http://en.wikipedia.org/wiki/Random_number_generation
http://en.wikipedia.org/wiki/Pseudorandom_number_generator
http://en.wikipedia.org/wiki/Cryptographically_secure_pseudorandom_number_generator

Random numbers
- Truly random or un-discernible patterns?

- Simulation or unpredictable?
- Deck has 52 cards... shuffle : -> permutation(52)

- Appearing random: Do you want "shuffle" to play "Let it Go" 10 times
  in a row?
  
- Local randomness: frequency (counts of possibilities), serial test
(two at a time), poker test (five at a time), gap test (gaps between
same numbers) [Tetris example]

0 2 3 5 7 8 9 1 8 5 4 7 0 3 1 -- 15 values for 10 times.. 1.5 value times
measure the information... you can try to compress it
0 - 2 1 - 2 2 - 1 3 - 2 4 - 1 5 - 2 6 - 0 7 - 2 8 - 2 9 - 1
00 01 56 38

True random:
- Harvest "entropy" from reality... with dice and coins (dynamical
systems) and radioactive decay
- RAND Corporation
- (often blocking)
- Reality may have biases that are hard to detect (dice, hard drive
arms, and human arms [mice & keyboard])
- Humans often alternate "too much"

Cryptographic:
- Next bit test: For all k, If you have all K bits, then you can't guess the
(k+1) bit with more than 50% accuracy in polynomial time. (O(n^m))

rand : State -> (Num x State)

rand n = n, n+1
0 1 2 3 4 5 

- State compromise: If you know the internal state at bit k, you can't compute
the "old" bits (i < k) [enumerate bits of pi]

--------------- ALGORITHMS -----------------------

Pseudo-random:
- The sequence WILL repeat, it is just a matter of when. (If its state
is n bits then at most 2^n results are possible.)

exists i j k,
rand State_i = n x State_j s.t. State_j = State_k where k < i
assuming State is Finite

- Not all seeds have the same (good) period length

9 8 7 6 5 4 3 2 1 0 0 0 0 0 0 0 0 0

Starting at n makes the period n.

- Some numbers and patterns are inherently unlikely

If the period is 16, and you want a 32 length sequence with no
repeats... it doesn't exist

- Successive values are often correlated
- Often of the form:

X_n+1 = (aX_n + b) mod m

- Often initialized with clock

Middle square:
- von Neumann
- Square a number and take the middle digits (extending the left if
necessary)

4 -> 16 -> 016 -> 1

4789 -> 22 9345 21 -> 9345 -> 87 3290 25 -> 3290 -> 10824100

Linear feedback shift register:

--->---->--->--->-->->
0 0 1 0 1 0 1 0 1 0 1
                    ^--- output

--->---->--->--->-->->
_ 0 0 1 0 1 0 1 0 1 0 
^--- new value

new value = output XOR bit5 + output XOR bit3

- Shift values to the right
- Compute the new value on the left with an XOR of the other values
- Some interesting theory to get good cycles of maximal-length

Multiply-with-Carry:
- A generalization of the form above.

X_0 ... X_n, c(arry)

X_n+1 = (a * X_n + b ) mod m

X_2 = f(X_1, c)
c = f(X_0, ..., X_n)

Blum Blum Shub:
- x_{n+1} = (x_{n})^2 mod M

where M = pq and p, q are large primes

- x_0 should be co-prime to M (p and q are not factors) and not 0 or 1
- p and q should be congruent to 3 (mod 4)
- gcd( phi(p - 1), phi(q - 1) ) should be small (to have a large cycle)

--- Example:
p = 11, q = 19, and x_0 = 3
gcd is 2

9 81 82 36 42 92 104 157 196 169 137 168 --- 12
9 81 82 36 42 92 104 157 196 169 137 168

observation function or output function

1 1 0 0 0 0 0 1 0 1 1 0

Mersenne Twister:
- A generalization of LFS
- Most popular now
- Period is 2^19937 − 1
- The beginning is "highly non-random"
- It passes a generalization of the poker test and many other tests

05467 - 1/10^5

- Not acceptable for crypto, stats, or numerical analysis (it only
takes 624 numbers to know where in the sequence you are)
- I have no idea why it works. Reading the code is not particularly instructive.

.........................

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