pseudo random number generators

©TechKnowXpress

PSEUDO RANDOM NUMBER GENERATION

-DARSHINI PARIKH

(TechKnowXpress)

©TechKnowXpress

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.

©TechKnowXpress

RANDOMNESS

• Uniform Distribution – frequency of occurrence of numbers

• Independence – inference of a subsequence should not be possible

©TechKnowXpress

CONGRUENTIAL GENERATOR

FOUR TYPES LINEAR CONGRUENTIAL GENERATOR(LCG)

MULTIPLICATIVE CONGRUENTIAL GENERATOR(MCG)

QUADRATIC CONGRUENTIAL GENERATOR(QCG)

INVERSIVE CONGRUENTIAL GENERATOR(ICG)

©TechKnowXpress

MCG

• Recurrence Relation:

Xn+1 = (a* Xn +c) mod m

a=Multiplier

c= Increment = 0 (ZERO)m=modulus

• Xn+1 = (a* Xn ) mod m

©TechKnowXpress

MCG EXAMPLE

Eg: X0 = a = c = 7m = 10

PRNs generated:

7, 6, 9, 0, 7, 6, 9, 0, ...

Eg: m=231

Range of PRNs – {0 - 231}

©TechKnowXpress

MCG Example (conti…)

a = 13

c = 0

m = 64

©TechKnowXpress

LCG

Recurrence Relation:

Xn+1 = (a* Xn +c) mod m

a=Multiplierc= Incrementm=modulus

©TechKnowXpress

SELECTING ‘a’ IN LCG

FOR GENERATING ANY 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

©TechKnowXpress

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}

©TechKnowXpress

LCG Example

Xn+1 =65539Xn mod 231

This PRNG generates a full period sequence

©TechKnowXpress

QCG

RECURRENCE RELATION:

Xn+1 = (a* X2n + b*Xn + c) mod m

a, b – multipliers

c - increment

m - modulus

©TechKnowXpress

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)

©TechKnowXpress

QCG Example

Xn+1 = (12*Xn2 + 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

©TechKnowXpress

ICG


X(n+1) = a*X-1n + c (mod m)

a – multiplier

c – increment

m - modulus

©TechKnowXpress

CRITERIA FOR FULL PERIOD SEQUENCE

POLYNOMIAL:

X2 - c*X – a

should be a primitive polynomial over Fm.

(Inversive Maximum Polynomial (IMP).)

©TechKnowXpress

ICG Example

Eg: X(n+1) = 2*X-1n + 3 (mod m)

Corresponding Equation: X(n+1) = a*X-1n + c (mod m)

IMP : Xn2 -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…

©TechKnowXpress

Lagged Fibonacci Generator (LFG)


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)

©TechKnowXpress

LFG Example

Eg: LFG (17,5,31)

So the period of this sequence will be approx. 247

247 = (217) * (2(31-1))

©TechKnowXpress

LFSR

©TechKnowXpress

Mersenne Twister

RECURRENCE RELATION

X(k+n) = X(k+m) ⊕ (Xuk | 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

©TechKnowXpress

BLUM BLUM SHUB GENERATOR


Xn+1 = X2n % 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

©TechKnowXpress

ANSI X9.17 (conti…)


Ri = EDE([K1,K2], [Vi ⊕ EDE([K1,K2],DTi)])

Vi+1 = EDE([K1,K2], [Ri⊕ EDE([K1,K2],DTi)])

©TechKnowXpress

APPLICATIONS OF RANDOM NUMBERS

CRYPTOGRAPHY

STATISTICAL SAMPLING

GENERATION OF

INITIALIZATION VECTORS

SIMULATIONS

GAMBLING&

LUCKY DRAWS

©TechKnowXpress

APPLICATIONS OF PRNGSs

GENERATION OF SESSION

KEYS

GENERATION OF PUBLIC

KEYS

GENERATION OF NONCE TO AVOID REPLAY ATTACKS

pseudo random number generators

Engineering