upto 5000
digits, I can not possibly fit naturally inside my alorithm
for the generator of this sequence ??

Well, for the part of how long a sub sequence should we consider, sampling
theory plays a major role, and all we need to do is
to have a fairly large sample, and many samples from the generator to
statiscally justify that the gererators generate a
statistically random sequence. FAIRLY LARGE, AND FAIRLY MANY are not really
interesting, since it has been well studied in the
area of statistics and DATA ANALYSIS. OKAY THEN, WE HAVE SOME REASONALBLY
LARGE RUNS OF SEQUENCES, AND WE HAVE REASONABLY MANY
INSTANCES OF THOSE RUNS. But what about those single very large number !,
how could we handle those in our algorithms?. THIS IS
ONE OF THE MANY REASONS THE TERM PSEUDO-RANDOM WAS COINED I THINK, but pls
check.

Now we can see that an important aspect, cardinality of the run ( or set) is
brushed away quite nicely for our practical purpose !.
But one thing to note is that generation of random events has been under
rigorous studies for long time due to various curiosities
about the enviroment&nature around us … So they discoverd countable and
un-countable infinites. When we look at most of the
literature we almost always see that the definitions and descriptions,
almost always, encompasses (0, 1), an uncountably infite
set, as the domain of discourse.

AND THIS SET IS MOST PROBABLY CONSIDERED FROM TWO POINTS OF VIEW -

  1. Without loss of generality it captures the sense of time in an
    apparently small subsets
  2. And there is a one to one and onto map between (0,1) and (0, infinity),
    hence cardinality is preserved !!!

DO YOU STILL THINK YOU COULD BE A FAN OF CARDINALS !. Not me for sure.

What is Independent business ?

Given today is saturday, we know tomorrow must be sunday !. This is not
independent. Essentially knowing any subsequence of any
length of the run, does not help in predicting what would be the next number
of the sequence when the genrator produces the
next number. Any geometic sequence, or arithmatic sequence represents
examples of dependent sequences… Most of the functions
we know of are not independent ( sin, cos, cosec, tan all are not
independent …).

OKAY, WE CAN BELIVE THAT, THEN WHAT IS THE BUSINESS OF IDENTICALLY
DISTRIBUTED ?

If we roll a dice, and say that we win when ever we get a face up that is
not marked “Occured before” then the roles are not
identically distributed. First roll has 6 equally prossible event, 2nd one
has 5 equally …, so on and so forth…

Instead, if we considered all faces on all trials, then each face is
identically distributed from the uniform distribution in
[1 … 6]. Here the distribution happens to be uniform based on the trust we
have on the maker of the dice ! No bias !

Given a random sequence of numbers, the underlying distribution ( that is
the probability mass/density function ) could be
different. It could be exponential, it could be normal, poissson, and
others. BUT WE ONLY CONSIDER UNIFORM DISTRIBUTION, SINCE
having a good analysis and a subsequent generator for a pseudo-random
sequence of numbers within (0,1) is enough to have a
capability to generates other important distribution(s). Since the others
are mere trasormation from uniform !!!

Today, most of the generator(s) of ordinary uses are based on :: Fast
algorithm, large periods, indipendent and identically distributed
over (0, 1) following uniform distribution. AND THERE ARE SOME STANDARD
TESTS TO FIND HOW GOOD IS A RANDOM GENRATOR. The tests
can be found in any good lituratures for Applied probability and Data
Analysis.

Conclusion::

So giving a wide variety of algorithms to generate a pseudo-random sequence
as well as different function(s) for different
use, one can easily pick one over the other by looking at the analysis made
on those generators, or if very adventourous we
can always do those analysis taking samples from the generators…
Alternatively, … trust someone or something.!!

Finally congruence is a favorite base to use in those generator, including
symmetric crypto. Mainly due to its fastness, and already
proven capability to gernerate good enough rand(). Also some amount of
mathematical simplicity is important to make it a
subject under analysis. CONGRUENCE IS WELL STUDIED UNDER NUMBER THEORY.
CLASSES COMING OUT OF CONGRUNCES FORMS FIELDS
(AN ABSTRACT MATHEMATICAL OBJECT ) that obeys our common notion of algebra,
and it is one of the method of coming up with lots
of finte fields. Then the immediate high in the hierarchy is the field of
finite polinomials, and assymetric crypto graphy
depends on this by using finite polYnomials of large prime number. SO
FACTORING POLYNOMIALS IS THE CRUX OF ITS BEAUTY.

Happy Halloween !

-pro


Questions? First check the Kernel Driver FAQ at
http://www.osronline.com/article.cfm?id=256

You are currently subscribed to ntdev as: unknown lmsubst tag argument: ‘’
To unsubscribe send a blank email to xxxxx@lists.osr.com

__________ NOD32 1.860 (20040903) Information __________

This message was checked by NOD32 antivirus system.
http://www.nod32.com

Honestly I did not find the time to read it much. May be a page I read when
the book came :). But most of what I jotted down came from my understanding
when I was taking bunch of probablility class at Duke school of math under
prof. Lawler. One of the finest professor who can dilute a very stiff
subject to common audience. At that time he was reseraching on Fractal
compression, a very efficient probabilistic method of compression using
Brownian motion…

I will read Wolfram’s book during christmas holidays !

-pro
-----Original Message-----
From: xxxxx@lists.osr.com
[mailto:xxxxx@lists.osr.com]On Behalf Of Jamey Kirby
Sent: Saturday, October 16, 2004 3:08 PM
To: Windows System Software Devs Interest List
Subject: RE: [ntdev] A Random essay on Random Numbers

I see you have been reading Wolfram’s book. J

Jamey


From: xxxxx@lists.osr.com
[mailto:xxxxx@lists.osr.com] On Behalf Of Prokash Sinha
Sent: Saturday, October 16, 2004 11:23 AM
To: Windows System Software Devs Interest List
Subject: [ntdev] A Random essay on Random Numbers

A Random essay on Random Numbers

What do we mean by Random Number ?

Conventioanlly a set of numbers that are independent and identically
distributed is called a random set of numbers. There are a
lot of hidden properties not very clearly defined here !!!. Huh!, what do
you mean by a set of numbers, what is the cardinality
of the set, what do you mean by independent, and what is identically
distributed ???.

A set of random numbers:: Usually in this context, it is any run of
numbers. As an example, 1,1,1,1, 2, 901, 357, … Yes in this case
it is an infinite sequence, and might very well be a random sequence of
numbers. Just by looking at first 4 or 5 terms, eyes tell
us that it is definitely not a random sequence, at least intuitively we
can not say they are random, can we ! So in this infinte
sequence of number, how long do we need to wait and how long should the
sequence be to be considered for us to determine if
it is a random sequence or not. Moreover, what do we do with very very
large numbers that pops up once in a while in that rundom
sequence, for example 89123490812349809871098138973456710… upto 5000
digits, I can not possibly fit naturally inside my alorithm
for the generator of this sequence ??

Well, for the part of how long a sub sequence should we consider, sampling
theory plays a major role, and all we need to do is
to have a fairly large sample, and many samples from the generator to
statiscally justify that the gererators generate a
statistically random sequence. FAIRLY LARGE, AND FAIRLY MANY are not
really interesting, since it has been well studied in the
area of statistics and DATA ANALYSIS. OKAY THEN, WE HAVE SOME REASONALBLY
LARGE RUNS OF SEQUENCES, AND WE HAVE REASONABLY MANY
INSTANCES OF THOSE RUNS. But what about those single very large number !,
how could we handle those in our algorithms?. THIS IS
ONE OF THE MANY REASONS THE TERM PSEUDO-RANDOM WAS COINED I THINK, but pls
check.

Now we can see that an important aspect, cardinality of the run ( or set)
is brushed away quite nicely for our practical purpose !.
But one thing to note is that generation of random events has been under
rigorous studies for long time due to various curiosities
about the enviroment&nature around us … So they discoverd countable and
un-countable infinites. When we look at most of the
literature we almost always see that the definitions and descriptions,
almost always, encompasses (0, 1), an uncountably infite
set, as the domain of discourse.

AND THIS SET IS MOST PROBABLY CONSIDERED FROM TWO POINTS OF VIEW -

  1. Without loss of generality it captures the sense of time in an
    apparently small subsets
  2. And there is a one to one and onto map between (0,1) and (0,
    infinity), hence cardinality is preserved !!!

DO YOU STILL THINK YOU COULD BE A FAN OF CARDINALS !. Not me for sure.

What is Independent business ?

Given today is saturday, we know tomorrow must be sunday !. This is not
independent. Essentially knowing any subsequence of any
length of the run, does not help in predicting what would be the next
number of the sequence when the genrator produces the
next number. Any geometic sequence, or arithmatic sequence represents
examples of dependent sequences… Most of the functions
we know of are not independent ( sin, cos, cosec, tan all are not
independent …).

OKAY, WE CAN BELIVE THAT, THEN WHAT IS THE BUSINESS OF IDENTICALLY
DISTRIBUTED ?

If we roll a dice, and say that we win when ever we get a face up that is
not marked “Occured before” then the roles are not
identically distributed. First roll has 6 equally prossible event, 2nd one
has 5 equally …, so on and so forth…

Instead, if we considered all faces on all trials, then each face is
identically distributed from the uniform distribution in
[1 … 6]. Here the distribution happens to be uniform based on the trust
we have on the maker of the dice ! No bias !

Given a random sequence of numbers, the underlying distribution ( that is
the probability mass/density function ) could be
different. It could be exponential, it could be normal, poissson, and
others. BUT WE ONLY CONSIDER UNIFORM DISTRIBUTION, SINCE
having a good analysis and a subsequent generator for a pseudo-random
sequence of numbers within (0,1) is enough to have a
capability to generates other important distribution(s). Since the others
are mere trasormation from uniform !!!

Today, most of the generator(s) of ordinary uses are based on :: Fast
algorithm, large periods, indipendent and identically distributed
over (0, 1) following uniform distribution. AND THERE ARE SOME STANDARD
TESTS TO FIND HOW GOOD IS A RANDOM GENRATOR. The tests
can be found in any good lituratures for Applied probability and Data
Analysis.

Conclusion::

So giving a wide variety of algorithms to generate a pseudo-random
sequence as well as different function(s) for different
use, one can easily pick one over the other by looking at the analysis
made on those generators, or if very adventourous we
can always do those analysis taking samples from the generators…
Alternatively, … trust someone or something.!!

Finally congruence is a favorite base to use in those generator, including
symmetric crypto. Mainly due to its fastness, and already
proven capability to gernerate good enough rand(). Also some amount of
mathematical simplicity is important to make it a
subject under analysis. CONGRUENCE IS WELL STUDIED UNDER NUMBER THEORY.
CLASSES COMING OUT OF CONGRUNCES FORMS FIELDS
(AN ABSTRACT MATHEMATICAL OBJECT ) that obeys our common notion of
algebra, and it is one of the method of coming up with lots
of finte fields. Then the immediate high in the hierarchy is the field of
finite polinomials, and assymetric crypto graphy
depends on this by using finite polYnomials of large prime number. SO
FACTORING POLYNOMIALS IS THE CRUX OF ITS BEAUTY.

Happy Halloween !

-pro


Questions? First check the Kernel Driver FAQ at
http://www.osronline.com/article.cfm?id=256

You are currently subscribed to ntdev as: unknown lmsubst tag argument: ‘’
To unsubscribe send a blank email to xxxxx@lists.osr.com

__________ NOD32 1.860 (20040903) Information __________

This message was checked by NOD32 antivirus system.
http://www.nod32.com


Questions? First check the Kernel Driver FAQ at
http://www.osronline.com/article.cfm?id=256

You are currently subscribed to ntdev as: unknown lmsubst tag argument: ‘’
To unsubscribe send a blank email to xxxxx@lists.osr.com

Speaking of compression in general,
Kolmogorov-Solomoroff-Chaitin complexity (i.e.
algorithmic information theory) has some nice
applications witnessed by Vitanyi’s and Li’s book:
http://www.amazon.com/exec/obidos/tg/detail/-/0387940537/qid=1098118087/sr=1-3/ref=sr_1_3/002-8995064-9203248?v=glance&s=books

Regards, Vasili

— Prokash Sinha wrote:

> Honestly I did not find the time to read it much.
> May be a page I read when
> the book came :). But most of what I jotted down
> came from my understanding
> when I was taking bunch of probablility class at
> Duke school of math under
> prof. Lawler. One of the finest professor who can
> dilute a very stiff
> subject to common audience. At that time he was
> reseraching on Fractal
> compression, a very efficient probabilistic method
> of compression using
> Brownian motion…
>
> I will read Wolfram’s book during christmas holidays
> !
>
> -pro
> -----Original Message-----
> From: xxxxx@lists.osr.com
> [mailto:xxxxx@lists.osr.com]On Behalf
> Of Jamey Kirby
> Sent: Saturday, October 16, 2004 3:08 PM
> To: Windows System Software Devs Interest List
> Subject: RE: [ntdev] A Random essay on Random
> Numbers
>
>
> I see you have been reading Wolfram’s book. J
>
>
>
> Jamey
>
>
>
>
>
----------------------------------------------------------------------------
> –
>
> From: xxxxx@lists.osr.com
> [mailto:xxxxx@lists.osr.com] On Behalf
> Of Prokash Sinha
> Sent: Saturday, October 16, 2004 11:23 AM
> To: Windows System Software Devs Interest List
> Subject: [ntdev] A Random essay on Random Numbers
>
>
>
>
> A Random essay on Random Numbers
> ---------------------------------
>
>
>
> What do we mean by Random Number ?
>
>
>
> Conventioanlly a set of numbers that are
> independent and identically
> distributed is called a random set of numbers. There
> are a
> lot of hidden properties not very clearly defined
> here !!!. Huh!, what do
> you mean by a set of numbers, what is the
> cardinality
> of the set, what do you mean by independent, and
> what is identically
> distributed ???.
>
>
>
> A set of random numbers:: Usually in this context,
> it is any run of
> numbers. As an example, 1,1,1,1, 2, 901, 357, …
> Yes in this case
> it is an infinite sequence, and might very well be
> a random sequence of
> numbers. Just by looking at first 4 or 5 terms, eyes
> tell
> us that it is definitely not a random sequence, at
> least intuitively we
> can not say they are random, can we ! So in this
> infinte
> sequence of number, how long do we need to wait
> and how long should the
> sequence be to be considered for us to determine if
> it is a random sequence or not. Moreover, what do
> we do with very very
> large numbers that pops up once in a while in that
> rundom
> sequence, for example
> 89123490812349809871098138973456710… upto 5000
> digits, I can not possibly fit naturally inside my
> alorithm
> for the generator of this sequence ??
>
>
>
> Well, for the part of how long a sub sequence
> should we consider, sampling
> theory plays a major role, and all we need to do is
> to have a fairly large sample, and many samples
> from the generator to
> statiscally justify that the gererators generate a
> statistically random sequence. FAIRLY LARGE, AND
> FAIRLY MANY are not
> really interesting, since it has been well studied
> in the
> area of statistics and DATA ANALYSIS. OKAY THEN,
> WE HAVE SOME REASONALBLY
> LARGE RUNS OF SEQUENCES, AND WE HAVE REASONABLY MANY
> INSTANCES OF THOSE RUNS. But what about those
> single very large number !,
> how could we handle those in our algorithms?. THIS
> IS
> ONE OF THE MANY REASONS THE TERM PSEUDO-RANDOM WAS
> COINED I THINK, but pls
> check.
>
>
>
>
>
>
>
> Now we can see that an important aspect,
> cardinality of the run ( or set)
> is brushed away quite nicely for our practical
> purpose !.
> But one thing to note is that generation of random
> events has been under
> rigorous studies for long time due to various
> curiosities
> about the enviroment&nature around us … So they
> discoverd countable and
> un-countable infinites. When we look at most of the
> literature we almost always see that the
> definitions and descriptions,
> almost always, encompasses (0, 1), an uncountably
> infite
> set, as the domain of discourse.
>
>
>
> AND THIS SET IS MOST PROBABLY CONSIDERED FROM TWO
> POINTS OF VIEW -
>
>
>
> 1) Without loss of generality it captures the
> sense of time in an
> apparently small subsets
> 2) And there is a one to one and onto map between
> (0,1) and (0,
> infinity), hence cardinality is preserved !!!
>
>
>
> DO YOU STILL THINK YOU COULD BE A FAN OF CARDINALS
> !. Not me for sure.
>
>
>
>
> What is Independent business ?
>
>
>
> Given today is saturday, we know tomorrow must be
> sunday !. This is not
> independent. Essentially knowing any subsequence of
> any
> length of the run, does not help in predicting
> what would be the next
> number of the sequence when the genrator produces
> the
> next number. Any geometic sequence, or arithmatic
> sequence represents
> examples of dependent sequences… Most of the
> functions
> we know of are not independent ( sin, cos, cosec,
> tan all are not
> independent …).
>
>
>
> OKAY, WE CAN BELIVE THAT, THEN WHAT IS THE
> BUSINESS OF IDENTICALLY
> DISTRIBUTED ?
>
>
>
> If we roll a dice, and say that we win when ever
> we get a face up that is
> not marked “Occured before” then the roles are not
> identically distributed. First roll has 6 equally
> prossible event, 2nd one
> has 5 equally …, so on and so forth…
>
>
=== message truncated ===

_______________________________
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Speaking of compression in general,
Kolmogorov-Solomoroff-Chaitin complexity (i.e.
algorithmic information theory) has some nice
applications witnessed by Vitanyi’s and Li’s book:
http://www.amazon.com/exec/obidos/tg/detail/-/0387940537/qid=1098118087/sr=1-3/ref=sr_1_3/002-8995064-9203248?v=glance&s=books

Regards, Vasili

— Prokash Sinha wrote:

> Honestly I did not find the time to read it much.
> May be a page I read when
> the book came :). But most of what I jotted down
> came from my understanding
> when I was taking bunch of probablility class at
> Duke school of math under
> prof. Lawler. One of the finest professor who can
> dilute a very stiff
> subject to common audience. At that time he was
> reseraching on Fractal
> compression, a very efficient probabilistic method
> of compression using
> Brownian motion…
>
> I will read Wolfram’s book during christmas holidays
> !
>
> -pro
> -----Original Message-----
> From: xxxxx@lists.osr.com
> [mailto:xxxxx@lists.osr.com]On Behalf
> Of Jamey Kirby
> Sent: Saturday, October 16, 2004 3:08 PM
> To: Windows System Software Devs Interest List
> Subject: RE: [ntdev] A Random essay on Random
> Numbers
>
>
> I see you have been reading Wolfram’s book. J
>
>
>
> Jamey
>
>
>
>
>
----------------------------------------------------------------------------
> –
>
> From: xxxxx@lists.osr.com
> [mailto:xxxxx@lists.osr.com] On Behalf
> Of Prokash Sinha
> Sent: Saturday, October 16, 2004 11:23 AM
> To: Windows System Software Devs Interest List
> Subject: [ntdev] A Random essay on Random Numbers
>
>
>
>
> A Random essay on Random Numbers
> ---------------------------------
>
>
>
> What do we mean by Random Number ?
>
>
>
> Conventioanlly a set of numbers that are
> independent and identically
> distributed is called a random set of numbers. There
> are a
> lot of hidden properties not very clearly defined
> here !!!. Huh!, what do
> you mean by a set of numbers, what is the
> cardinality
> of the set, what do you mean by independent, and
> what is identically
> distributed ???.
>
>
>
> A set of random numbers:: Usually in this context,
> it is any run of
> numbers. As an example, 1,1,1,1, 2, 901, 357, …
> Yes in this case
> it is an infinite sequence, and might very well be
> a random sequence of
> numbers. Just by looking at first 4 or 5 terms, eyes
> tell
> us that it is definitely not a random sequence, at
> least intuitively we
> can not say they are random, can we ! So in this
> infinte
> sequence of number, how long do we need to wait
> and how long should the
> sequence be to be considered for us to determine if
> it is a random sequence or not. Moreover, what do
> we do with very very
> large numbers that pops up once in a while in that
> rundom
> sequence, for example
> 89123490812349809871098138973456710… upto 5000
> digits, I can not possibly fit naturally inside my
> alorithm
> for the generator of this sequence ??
>
>
>
> Well, for the part of how long a sub sequence
> should we consider, sampling
> theory plays a major role, and all we need to do is
> to have a fairly large sample, and many samples
> from the generator to
> statiscally justify that the gererators generate a
> statistically random sequence. FAIRLY LARGE, AND
> FAIRLY MANY are not
> really interesting, since it has been well studied
> in the
> area of statistics and DATA ANALYSIS. OKAY THEN,
> WE HAVE SOME REASONALBLY
> LARGE RUNS OF SEQUENCES, AND WE HAVE REASONABLY MANY
> INSTANCES OF THOSE RUNS. But what about those
> single very large number !,
> how could we handle those in our algorithms?. THIS
> IS
> ONE OF THE MANY REASONS THE TERM PSEUDO-RANDOM WAS
> COINED I THINK, but pls
> check.
>
>
>
>
>
>
>
> Now we can see that an important aspect,
> cardinality of the run ( or set)
> is brushed away quite nicely for our practical
> purpose !.
> But one thing to note is that generation of random
> events has been under
> rigorous studies for long time due to various
> curiosities
> about the enviroment&nature around us … So they
> discoverd countable and
> un-countable infinites. When we look at most of the
> literature we almost always see that the
> definitions and descriptions,
> almost always, encompasses (0, 1), an uncountably
> infite
> set, as the domain of discourse.
>
>
>
> AND THIS SET IS MOST PROBABLY CONSIDERED FROM TWO
> POINTS OF VIEW -
>
>
>
> 1) Without loss of generality it captures the
> sense of time in an
> apparently small subsets
> 2) And there is a one to one and onto map between
> (0,1) and (0,
> infinity), hence cardinality is preserved !!!
>
>
>
> DO YOU STILL THINK YOU COULD BE A FAN OF CARDINALS
> !. Not me for sure.
>
>
>
>
> What is Independent business ?
>
>
>
> Given today is saturday, we know tomorrow must be
> sunday !. This is not
> independent. Essentially knowing any subsequence of
> any
> length of the run, does not help in predicting
> what would be the next
> number of the sequence when the genrator produces
> the
> next number. Any geometic sequence, or arithmatic
> sequence represents
> examples of dependent sequences… Most of the
> functions
> we know of are not independent ( sin, cos, cosec,
> tan all are not
> independent …).
>
>
>
> OKAY, WE CAN BELIVE THAT, THEN WHAT IS THE
> BUSINESS OF IDENTICALLY
> DISTRIBUTED ?
>
>
>
> If we roll a dice, and say that we win when ever
> we get a face up that is
> not marked “Occured before” then the roles are not
> identically distributed. First roll has 6 equally
> prossible event, 2nd one
> has 5 equally …, so on and so forth…
>
>
=== message truncated ===

__________________________________
Do you Yahoo!?
Y! Messenger - Communicate in real time. Download now.
http://messenger.yahoo.com

Thanx Vasili, for the infos :). There are some interesting readings for sure …, and I will look into those, just in case …

thanx again
-pro

Depends on purpose.

If the purpose is crypto (“unguessable” is a key) - use CryptGenRandom, or
calculate SHA1 or MD5 hash of some high-enthropy data like the current system
load.

If the purpose is Monte-Carlo style math modeling of something - then “good
distribution” is a key, and yes, Vasili is correct for this.

Maxim Shatskih, Windows DDK MVP
StorageCraft Corporation
xxxxx@storagecraft.com
http://www.storagecraft.com

----- Original Message -----
From: “Galchin Vasili”
To: “Windows System Software Devs Interest List”
Sent: Sunday, October 17, 2004 12:14 AM
Subject: Re: [ntdev] A Random essay on Random Numbers

> Hello,
>
> Speaking of random numbers, randomness (a very
> subtle and tricky topic of philosophy, I suggest
> looking at Kolmogorov-Solomorov-Chaitin complexity:
>
> http://www.cs.auckland.ac.nz/CDMTCS/chaitin/
>
> http://homepages.cwi.nl/~tromp/cl/cl.html
>
> http://homepages.cwi.nl/~tromp/cl/LC.pdf
>
> Regards, Vasili
>
> — Prokash Sinha wrote:
>
> >
> > A Random essay on Random Numbers
> > ---------------------------------
> >
> > What do we mean by Random Number ?
> >
> > Conventioanlly a set of numbers that are independent
> > and identically distributed is called a random set
> > of numbers. There are a
> > lot of hidden properties not very clearly defined
> > here !!!. Huh!, what do you mean by a set of
> > numbers, what is the cardinality
> > of the set, what do you mean by independent, and
> > what is identically distributed ???.
> >
> > A set of random numbers:: Usually in this context,
> > it is any run of numbers. As an example, 1,1,1,1, 2,
> > 901, 357, … Yes in this case
> > it is an infinite sequence, and might very well be a
> > random sequence of numbers. Just by looking at first
> > 4 or 5 terms, eyes tell
> > us that it is definitely not a random sequence, at
> > least intuitively we can not say they are random,
> > can we ! So in this infinte
> > sequence of number, how long do we need to wait and
> > how long should the sequence be to be considered
> > for us to determine if
> > it is a random sequence or not. Moreover, what do we
> > do with very very large numbers that pops up once in
> > a while in that rundom
> > sequence, for example
> > 89123490812349809871098138973456710… upto 5000
> > digits, I can not possibly fit naturally inside my
> > alorithm
> > for the generator of this sequence ??
> >
> > Well, for the part of how long a sub sequence should
> > we consider, sampling theory plays a major role, and
> > all we need to do is
> > to have a fairly large sample, and many samples from
> > the generator to statiscally justify that the
> > gererators generate a
> > statistically random sequence. FAIRLY LARGE, AND
> > FAIRLY MANY are not really interesting, since it has
> > been well studied in the
> > area of statistics and DATA ANALYSIS. OKAY THEN, WE
> > HAVE SOME REASONALBLY LARGE RUNS OF SEQUENCES, AND
> > WE HAVE REASONABLY MANY
> > INSTANCES OF THOSE RUNS. But what about those single
> > very large number !, how could we handle those in
> > our algorithms?. THIS IS
> > ONE OF THE MANY REASONS THE TERM PSEUDO-RANDOM WAS
> > COINED I THINK, but pls check.
> >
> >
> >
> > Now we can see that an important aspect, cardinality
> > of the run ( or set) is brushed away quite nicely
> > for our practical purpose !.
> > But one thing to note is that generation of random
> > events has been under rigorous studies for long time
> > due to various curiosities
> > about the enviroment&nature around us … So they
> > discoverd countable and un-countable infinites. When
> > we look at most of the
> > literature we almost always see that the definitions
> > and descriptions, almost always, encompasses (0, 1),
> > an uncountably infite
> > set, as the domain of discourse.
> >
> > AND THIS SET IS MOST PROBABLY CONSIDERED FROM TWO
> > POINTS OF VIEW -
> >
> > 1) Without loss of generality it captures the sense
> > of time in an apparently small subsets
> > 2) And there is a one to one and onto map between
> > (0,1) and (0, infinity), hence cardinality is
> > preserved !!!
> >
> > DO YOU STILL THINK YOU COULD BE A FAN OF CARDINALS
> > !. Not me for sure.
> >
> >
> > What is Independent business ?
> >
> > Given today is saturday, we know tomorrow must be
> > sunday !. This is not independent. Essentially
> > knowing any subsequence of any
> > length of the run, does not help in predicting what
> > would be the next number of the sequence when the
> > genrator produces the
> > next number. Any geometic sequence, or arithmatic
> > sequence represents examples of dependent
> > sequences… Most of the functions
> > we know of are not independent ( sin, cos, cosec,
> > tan all are not independent …).
> >
> > OKAY, WE CAN BELIVE THAT, THEN WHAT IS THE BUSINESS
> > OF IDENTICALLY DISTRIBUTED ?
> >
> > If we roll a dice, and say that we win when ever we
> > get a face up that is not marked “Occured before”
> > then the roles are not
> > identically distributed. First roll has 6 equally
> > prossible event, 2nd one has 5 equally …, so on
> > and so forth…
> >
> > Instead, if we considered all faces on all trials,
> > then each face is identically distributed from the
> > uniform distribution in
> > [1 … 6]. Here the distribution happens to be
> > uniform based on the trust we have on the maker of
> > the dice ! No bias !
> >
> >
> > Given a random sequence of numbers, the underlying
> > distribution ( that is the probability mass/density
> > function ) could be
> > different. It could be exponential, it could be
> > normal, poissson, and others. BUT WE ONLY CONSIDER
> > UNIFORM DISTRIBUTION, SINCE
> > having a good analysis and a subsequent generator
> > for a pseudo-random sequence of numbers within (0,1)
> > is enough to have a
> > capability to generates other important
> > distribution(s). Since the others are mere
> > trasormation from uniform !!!
> >
> >
> > Today, most of the generator(s) of ordinary uses are
> > based on :: Fast algorithm, large periods,
> > indipendent and identically distributed
> > over (0, 1) following uniform distribution. AND
> > THERE ARE SOME STANDARD TESTS TO FIND HOW GOOD IS A
> > RANDOM GENRATOR. The tests
> > can be found in any good lituratures for Applied
> > probability and Data Analysis.
> >
> >
> >
> > Conclusion::
> > -----------
> >
> > So giving a wide variety of algorithms to generate a
> > pseudo-random sequence as well as different
> > function(s) for different
> > use, one can easily pick one over the other by
> > looking at the analysis made on those generators, or
> > if very adventourous we
> > can always do those analysis taking samples from the
> > generators… Alternatively, … trust someone or
> > something.!!
> >
> > Finally congruence is a favorite base to use in
> > those generator, including symmetric crypto. Mainly
> > due to its fastness, and already
> > proven capability to gernerate good enough rand().
> > Also some amount of mathematical simplicity is
> > important to make it a
> > subject under analysis. CONGRUENCE IS WELL STUDIED
> > UNDER NUMBER THEORY. CLASSES COMING OUT OF
> > CONGRUNCES FORMS FIELDS
> > (AN ABSTRACT MATHEMATICAL OBJECT ) that obeys our
> > common notion of algebra, and it is one of the
> > method of coming up with lots
> > of finte fields. Then the immediate high in the
> > hierarchy is the field of finite polinomials, and
> > assymetric crypto graphy
> > depends on this by using finite polYnomials of large
> > prime number. SO FACTORING POLYNOMIALS IS THE CRUX
> > OF ITS BEAUTY.
> >
> >
> >
> > Happy Halloween !
> >
> > -pro
> >
> >
> >
> >
> >
> >
> >
> >
> > —
> > Questions? First check the Kernel Driver FAQ at
> > http://www.osronline.com/article.cfm?id=256
> >
> > You are currently subscribed to ntdev as: unknown
> > lmsubst tag argument: ‘’
> > To unsubscribe send a blank email to
> xxxxx@lists.osr.com
>
>
>
>
> _______________________________
> Do you Yahoo!?
> Declare Yourself - Register online to vote today!
> http://vote.yahoo.com
>
>
> —
> Questions? First check the Kernel Driver FAQ at
http://www.osronline.com/article.cfm?id=256
>
> You are currently subscribed to ntdev as: xxxxx@storagecraft.com
> To unsubscribe send a blank email to xxxxx@lists.osr.com

I know what you were trying to say, but to put in a few extra cents: the
current system load is nowhere nearly high enough entropy to use for
crypto. Read the docs for CryptGenRandom to get some idea of the
complexity of finding sufficiently high entropy bits.

Maxim S. Shatskih wrote:

Depends on purpose.

If the purpose is crypto (“unguessable” is a key) - use CryptGenRandom, or
calculate SHA1 or MD5 hash of some high-enthropy data like the current system
load.

If the purpose is Monte-Carlo style math modeling of something - then “good
distribution” is a key, and yes, Vasili is correct for this.

Maxim Shatskih, Windows DDK MVP
StorageCraft Corporation
xxxxx@storagecraft.com
http://www.storagecraft.com

----- Original Message -----
From: “Galchin Vasili”
> To: “Windows System Software Devs Interest List”
> Sent: Sunday, October 17, 2004 12:14 AM
> Subject: Re: [ntdev] A Random essay on Random Numbers
>
>
>
>>Hello,
>>
>> Speaking of random numbers, randomness (a very
>>subtle and tricky topic of philosophy, I suggest
>>looking at Kolmogorov-Solomorov-Chaitin complexity:
>>
>>http://www.cs.auckland.ac.nz/CDMTCS/chaitin/
>>
>>http://homepages.cwi.nl/~tromp/cl/cl.html
>>
>>http://homepages.cwi.nl/~tromp/cl/LC.pdf
>>
>>Regards, Vasili
>>
>>— Prokash Sinha wrote:
>>
>>
>>>A Random essay on Random Numbers
>>>---------------------------------
>>>
>>>What do we mean by Random Number ?
>>>
>>>Conventioanlly a set of numbers that are independent
>>>and identically distributed is called a random set
>>>of numbers. There are a
>>>lot of hidden properties not very clearly defined
>>>here !!!. Huh!, what do you mean by a set of
>>>numbers, what is the cardinality
>>>of the set, what do you mean by independent, and
>>>what is identically distributed ???.
>>>
>>>A set of random numbers:: Usually in this context,
>>>it is any run of numbers. As an example, 1,1,1,1, 2,
>>>901, 357, … Yes in this case
>>>it is an infinite sequence, and might very well be a
>>>random sequence of numbers. Just by looking at first
>>>4 or 5 terms, eyes tell
>>>us that it is definitely not a random sequence, at
>>>least intuitively we can not say they are random,
>>>can we ! So in this infinte
>>>sequence of number, how long do we need to wait and
>>>how long should the sequence be to be considered
>>>for us to determine if
>>>it is a random sequence or not. Moreover, what do we
>>>do with very very large numbers that pops up once in
>>>a while in that rundom
>>>sequence, for example
>>>89123490812349809871098138973456710… upto 5000
>>>digits, I can not possibly fit naturally inside my
>>>alorithm
>>>for the generator of this sequence ??
>>>
>>>Well, for the part of how long a sub sequence should
>>>we consider, sampling theory plays a major role, and
>>>all we need to do is
>>>to have a fairly large sample, and many samples from
>>>the generator to statiscally justify that the
>>>gererators generate a
>>>statistically random sequence. FAIRLY LARGE, AND
>>>FAIRLY MANY are not really interesting, since it has
>>>been well studied in the
>>>area of statistics and DATA ANALYSIS. OKAY THEN, WE
>>>HAVE SOME REASONALBLY LARGE RUNS OF SEQUENCES, AND
>>>WE HAVE REASONABLY MANY
>>>INSTANCES OF THOSE RUNS. But what about those single
>>>very large number !, how could we handle those in
>>>our algorithms?. THIS IS
>>>ONE OF THE MANY REASONS THE TERM PSEUDO-RANDOM WAS
>>>COINED I THINK, but pls check.
>>>
>>>
>>>
>>>Now we can see that an important aspect, cardinality
>>>of the run ( or set) is brushed away quite nicely
>>>for our practical purpose !.
>>>But one thing to note is that generation of random
>>>events has been under rigorous studies for long time
>>>due to various curiosities
>>>about the enviroment&nature around us … So they
>>>discoverd countable and un-countable infinites. When
>>>we look at most of the
>>>literature we almost always see that the definitions
>>>and descriptions, almost always, encompasses (0, 1),
>>>an uncountably infite
>>>set, as the domain of discourse.
>>>
>>> AND THIS SET IS MOST PROBABLY CONSIDERED FROM TWO
>>>POINTS OF VIEW -
>>>
>>> 1) Without loss of generality it captures the sense
>>>of time in an apparently small subsets
>>> 2) And there is a one to one and onto map between
>>>(0,1) and (0, infinity), hence cardinality is
>>>preserved !!!
>>>
>>>DO YOU STILL THINK YOU COULD BE A FAN OF CARDINALS
>>>!. Not me for sure.
>>>
>>>
>>>What is Independent business ?
>>>
>>>Given today is saturday, we know tomorrow must be
>>>sunday !. This is not independent. Essentially
>>>knowing any subsequence of any
>>>length of the run, does not help in predicting what
>>>would be the next number of the sequence when the
>>>genrator produces the
>>>next number. Any geometic sequence, or arithmatic
>>>sequence represents examples of dependent
>>>sequences… Most of the functions
>>>we know of are not independent ( sin, cos, cosec,
>>>tan all are not independent …).
>>>
>>>OKAY, WE CAN BELIVE THAT, THEN WHAT IS THE BUSINESS
>>>OF IDENTICALLY DISTRIBUTED ?
>>>
>>>If we roll a dice, and say that we win when ever we
>>>get a face up that is not marked “Occured before”
>>>then the roles are not
>>>identically distributed. First roll has 6 equally
>>>prossible event, 2nd one has 5 equally …, so on
>>>and so forth…
>>>
>>>Instead, if we considered all faces on all trials,
>>>then each face is identically distributed from the
>>>uniform distribution in
>>>[1 … 6]. Here the distribution happens to be
>>>uniform based on the trust we have on the maker of
>>>the dice ! No bias !
>>>
>>>
>>>Given a random sequence of numbers, the underlying
>>>distribution ( that is the probability mass/density
>>>function ) could be
>>>different. It could be exponential, it could be
>>>normal, poissson, and others. BUT WE ONLY CONSIDER
>>>UNIFORM DISTRIBUTION, SINCE
>>>having a good analysis and a subsequent generator
>>>for a pseudo-random sequence of numbers within (0,1)
>>>is enough to have a
>>>capability to generates other important
>>>distribution(s). Since the others are mere
>>>trasormation from uniform !!!
>>>
>>>
>>>Today, most of the generator(s) of ordinary uses are
>>>based on :: Fast algorithm, large periods,
>>>indipendent and identically distributed
>>>over (0, 1) following uniform distribution. AND
>>>THERE ARE SOME STANDARD TESTS TO FIND HOW GOOD IS A
>>>RANDOM GENRATOR. The tests
>>>can be found in any good lituratures for Applied
>>>probability and Data Analysis.
>>>
>>>
>>>
>>>Conclusion::
>>>-----------
>>>
>>>So giving a wide variety of algorithms to generate a
>>>pseudo-random sequence as well as different
>>>function(s) for different
>>>use, one can easily pick one over the other by
>>>looking at the analysis made on those generators, or
>>>if very adventourous we
>>>can always do those analysis taking samples from the
>>>generators… Alternatively, … trust someone or
>>>something.!!
>>>
>>>Finally congruence is a favorite base to use in
>>>those generator, including symmetric crypto. Mainly
>>>due to its fastness, and already
>>>proven capability to gernerate good enough rand().
>>>Also some amount of mathematical simplicity is
>>>important to make it a
>>>subject under analysis. CONGRUENCE IS WELL STUDIED
>>>UNDER NUMBER THEORY. CLASSES COMING OUT OF
>>>CONGRUNCES FORMS FIELDS
>>>(AN ABSTRACT MATHEMATICAL OBJECT ) that obeys our
>>>common notion of algebra, and it is one of the
>>>method of coming up with lots
>>>of finte fields. Then the immediate high in the
>>>hierarchy is the field of finite polinomials, and
>>>assymetric crypto graphy
>>>depends on this by using finite polYnomials of large
>>>prime number. SO FACTORING POLYNOMIALS IS THE CRUX
>>>OF ITS BEAUTY.
>>>
>>>
>>>
>>> Happy Halloween !
>>>
>>>-pro
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>—
>>>Questions? First check the Kernel Driver FAQ at
>>>http://www.osronline.com/article.cfm?id=256
>>>
>>>You are currently subscribed to ntdev as: unknown
>>>lmsubst tag argument: ‘’
>>>To unsubscribe send a blank email to
>>
>>xxxxx@lists.osr.com
>>
>>
>>
>>
>> _______________________________
>>Do you Yahoo!?
>>Declare Yourself - Register online to vote today!
>>http://vote.yahoo.com
>>
>>
>>—
>>Questions? First check the Kernel Driver FAQ at
>
> http://www.osronline.com/article.cfm?id=256
>
>>You are currently subscribed to ntdev as: xxxxx@storagecraft.com
>>To unsubscribe send a blank email to xxxxx@lists.osr.com
>
>
>
>


…/ray..

Please remove “.spamblock” from my email address if you need to contact
me outside the newsgroup.

Just the detailed stats on OS load.

Maxim Shatskih, Windows DDK MVP
StorageCraft Corporation
xxxxx@storagecraft.com
http://www.storagecraft.com

----- Original Message -----
From: “Ray Trent”
Newsgroups: ntdev
To: “Windows System Software Devs Interest List”
Sent: Wednesday, October 20, 2004 11:48 PM
Subject: Re:[ntdev] A Random essay on Random Numbers

> I know what you were trying to say, but to put in a few extra cents: the
> current system load is nowhere nearly high enough entropy to use for
> crypto. Read the docs for CryptGenRandom to get some idea of the
> complexity of finding sufficiently high entropy bits.
>
> Maxim S. Shatskih wrote:
> > Depends on purpose.
> >
> > If the purpose is crypto (“unguessable” is a key) - use CryptGenRandom,
or
> > calculate SHA1 or MD5 hash of some high-enthropy data like the current
system
> > load.
> >
> > If the purpose is Monte-Carlo style math modeling of something - then
“good
> > distribution” is a key, and yes, Vasili is correct for this.
> >
> > Maxim Shatskih, Windows DDK MVP
> > StorageCraft Corporation
> > xxxxx@storagecraft.com
> > http://www.storagecraft.com
> >
> > ----- Original Message -----
> > From: “Galchin Vasili”
> > To: “Windows System Software Devs Interest List”
> > Sent: Sunday, October 17, 2004 12:14 AM
> > Subject: Re: [ntdev] A Random essay on Random Numbers
> >
> >
> >
> >>Hello,
> >>
> >> Speaking of random numbers, randomness (a very
> >>subtle and tricky topic of philosophy, I suggest
> >>looking at Kolmogorov-Solomorov-Chaitin complexity:
> >>
> >>http://www.cs.auckland.ac.nz/CDMTCS/chaitin/
> >>
> >>http://homepages.cwi.nl/~tromp/cl/cl.html
> >>
> >>http://homepages.cwi.nl/~tromp/cl/LC.pdf
> >>
> >>Regards, Vasili
> >>
> >>— Prokash Sinha wrote:
> >>
> >>
> >>>A Random essay on Random Numbers
> >>>---------------------------------
> >>>
> >>>What do we mean by Random Number ?
> >>>
> >>>Conventioanlly a set of numbers that are independent
> >>>and identically distributed is called a random set
> >>>of numbers. There are a
> >>>lot of hidden properties not very clearly defined
> >>>here !!!. Huh!, what do you mean by a set of
> >>>numbers, what is the cardinality
> >>>of the set, what do you mean by independent, and
> >>>what is identically distributed ???.
> >>>
> >>>A set of random numbers:: Usually in this context,
> >>>it is any run of numbers. As an example, 1,1,1,1, 2,
> >>>901, 357, … Yes in this case
> >>>it is an infinite sequence, and might very well be a
> >>>random sequence of numbers. Just by looking at first
> >>>4 or 5 terms, eyes tell
> >>>us that it is definitely not a random sequence, at
> >>>least intuitively we can not say they are random,
> >>>can we ! So in this infinte
> >>>sequence of number, how long do we need to wait and
> >>>how long should the sequence be to be considered
> >>>for us to determine if
> >>>it is a random sequence or not. Moreover, what do we
> >>>do with very very large numbers that pops up once in
> >>>a while in that rundom
> >>>sequence, for example
> >>>89123490812349809871098138973456710… upto 5000
> >>>digits, I can not possibly fit naturally inside my
> >>>alorithm
> >>>for the generator of this sequence ??
> >>>
> >>>Well, for the part of how long a sub sequence should
> >>>we consider, sampling theory plays a major role, and
> >>>all we need to do is
> >>>to have a fairly large sample, and many samples from
> >>>the generator to statiscally justify that the
> >>>gererators generate a
> >>>statistically random sequence. FAIRLY LARGE, AND
> >>>FAIRLY MANY are not really interesting, since it has
> >>>been well studied in the
> >>>area of statistics and DATA ANALYSIS. OKAY THEN, WE
> >>>HAVE SOME REASONALBLY LARGE RUNS OF SEQUENCES, AND
> >>>WE HAVE REASONABLY MANY
> >>>INSTANCES OF THOSE RUNS. But what about those single
> >>>very large number !, how could we handle those in
> >>>our algorithms?. THIS IS
> >>>ONE OF THE MANY REASONS THE TERM PSEUDO-RANDOM WAS
> >>>COINED I THINK, but pls check.
> >>>
> >>>
> >>>
> >>>Now we can see that an important aspect, cardinality
> >>>of the run ( or set) is brushed away quite nicely
> >>>for our practical purpose !.
> >>>But one thing to note is that generation of random
> >>>events has been under rigorous studies for long time
> >>>due to various curiosities
> >>>about the enviroment&nature around us … So they
> >>>discoverd countable and un-countable infinites. When
> >>>we look at most of the
> >>>literature we almost always see that the definitions
> >>>and descriptions, almost always, encompasses (0, 1),
> >>>an uncountably infite
> >>>set, as the domain of discourse.
> >>>
> >>> AND THIS SET IS MOST PROBABLY CONSIDERED FROM TWO
> >>>POINTS OF VIEW -
> >>>
> >>> 1) Without loss of generality it captures the sense
> >>>of time in an apparently small subsets
> >>> 2) And there is a one to one and onto map between
> >>>(0,1) and (0, infinity), hence cardinality is
> >>>preserved !!!
> >>>
> >>>DO YOU STILL THINK YOU COULD BE A FAN OF CARDINALS
> >>>!. Not me for sure.
> >>>
> >>>
> >>>What is Independent business ?
> >>>
> >>>Given today is saturday, we know tomorrow must be
> >>>sunday !. This is not independent. Essentially
> >>>knowing any subsequence of any
> >>>length of the run, does not help in predicting what
> >>>would be the next number of the sequence when the
> >>>genrator produces the
> >>>next number. Any geometic sequence, or arithmatic
> >>>sequence represents examples of dependent
> >>>sequences… Most of the functions
> >>>we know of are not independent ( sin, cos, cosec,
> >>>tan all are not independent …).
> >>>
> >>>OKAY, WE CAN BELIVE THAT, THEN WHAT IS THE BUSINESS
> >>>OF IDENTICALLY DISTRIBUTED ?
> >>>
> >>>If we roll a dice, and say that we win when ever we
> >>>get a face up that is not marked “Occured before”
> >>>then the roles are not
> >>>identically distributed. First roll has 6 equally
> >>>prossible event, 2nd one has 5 equally …, so on
> >>>and so forth…
> >>>
> >>>Instead, if we considered all faces on all trials,
> >>>then each face is identically distributed from the
> >>>uniform distribution in
> >>>[1 … 6]. Here the distribution happens to be
> >>>uniform based on the trust we have on the maker of
> >>>the dice ! No bias !
> >>>
> >>>
> >>>Given a random sequence of numbers, the underlying
> >>>distribution ( that is the probability mass/density
> >>>function ) could be
> >>>different. It could be exponential, it could be
> >>>normal, poissson, and others. BUT WE ONLY CONSIDER
> >>>UNIFORM DISTRIBUTION, SINCE
> >>>having a good analysis and a subsequent generator
> >>>for a pseudo-random sequence of numbers within (0,1)
> >>>is enough to have a
> >>>capability to generates other important
> >>>distribution(s). Since the others are mere
> >>>trasormation from uniform !!!
> >>>
> >>>
> >>>Today, most of the generator(s) of ordinary uses are
> >>>based on :: Fast algorithm, large periods,
> >>>indipendent and identically distributed
> >>>over (0, 1) following uniform distribution. AND
> >>>THERE ARE SOME STANDARD TESTS TO FIND HOW GOOD IS A
> >>>RANDOM GENRATOR. The tests
> >>>can be found in any good lituratures for Applied
> >>>probability and Data Analysis.
> >>>
> >>>
> >>>
> >>>Conclusion::
> >>>-----------
> >>>
> >>>So giving a wide variety of algorithms to generate a
> >>>pseudo-random sequence as well as different
> >>>function(s) for different
> >>>use, one can easily pick one over the other by
> >>>looking at the analysis made on those generators, or
> >>>if very adventourous we
> >>>can always do those analysis taking samples from the
> >>>generators… Alternatively, … trust someone or
> >>>something.!!
> >>>
> >>>Finally congruence is a favorite base to use in
> >>>those generator, including symmetric crypto. Mainly
> >>>due to its fastness, and already
> >>>proven capability to gernerate good enough rand().
> >>>Also some amount of mathematical simplicity is
> >>>important to make it a
> >>>subject under analysis. CONGRUENCE IS WELL STUDIED
> >>>UNDER NUMBER THEORY. CLASSES COMING OUT OF
> >>>CONGRUNCES FORMS FIELDS
> >>>(AN ABSTRACT MATHEMATICAL OBJECT ) that obeys our
> >>>common notion of algebra, and it is one of the
> >>>method of coming up with lots
> >>>of finte fields. Then the immediate high in the
> >>>hierarchy is the field of finite polinomials, and
> >>>assymetric crypto graphy
> >>>depends on this by using finite polYnomials of large
> >>>prime number. SO FACTORING POLYNOMIALS IS THE CRUX
> >>>OF ITS BEAUTY.
> >>>
> >>>
> >>>
> >>> Happy Halloween !
> >>>
> >>>-pro
> >>>
> >>>
> >>>
> >>>
> >>>
> >>>
> >>>
> >>>
> >>>—
> >>>Questions? First check the Kernel Driver FAQ at
> >>>http://www.osronline.com/article.cfm?id=256
> >>>
> >>>You are currently subscribed to ntdev as: unknown
> >>>lmsubst tag argument: ‘’
> >>>To unsubscribe send a blank email to
> >>
> >>xxxxx@lists.osr.com
> >>
> >>
> >>
> >>
> >> _______________________________
> >>Do you Yahoo!?
> >>Declare Yourself - Register online to vote today!
> >>http://vote.yahoo.com
> >>
> >>
> >>—
> >>Questions? First check the Kernel Driver FAQ at
> >
> > http://www.osronline.com/article.cfm?id=256
> >
> >>You are currently subscribed to ntdev as: xxxxx@storagecraft.com
> >>To unsubscribe send a blank email to xxxxx@lists.osr.com
> >
> >
> >
> >
>
> –
> …/ray..
>
> Please remove “.spamblock” from my email address if you need to contact
> me outside the newsgroup.
>
> —
> Questions? First check the Kernel Driver FAQ at
http://www.osronline.com/article.cfm?id=256
>
> You are currently subscribed to ntdev as: xxxxx@storagecraft.com
> To unsubscribe send a blank email to xxxxx@lists.osr.com