1. Random-Number Generation
Andy Wang
CIS
Computer Systems
Performance Analysis

2. Generate Random Values
Two steps
Random-number generation
Get a sequence of random numbers distributed uniformly between 0 and 1
Random-variate generation
Transform the sequence to produce random values satisfying the desired distribution

3. Background The most common method Use a recursive function
xn = f(xn-1, xn-2, …)

4. Example
xn = (5xn-1 + 1) %16
Suppose x0 = 5
The first 32 numbers are between 0 and 15
Divide xn by 15 to get numbers between 0 and 1

5. Basic Terms x0 = seed Generated numbers are pseudo random
Given a function, the entire sequence can be regenerated with x0
Generated numbers are pseudo random
Deterministic
Can pass statistical tests for randomness
Preferred to fully random numbers so that simulated results can be repeated

6. Cycle Length
Note that starting with the 17th number, the sequence repeats
Cycle length of 16

7. More Terms
Some generators do not repeat the initial part (tail) of the sequence
Period of a generator
= tail + cycle length
tail
cycle length
period

8. Question How to choose seeds and random-number generation functions?
Efficiently computable
Heavily used in simulations
The period should be large
Successive values should be independent and uniformly distributed

9. Types of Random-Number Generators
Linear-congruential generators
Tausworth generators
Extended Fibonacci generators
Combined generators
Others

10. Linear-Congruential Generators
In 1951, Lehmer found residues of successive powers of a number have good randomness properties
xn = an % m = aan-1 % m = axn-1 % m
Lehmer’s choices of a and m
a = 23 (multiplier)
m = (modulus)
Implemented on ENIAC

11. (Mixed) Linear-Congruential Generators (LCG)
xn = (axn-1 + b) % m
xn is between 0 and m – 1
a and b are non-negative integers
“Mixed”  using both multiplication by a and addition by b

12. The Choice of a, b, and m m should be large To compute % m efficiently
Period is never longer than m
To compute % m efficiently
Make m = 2k
Just truncate the result by k bits

13. The Choice of a, b, and m
If b > 0, maximum period m is obtained when
m = 2k
a = 4c + 1
b is odd
c, b, and k are positive integers

14. Full-Period Generators
Generators with maximum possible periods
Not equally good
Look for low autocorrelations between successive numbers
xn = (( )xn-1 + 1) % 235 has an autocorrelation of 0.25
xn = (( )xn-1 + 1) % 235 has an autocorrelation of 2-18

15. Multiplicative LCG xn = axn-1 % m, b = 0
Can compute more efficiently when m = 2k
However, maximum period is only 2k-2
Problem: Cyclic patterns with lower bits

16. Multiplicative LCG with m = 2k
When a = 8i ± 3
E.g., xn = 5xn-1 % 25
Period is only 8
Which is ¼ of 25
When a ≠ 8i ± 3
E.g., xn = 7xn-1 % 25
Period is only 4

17. Multiplicative LCG with m ≠ 2k
To get a longer period, use m = prime number
With proper choice of a, it is possible to get a period of m – 1
a needs to be a prime root of m
If and only if an % m ≠ 1 for n = 1..m - 2

18. Multiplicative LCG with m ≠ 2k
xn = 3xn-1 % 31
x0 = 1
Period is 30
3 is a prime root of 31

19. Multiplicative LCG with m ≠ 2k
xn = 75xn-1 % (231 – 1)
75 is a prime root of 231 – 1
But watch out for computational errors
Multiplication overflow
Need to apply tricks mentioned in p. 442
Truncation due to the number of digits available

20. Tausworthe Generations
How to generate large random numbers?
The Tausworthe generator produces a random sequence of binary digits
The generator then divides the sequence into strings of desired lengths
Based on a characteristic polynomial

21. Tausworthe Example
Suppose we use the following characteristic polynomial
x7 + x3 + 1
The corresponding generation function is
bn+7  bn+3  bn = 0 Or
bn = bn-4  bn-7
Need a 7-bit seed

22. Tausworthe Example The bit stream sequence….
Convert to random numbers between 0 and 1, with 8-bit numbers
x0 = =
x1 = =
x2 = = …

23. Tausworthe Generator Characteristics
For the L-bit numbers generated
+E[xn] = ½
+V[xn] = 1/12
+The serial correlation is zero
+ Good results over the complete cycle
- Poor local behavior within a sequence

24. Tausworthe Example
If a characteristic polynomial of order q has a period of 2q – 1, it is a primitive polynomial
For x7 + x3 + 1
q = 7
Sequence repeats after 127 bits =
A primitive polynomial

25. Tausworthe Implementation
Can be easily generated via linear-feedback shift-registers
For x5 + x3 + 1 bn
bn-1
bn-2
bn-3
bn-4
bn-5 

26. Extended Fibonacci Generators
xn = (xn-1 + xn-2) % m
Does not have good randomness properties
High serial correlation
An extension
xn = (xn-5 + xn-17) % 2k

27. Combined Generations Add random numbers by two or more generators
Can considerably increase the period and randomness
xn = 40014xn-1 %
yn = 40692yn-1 %
wn = (xn - yn) %
This generator has a period of 2.3 x 1018

28. Combined Generators wn = 157wn-1 % 32363 xn = 146xn-1 % 31727
yn = 142yn-1 % 31657
vn = (wn - xn + yn) % 32362
This generator has a period of 8.1 x 1012
Can avoid the multiplication overflow problem

29. Combined Generators
XOR random numbers by two or more generators

30. Combined Generators Shuffle One sequence as an index
To an array filled with random numbers generated by the second sequence
The chosen number in the second sequence is replaced by a new random number
Problem
Cannot skip to the nth random number

31. A Survey of Random-number Generators
Some published generator functions
xn = 75xn-1 % (231 – 1)
Full period of 231 – 2
Low-order bits are randomly distributed
Many others (see textbook)
All have problems
General lessons: Use established ones; Do not invent your own

32. Seed Selection If the generator has a full period
Only one random variable is required
Any seed value is good
However, with more than one random variable, the story is different for multistream simulations
E.g., random arrival and service times
Should use two streams of random numbers

33. Seed Selection Guidelines
Do not use zero
Not good for multiplicative LCGs and Tausworthe generators
Avoid even values
Not good if a generator does not have a full period
Do not use one stream for all variables
May yield strong correlations among variables

34. Seed Selection Guidelines
Use nonoverlapping streams
Each stream requires a separate seed
Otherwise…
A long interarrival time may correlate with a long service time
Suppose we need 10,000 random numbers for interarrival times; 10,000 for service times, use seeds 1 and 10,001
xn = [anx0 + c(an – 1)/(a – 1)] % m
For multiplicative LCGs, c = 0

35. Seed Selection Guidelines
Not to reuse seeds in successive simulation runs
No point to run a simulation again with the same seed
Just continue with the last random number as the seed for the successive runs

36. Seed Selection Guidelines
Do not use random random-number generator seeds
E.g., do not use the time of day, or /dev/random to seed simulations
Simulations should be repeatable
Cannot guarantee that multiple streams will not overlap
Do not use numbers generated by random-number generators as seeds

37. Myths About Random-number Generation
A complex set of operations leads to random results
Hard to guess does not mean random
Random numbers are not predictable
Given a few successive numbers from an LCG
Can solve a, c, and m
Not suitable for cryptographic applications

38. Myths about Random- number Generation
Some seeds are better than others
True
Avoid generators whose period and randomness depend on the seed
Accurate implementation is not important
Watch out for overflows and truncations

39. Myths about Random- number Generation
Bits of successive words generated by a random-number generator are equally randomly distributed
Nope

40. Myths about Random- number Generation
xn = (25173xn ) % 216
x0 = 1
Least significant bit is always 1
Bit 2 is always 0
Bit 3 has a cycle of 2
Bit 4 has a cycle of 4
Bit 5 has a cycle of 8 n
decimal
binary 1
25173 2
12345 3
54509 4
27825 5
55493 6
25449 7
13277

41. Myths about Random- number Generation
For all multiplicative LCGs
The Lth bit has a period that is at most 2L
For LCGs, with the form
xn = axn-1 % 2k
The least significant bit is always 0 or 1
High-order bits are more random

42. More on Random Number Generations
Mersenne twister
Period =~
/dev/random
Extract randomness from physical devices
Truly random

43. White Slide