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Random- number generatorS
& reVIEW ON Intel rng
Presented by syed atif chishti
IE 508 – SYSTEM SIMULATION & MODELING
overview
Topics to be covered
1. Introduction
2.Linear Congruential Generators
3. Composite Generators
4.Testing Random-Number Generator
5. Intel Random Number Generator
Introduction
A simulation of process in which random Component requires
A method of generating Numbers that are random
Methods of generating random variates from uniform distribution
On the interval [0 1] denoted as U(0,1)
Random variates generated from U(0,1) distribution will be
Called as random numbers.
Introduction
Casting lots, throwing dice, dealing out cards
Electronic random number indicator equipment(ERNIE)
Was used by British GPO to pick winners in lottery
.
Mid square method
.
Properties of Arithmetic rng
It is distributed uniformly on U(0,1).No correlation.
Fast and avoid the need of storage.
Stream reproduce second times.
Produce separate streams easily
Generator to be portable i.e produce the number up to
Machine accuracy.
.
Linear congruential generators
It can be defined by the recursive formula
Z=(az+c)(mod m)
M= modulus
A= multiplier
C= increment
Z= seed or starting value
U= Z/m
.
Objection on lcg
Pseudo random number generator as Z is completely
Determined by m,a,c,z parameter.
U’s can take only the rational values 0,1/m,2/m,…,(m-1)/m
example
M=16, a=5,c=3 & Z=7.
Mixed generator
Conditions to get full period of a generator.
Only positive integer that divides both m and c is 1.
If q is a prime number that divides m, then q divides a-1.
If 4 divides m,then 4 divides a-1.
For C>0 ,condition 1 is possible and we get full period.
M=2expb, since b is the bits(binary digits) to store the data.
Multiplicative generators
C is not needed.
Don’t have full period, condition 1 not satisfied.
M=2expb-2, only one fourth of integers 0-m-1 can be obtain
If a= 2exp l + J then it is called RANDU.
Prime modulus multiplicative LCG(PMMLCG)
M is prime and the period is m-1 and if a is primitive
Element modulo m i.e smallest integer l for which al-1/m
Gives l=m-1.
Alternatives types to LCG
It can be expressed as
Z=g(Zi-1,Zi-2,…)(mod m) = a’z^2+az+c
Similar to midsquare method
Better statistical properties
Period of QCG=m
Quadratic Congruential Generator
Alternatives types to LCG
Two or more separate generators and combine them to
Generate the final random numbers.
Second LCG to shuffle the output from the first LCG
Initially a vector V=(v1,v2,,,,,vk) is filled sequentially with the
First KU from the first LCG where k=128 and second LCG is
Used to generate a random integer I distributed uniformly.
V1 returns as first U(0,1) variate ,first LCG replaces its Ith
Location in V with the next U and second LCG randomly
Chooses the random number from this updated V
COMPOSITE GENERATORS
Alternatives types to LCG
These generators are called cryptographic ,operate directly
On bits to form random numbers.
Bi=(c1bi-1+c2bi-2+….+cqbi-q)(mod 2)
c1=c2=cq-1= 0 or 1
Cq=1
In most application c =0 thus it become
Bi=(bi-r+bi-q)(mod2)
Or
Bi=[ 0 if bi-r = bi-q , 1 if bi-r not= bi-q]
FEEDBACK SHIFT REGISTER GENERATORS
Testing Random-Number Generators
Empirical test are the kinds of statistical tests and are
based on U’s Produced by generator.
Theoretical test use numerical parameters of a generator
To assess it globally without actually generating U’s
EMPIRICAL TESTS VS THEORETICAL TEST
EMPIRICAL TESTS
The direct way to test any generator is to generate some
U’s and then statistically examined to see the result to
IID U(0,1)
Test 1 Chi – Square method:
Check whether U’s appear to be uniformly distributed b/w 0 & 1
Divide the [0,1] into k sub intervals of equal length &
Generate U1,U2,U3….Un
EMPIRICAL TESTS
Test 2 Serial Test method:
Generalization of Chi-square test to higher dimension.
If the U’s are really from IID U(0,1) random variates, the overlap
D –tuples is
U1=(U1,U2,…,Ud), U2=(Ud+1,Ud+2,…,U2d) …..
Should be IID random vectors distributed uniformly on the d
Dimensional unit hypercube [0,1]d.
Divide [0,1] into k subintervals of equal size and generate U1,
U2,….Un.
EMPIRICAL TESTS
Test 3 runs (runs –up) test:
Examine the Ui sequence for unbroken sequence of maximal
Length with in which the Ui’s increase monotonically Such
Subsequence is called a run up.
Let U1,U2…U10 : 0.86,0.11,0.23,0.03,0.13,0.06,0.55,0.64,0.87
0.10. the sequence starts with run up of length 1(0.86) followed
By run up of length 2(0.11,0.23) then run up of length 2(0.03,0.13)
Then a run up of length 4(0.06,0.55,0.64,0.87) and finally run up
Of length 1(0.10)
R=1/n ∑∑aij(ri-nbi)(rj-nbj)
EMPIRICAL TESTS
Test 4 Discernible correlation:
Estimate the generated Ui’s correlation at lags j=1,2…l.
It is defined as Pj= Cj/Co
Where Cj =COV (Xi,Xi+j)= E(XiXi+J)-E(Xi)E(Xi+j)
Covariance between entries in the sequence separated by j.
THEORETICAL TESTS
Best known theoretical test are based on upsetting
observation that random numbers fall mainly in the planes.
.
True Random Number Generator
Uses a non deterministic source to produce randomness.
It measuring unpredictable natural process such as thermal
(resistance or shot) noise or nuclear decay.
INTEL RANDOM NUMBER GENERATOR
True Random Number Generator
Uses a non deterministic source to produce randomness.
It measuring unpredictable natural process such as thermal
(resistance or shot) noise or nuclear decay.
Through mouse movement ,keys can be generated.
INTEL RANDOM NUMBER GENERATOR
Architecture Analysis Of Intel RNG
Noise Source:
Johnson noise also called thermal noise ,shot noise and
Flicker noise are all present in resistor.
They have electrically measurable characteristics and are the
Result of random electron & material behavior.
Intel RNG first samples thermal noise by amplifying the voltage
Measured across resistor.
Apart from large random component , this measurement are
Correlated to electromagnetic radiation,temperature and power
Supply fluctuation.
Intel RNG reduces the coupled component by subtract the
signals sampled from two adjacent resistor.
Dual Oscillator Architecture :
Intel RNG uses a random source that is derived
from two free –running oscillator. one is fast and
one is slow.
Thermal noise source use to modulate the
Frequency of slower clock
The variable, noise modulated slower clock is used
To trigger the measurement of fast clock.
Drift between the two clocks thus provides the
Source of random binary digits.
Digital Post Processing :
The initial random measurement are processed by hardware
Corrector based concept to produce a balanced distribution
Of 0 & 1 bits.
Statistical Evaluation :
Intel RNG uses a random source that is derived
from two free –running oscillator. one is fast and
one is slow.
Thermal noise source use to modulate the
Frequency of slower clock
The variable, noise modulated slower clock is used
To trigger the measurement of fast clock.
Drift between the two clocks thus provides the
Source of random binary digits.
Intel Software Library
Software Architecture
Random number generator
Thank you

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Random number generator

  • 1. Random- number generatorS & reVIEW ON Intel rng Presented by syed atif chishti IE 508 – SYSTEM SIMULATION & MODELING
  • 2. overview Topics to be covered 1. Introduction 2.Linear Congruential Generators 3. Composite Generators 4.Testing Random-Number Generator 5. Intel Random Number Generator
  • 3. Introduction A simulation of process in which random Component requires A method of generating Numbers that are random Methods of generating random variates from uniform distribution On the interval [0 1] denoted as U(0,1) Random variates generated from U(0,1) distribution will be Called as random numbers.
  • 4. Introduction Casting lots, throwing dice, dealing out cards Electronic random number indicator equipment(ERNIE) Was used by British GPO to pick winners in lottery .
  • 6. Properties of Arithmetic rng It is distributed uniformly on U(0,1).No correlation. Fast and avoid the need of storage. Stream reproduce second times. Produce separate streams easily Generator to be portable i.e produce the number up to Machine accuracy. .
  • 7. Linear congruential generators It can be defined by the recursive formula Z=(az+c)(mod m) M= modulus A= multiplier C= increment Z= seed or starting value U= Z/m .
  • 8. Objection on lcg Pseudo random number generator as Z is completely Determined by m,a,c,z parameter. U’s can take only the rational values 0,1/m,2/m,…,(m-1)/m
  • 10. Mixed generator Conditions to get full period of a generator. Only positive integer that divides both m and c is 1. If q is a prime number that divides m, then q divides a-1. If 4 divides m,then 4 divides a-1. For C>0 ,condition 1 is possible and we get full period. M=2expb, since b is the bits(binary digits) to store the data.
  • 11. Multiplicative generators C is not needed. Don’t have full period, condition 1 not satisfied. M=2expb-2, only one fourth of integers 0-m-1 can be obtain If a= 2exp l + J then it is called RANDU. Prime modulus multiplicative LCG(PMMLCG) M is prime and the period is m-1 and if a is primitive Element modulo m i.e smallest integer l for which al-1/m Gives l=m-1.
  • 12. Alternatives types to LCG It can be expressed as Z=g(Zi-1,Zi-2,…)(mod m) = a’z^2+az+c Similar to midsquare method Better statistical properties Period of QCG=m Quadratic Congruential Generator
  • 13. Alternatives types to LCG Two or more separate generators and combine them to Generate the final random numbers. Second LCG to shuffle the output from the first LCG Initially a vector V=(v1,v2,,,,,vk) is filled sequentially with the First KU from the first LCG where k=128 and second LCG is Used to generate a random integer I distributed uniformly. V1 returns as first U(0,1) variate ,first LCG replaces its Ith Location in V with the next U and second LCG randomly Chooses the random number from this updated V COMPOSITE GENERATORS
  • 14. Alternatives types to LCG These generators are called cryptographic ,operate directly On bits to form random numbers. Bi=(c1bi-1+c2bi-2+….+cqbi-q)(mod 2) c1=c2=cq-1= 0 or 1 Cq=1 In most application c =0 thus it become Bi=(bi-r+bi-q)(mod2) Or Bi=[ 0 if bi-r = bi-q , 1 if bi-r not= bi-q] FEEDBACK SHIFT REGISTER GENERATORS
  • 15. Testing Random-Number Generators Empirical test are the kinds of statistical tests and are based on U’s Produced by generator. Theoretical test use numerical parameters of a generator To assess it globally without actually generating U’s EMPIRICAL TESTS VS THEORETICAL TEST
  • 16. EMPIRICAL TESTS The direct way to test any generator is to generate some U’s and then statistically examined to see the result to IID U(0,1) Test 1 Chi – Square method: Check whether U’s appear to be uniformly distributed b/w 0 & 1 Divide the [0,1] into k sub intervals of equal length & Generate U1,U2,U3….Un
  • 17. EMPIRICAL TESTS Test 2 Serial Test method: Generalization of Chi-square test to higher dimension. If the U’s are really from IID U(0,1) random variates, the overlap D –tuples is U1=(U1,U2,…,Ud), U2=(Ud+1,Ud+2,…,U2d) ….. Should be IID random vectors distributed uniformly on the d Dimensional unit hypercube [0,1]d. Divide [0,1] into k subintervals of equal size and generate U1, U2,….Un.
  • 18. EMPIRICAL TESTS Test 3 runs (runs –up) test: Examine the Ui sequence for unbroken sequence of maximal Length with in which the Ui’s increase monotonically Such Subsequence is called a run up. Let U1,U2…U10 : 0.86,0.11,0.23,0.03,0.13,0.06,0.55,0.64,0.87 0.10. the sequence starts with run up of length 1(0.86) followed By run up of length 2(0.11,0.23) then run up of length 2(0.03,0.13) Then a run up of length 4(0.06,0.55,0.64,0.87) and finally run up Of length 1(0.10) R=1/n ∑∑aij(ri-nbi)(rj-nbj)
  • 19. EMPIRICAL TESTS Test 4 Discernible correlation: Estimate the generated Ui’s correlation at lags j=1,2…l. It is defined as Pj= Cj/Co Where Cj =COV (Xi,Xi+j)= E(XiXi+J)-E(Xi)E(Xi+j) Covariance between entries in the sequence separated by j.
  • 20. THEORETICAL TESTS Best known theoretical test are based on upsetting observation that random numbers fall mainly in the planes. .
  • 21. True Random Number Generator Uses a non deterministic source to produce randomness. It measuring unpredictable natural process such as thermal (resistance or shot) noise or nuclear decay. INTEL RANDOM NUMBER GENERATOR
  • 22. True Random Number Generator Uses a non deterministic source to produce randomness. It measuring unpredictable natural process such as thermal (resistance or shot) noise or nuclear decay. Through mouse movement ,keys can be generated. INTEL RANDOM NUMBER GENERATOR
  • 24. Noise Source: Johnson noise also called thermal noise ,shot noise and Flicker noise are all present in resistor. They have electrically measurable characteristics and are the Result of random electron & material behavior. Intel RNG first samples thermal noise by amplifying the voltage Measured across resistor. Apart from large random component , this measurement are Correlated to electromagnetic radiation,temperature and power Supply fluctuation. Intel RNG reduces the coupled component by subtract the signals sampled from two adjacent resistor.
  • 25. Dual Oscillator Architecture : Intel RNG uses a random source that is derived from two free –running oscillator. one is fast and one is slow. Thermal noise source use to modulate the Frequency of slower clock The variable, noise modulated slower clock is used To trigger the measurement of fast clock. Drift between the two clocks thus provides the Source of random binary digits.
  • 26. Digital Post Processing : The initial random measurement are processed by hardware Corrector based concept to produce a balanced distribution Of 0 & 1 bits.
  • 27. Statistical Evaluation : Intel RNG uses a random source that is derived from two free –running oscillator. one is fast and one is slow. Thermal noise source use to modulate the Frequency of slower clock The variable, noise modulated slower clock is used To trigger the measurement of fast clock. Drift between the two clocks thus provides the Source of random binary digits.