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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)])](https://image.slidesharecdn.com/prng-151218055833/75/Pseudo-Random-Number-Generators-26-2048.jpg)
![©TechKnowXpress
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)])](https://crownmelresort.com/image.slidesharecdn.com/prng-151218055833/75/Pseudo-Random-Number-Generators-26-2048.jpg)
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Pseudo Random Number Generators
- 1. ©TechKnowXpress PSEUDO RANDOM NUMBER GENERATION -DARSHINIPARIKH (TechKnowXpress)
- 2. ©TechKnowXpress WHAT ARE PSEUDORANDOM 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.
- 3. ©TechKnowXpress RANDOMNESS • Uniform Distribution– frequency of occurrence of numbers • Independence – inference of a subsequence should not be possible
- 4. ©TechKnowXpress CONGRUENTIAL GENERATOR FOUR TYPESLINEAR CONGRUENTIAL GENERATOR (LCG) MULTIPLICATIVE CONGRUENTIAL GENERATOR (MCG) QUADRATIC CONGRUENTIAL GENERATOR (QCG) INVERSIVE CONGRUENTIAL GENERATOR (ICG)
- 5. ©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
- 6. ©TechKnowXpress 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}
- 7. ©TechKnowXpress MCG Example (conti…) a= 13 c = 0 m = 64
- 8. ©TechKnowXpress LCG Recurrence Relation: Xn+1 =(a* Xn +c) mod m a=Multiplier c= Increment m=modulus
- 9. ©TechKnowXpress SELECTING ‘a’ INLCG 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
- 10. ©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}
- 11. ©TechKnowXpress LCG Example Xn+1 =65539Xnmod 231 This PRNG generates a full period sequence
- 12. ©TechKnowXpress QCG RECURRENCE RELATION: Xn+1 =(a* X2 n + b*Xn + c) mod m a, b – multipliers c - increment m - modulus
- 13. ©TechKnowXpress CRITERIA FOR FULLPERIOD 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)
- 14. ©TechKnowXpress 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
- 15. ©TechKnowXpress ICG RECURRENCE RELATION: X(n+1) =a*X-1 n + c (mod m) a – multiplier c – increment m - modulus
- 16. ©TechKnowXpress CRITERIA FOR FULLPERIOD SEQUENCE POLYNOMIAL: X2 - c*X – a should be a primitive polynomial over Fm. (Inversive Maximum Polynomial (IMP).)
- 17. ©TechKnowXpress 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…
- 18. ©TechKnowXpress 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)
- 19. ©TechKnowXpress LFG Example Eg: LFG(17,5,31) So the period of this sequence will be approx. 247 247 = (217) * (2(31-1))
- 20.
- 21. ©TechKnowXpress LFSR Example Suppose m– 24 -1 Initial value: 1000 Sequence: 1000, 1001, 1010, 1111,…
- 22. ©TechKnowXpress 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
- 23. ©TechKnowXpress BLUM BLUM SHUBGENERATOR 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
- 24. ©TechKnowXpress BBSG Example Eg: p-383, q – 503, S = 101355 m =192649 = 383 * 503 The sequence generated is:
- 25.
- 26. ©TechKnowXpress ANSI X9.17 (conti…) RECURRENCERELATION: Ri = EDE([K1,K2], [Vi ⊕ EDE([K1,K2],DTi)]) Vi+1 = EDE([K1,K2], [Ri ⊕ EDE([K1,K2],DTi)])
- 27. ©TechKnowXpress APPLICATIONS OF RANDOMNUMBERS CRYPTOGRAPHY STATISTICAL SAMPLING GENERATION OF INITIALIZATION VECTORS SIMULATIONS GAMBLING & LUCKY DRAWS
- 28. ©TechKnowXpress APPLICATIONS OF PRNGSs GENERATION OFSESSION KEYS GENERATION OF PUBLIC KEYS GENERATIONOF NONCETOAVOID REPLAYATTACKS
- 29.