Slide 1
SEMINAR ONGENERATION OF RANDOM NUMBERS AND ITS
APPLICATIONSGUIDED BYPROFESSOR P. K. MANDAL
PRESENTED BY
SAMSTUTI CHANDA
OVERVIEW INTRODUCTIONS WHAT ARE RANDOM NUMBERS NECESSITY OF
RANDOM NUMBERS TYPES OF RANDOM NUMBERS PRNGs versus TRNGs EVOLUTION
OF RANDOM NUMBERS SOME APPLICATIONS OF RANDOM NUMBERS SOME
EXPERIMENTS WITH RANDOM NUMBERS CONCLUSIONS
WHAT ARE RANDOM NUMBERS
A random number is a number generated by a process, whose
outcome is unpredictable, and which cannot be subsequently reliably
reproduced. This definition works fine provided that one has some
kind of a black box such a black box is usually called a random
number generator that fulfils this task.
However, if one were to be given a number, it is simply
impossible to verify whether it was produced by a random number
generator or not. In order to study the randomness of the output of
such a generator, it is hence absolutely essential to consider
sequences of numbers. It is quite straightforward to define whether
a sequence of infinite length is random or not. This sequence is
random if the quantity of information it contains in the sense of
Shannon's information theory is also infinite.
In the case of a finite sequence of numbers, it is formally
impossible to verify whether it is random or not. It is only
possible to check that it shares the statistical properties of a
random sequence like the equiprobability of all numbers but this a
difficult and tricky task.
In order to cope with this difficulty, definitions have been
proposed to characterize "practical" random number sequences.
According to Knuth [1], a sequence of random numbers is a sequence
of independent numbers with a specified distribution and a
specified probability of falling in any given range of values. For
Schneier [2], it is a sequence that has the same statistical
properties as random bit is unpredictable and cannot be reliably
reproduced. A concept that is present in both of these definitions
and that must be emphasized is the fact that numbers in a random
sequence must not be correlated. Knowing one of the numbers of a
sequence must not help predicting the other ones.
RANDOM NUMBERS : EXPECT THE UNEXPECTED
NECESSITY OF RANDOM NUMBERS
1.SIMULATION: TO SIMULATE SOME NATURAL PHENOMENA USING COMPUTERS
(LIKE RADIOACTIVITY,DIFFUSION etc.)2.SAMPLING: THE AVERAGE VALUE OF
A FUNCTION F(X) OVER AN INTERVAL MAY BE ESTIMATED FROM ITS AVERAGE
OVER A FINITE, RANDOM SUBSET OF POINTS IN THAT INTERVAL.
3.NUMERICAL ANALYSIS: SOLVING COMPLICATED NUMERICAL PROBLEMS
.4.COMPUTER PROGRAMING: GOOD SOURCE OF DATA FOR TESTING THE
EFFECTIVENESS OF COMPUTER ALGORITHMS.5.DECISION MAKING: TO MAKE A
COMPLETELY "UNBIASED" DECISION.
6. RECREATION: ROLLING DICE, SHUFFLING DECKS OF CARDS, SPINNING
ROULETTE WHEELS, ETC., ARE FASCINATING PASTIMES FOR JUST ABOUT
EVERYBODY. THESE TRADITIONAL USES OF RANDOM NUMBERS HAVE SUGGESTED
THE NAME "MONTE CARLO METHOD," A GENERAL TERM USED TO DESCRIBE ANY
ALGORITHM THAT EMPLOYS RANDOM NUMBERS.
TYPES OF RANDOM NUMBERSPSEUDO RANDOM NUMBERS: DEFINED AS HAVING
THE APPEARENCE OF RANDOMNESS. GENERATED DETERMINISTICALLY.
TRUE RANDOM NUMBERS : DEFINED AS EXHIBITING TRUE RANDOMNESS.
OBTAINED FROM NATURAL PROCESSES LIKE NOISE .
PRNGs : PSEUDO RANDOM NUMBER GENERATORS FOLLOWS PARTICULAR
ALGORITHM TO GENERATE RANDOM NUMBERS. HENCE THE GENETED NUMBERS ARE
CALLED PSEUDO RANDOM NUMBER. IT IS IMPOSSIBLE TO HAVE AN IDEAL
DETERMINISTIC RANDOM NUMBER GENERATOR. IN GENERAL, OUR RANDOM
NUMBER GENERATOR MUST SATISFY THE FOLLOWING BASIC CRITERIA: 1. THE
DISTRIBUTION OF THE NUMBERS SHOULD BE UNIFORM WITHIN A SPECIFIED
RANGE AND SHOULD SATISFY STATISTICAL TESTS FOR RANDOMNESS, SUCH AS
LACK OF PREDICTABILITY AND OF CORRELATIONS AMONG NEIGHBOURING
NUMBERS. 2. THE CALCULATION SHOULD PRODUCE A LARGE NUMBER OF UNIQUE
NUMBERS BEFORE REPEATING THE CYCLE. 3. THE CALCULATION SHOULD BE
VERY FAST
TRNGs: TRUE RANDOM NUMBER GENERATOR GENERATES RANDOM NUMBERS
FROM A PHYSICAL PROCESS.EXAMPLE : A HARDWARE RANDOM NUMBER
GENERATOR .SUCH DEVICES ARE OFTEN BASED ON MICROSCOPIC PHENOMENA
THAT GENERATE LOW LEVEL, STATISTICALLY RANDOM NOISE SIGNALS, SUCH
AS THERMAL NOISE, PHOTOELECTRIC EFFECT, INVOLVING A BEAM SPLITTER
AND OTHER QUANTUM PHENOMENA. THESE PROCESSES ARE, IN THEORY,
COMPLETELY UNPREDICTABLE.
Two TRNG schemes based on the (a) particle-like or (b) wave-like
nature of light
HOW RANDOM NUMBERAND ITS APPLICATIONS EVOLVE
In 1650, Blaise Pascal designed perpetual motion experiment.
MIDDLE SQUARE METHOD
IN PRACTICE IT IS NOT A GOOD METHOD, SINCE ITS PERIOD IS USUALLY
VERY SHORT.
A PROGRAM TO GENERATE 4 digit RANDOM NUMBERS USING MID SQURE
METHOD*********************************************************************************************WRITE(*,*)
'HOW MANY RANDOM NOS TO GENERATE'READ(*,*)
NOPEN(2,FILE='MIDSQ.txt',STATUS='UNKNOWN')WRITE(*,*)' 4 DIGIT SEED
VALUE='READ(*,*) DDO I=1,NX=D*DIF(X.LT.10000000.0)THEN
X0=X*0.00000001X1=100*X0-INT(100*X0)D=INT(X1*10000)ELSEX0=INT(X/100)D=X0-(INT(X0/10000))*10000ENDIFWRITE(2,8)
DENDDO8 FORMAT(F10.4) STOPEND
InputHOW MANY RANDOM NOS TO GENERATE104 DIGIT SEED
VALUE=1234Output5227.0000 3215.0000 3362.0000 3030.0000 1808.0000
2688.0000 2253.0000 760.0000 5775.0000 3506.0000 3 DIGIT NUMBER. IT
HAPPENS BECAUSE THE MIDDLE TERM HAS ZERO BEFORE IT.
CONGRUENTIAL GENERATOR
IN 1949 D. H. LEHMER INTRODUCED CONGRUENTIAL GENERATORS WHICH
ARE BY FAR THE MOST POPULAR RANDOM NUMBER GENERATORS IN USE
TODAY
RECURRENCE RELATION FOR CONGRUENTIAL GENERATORS :
1.LCG:
where X is thesequence of pseudorandom values, andm, m> 0
,the modulusa, 0
