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 -
- Without loss of generality it captures the sense of time in an apparently small subsets
- 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
