This presentation on Pseudo Random Number Generator enlists the different generators, their mechanisms and the various applications of random numbers and pseudo random numbers in different arenas.

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PSEUDO RANDOM
NUMBER GENERATION
-DARSHINI PARIKH
(TechKnowXpress)

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WHAT ARE PSEUDO RANDOM
NUMBERS(PRNs)?
• Deterministic Algorithms used to generate a sequence of numbers that are
not statistically random.
• Good algorithms pass a number of tests of randomness.

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RANDOMNESS
• Uniform Distribution – frequency of occurrence of numbers
• Independence – inference of a subsequence should not be possible

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CONGRUENTIAL GENERATOR
FOUR TYPES LINEAR CONGRUENTIAL GENERATOR
(LCG)
MULTIPLICATIVE CONGRUENTIAL GENERATOR
(MCG)
QUADRATIC CONGRUENTIAL GENERATOR
(QCG)
INVERSIVE CONGRUENTIAL GENERATOR
(ICG)

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MCG
• Recurrence Relation:
Xn+1 = (a* Xn +c) mod m
a=Multiplier
c= Increment = 0 (ZERO)
m=modulus
• Xn+1 = (a* Xn ) mod m

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MCG EXAMPLE
Eg: X0 = a = c = 7
m = 10
PRNs generated:
7, 6, 9, 0, 7, 6, 9, 0, ...
Eg: m=231
Range of PRNs – {0 - 231}

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MCG Example (conti…)
a = 13
c = 0
m = 64

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LCG
Recurrence Relation:
Xn+1 = (a* Xn +c) mod m
a=Multiplier
c= Increment
m=modulus

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SELECTING ‘a’ IN LCG
FOR GENERATINGANY LCG
a belongs to:
{0 – m}
FOR GENERATING FULL
PERIOD LCG
(i) (a-1) should be
divisible by all prime
numbers of m.
(ii) (a-1) should be
divisible by 4 if m is
divisible by 4

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SELECTING ‘m’ & ‘c’ IN LCG
SELECTING M
(i) M should be large
(ii) For efficient
computation; m should
be a power of 2.
SELECTING C
C belongs to {0 to m}

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LCG Example
Xn+1 =65539Xn mod 231
This PRNG generates a full period sequence

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QCG
RECURRENCE RELATION:
Xn+1 = (a* X2
n + b*Xn + c) mod m
a, b – multipliers
c - increment
m - modulus

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CRITERIA FOR FULL PERIOD SEQUENCE
 gcd(m,c) = 1; m and c are
relatively prime
 a,b =0 (mod p);p = odd
prime divisor of m
 a=0 (mod 2) and
b=(a+1) (mod 4) if 4|m or
b=(a+1) (mod 2) if 2|m
 if 9|m then either a=0
(mod 9) or b=1 (mod 9)
and ac=6 (mod 9).
 m=2p
 c = 1 (mod 2) => c is odd
 a = 0 (mod 2) => a is even
 b= (a+1) (mod 4)

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QCG Example
Xn+1 = (12*Xn
2 + 25* Xn + 11) % 36
X0 = 13
Corresponding equation:
Now , 36 – (22 * 32)
Criteria satisfied:
gcd (c,m) = 1 (gcd(11,36) = 1)
a % 2 = a % 3 =0 (a=12)
b % 2 = b % 3 = 1 (b=25)
b = a+1 (mod 4) (25=13 (mod 4))
a*c = 6 (mod 9) (12*11 = 6 (mod 9))
This PRNG will generate a full period sequence

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ICG
RECURRENCE RELATION:
X(n+1) = a*X-1
n + c (mod m)
a – multiplier
c – increment
m - modulus

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CRITERIA FOR FULL PERIOD SEQUENCE
POLYNOMIAL:
X2 - c*X – a
should be a primitive
polynomial over Fm.
(Inversive Maximum
Polynomial (IMP).)

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ICG Example
Eg: X(n+1) = 2*X-1
n + 3 (mod m)
Corresponding Equation: X(n+1) = a*X-1
n + c (mod m)
IMP : Xn
2 -3 * Xn -2= Xn
2 + 4* Xn + 5 (mod 7) is a primitive polynomial over F7.
This PRNG will generate a full period sequence
Sequence generated: 1,5,2,4,0,3,6,1…

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Lagged Fibonacci Generator (LFG)
RECURRENCE RELATION:
Xn = (X(n-L) * X(n-k)) mod m
Given – L bits of the sequence
k, L – lags
m = 2M
Period of the Generator = (2L-1)*(2M-1)
LFG Notation: LFG(L, k, M)

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LFG Example
Eg: LFG (17,5,31)
So the period of this sequence will be approx. 247
247 = (217) * (2(31-1))

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LFSR

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LFSR Example
Suppose m – 24 -1
Initial value: 1000
Sequence: 1000, 1001, 1010,
1111,…

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MersenneTwister
RECURRENCE RELATION
X(k+n) = X(k+m) ⊕ (Xu
k | XL
(k+1)) • A
A – w x w matrix
r - 0< r <w-1
m – 1< m <n
k – {0,1,….}
u – higher order bits = w-r bits
L – lower order r bits
| - Concatenation Operation

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BLUM BLUM SHUB GENERATOR
RECURRENCE RELATION:
Xn+1 = X2
n % m
X0 = S2 % m
Bn+1 = Xn+1 % 2
S – Seed value
m – modulus – p*q (p & q are large primes such that p=q=3 (mod 4))
B – BBS bit

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BBSG Example
Eg: p- 383, q – 503, S = 101355
m =192649 = 383 * 503
The sequence generated is:

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ANSI X9.17

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ANSI X9.17 (conti…)
RECURRENCE RELATION:
Ri = EDE([K1,K2], [Vi ⊕ EDE([K1,K2],DTi)])
Vi+1 = EDE([K1,K2], [Ri ⊕ EDE([K1,K2],DTi)])

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APPLICATIONS OF RANDOM NUMBERS
CRYPTOGRAPHY
STATISTICAL
SAMPLING
GENERATION
OF
INITIALIZATION
VECTORS
SIMULATIONS
GAMBLING
&
LUCKY
DRAWS

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APPLICATIONS OF PRNGSs
GENERATION
OF SESSION
KEYS
GENERATION
OF PUBLIC
KEYS
GENERATIONOF
NONCETOAVOID
REPLAYATTACKS

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