in reply to Re^3: How likely is rand() to repeat?
in thread How likely is rand() to repeat?

Let's say we were on a 2 bit processor and we had a 2-bit PRNG. There could only be 22 starting points (seeds).

But the (non-repeating) sequences it could produce are any permutation of the following 24 permutations of the 4 basic values it can produce:

{0, 1, 2, 3}  |  {0, 1, 3, 2}  |  {0, 2, 1, 3}  |  {0, 2, 3, 1}  |  
{0, 3, 1, 2}  |  {0, 3, 2, 1}  |  {1, 0, 2, 3}  |  {1, 0, 3, 2}  |  
{1, 2, 0, 3}  |  {1, 2, 3, 0}  |  {1, 3, 0, 2}  |  {1, 3, 2, 0}  |  
{2, 0, 1, 3}  |  {2, 0, 3, 1}  |  {2, 1, 0, 3}  |  {2, 1, 3, 0}  |  
{2, 3, 0, 1}  |  {2, 3, 1, 0}  |  {3, 0, 1, 2}  |  {3, 0, 2, 1}  |  
{3, 1, 0, 2}  |  {3, 1, 2, 0}  |  {3, 2, 0, 1}  |  {3, 2, 1, 0}

[download]

Hence, the 32-bit, Mersenne Twister MT19937 can produce 219937 - 1 values (from any given starting point) before it repeats itself exactly.

With the rise and rise of 'Social' network sites: 'Computers are making people easier to use everyday'
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Re^5: How likely is rand() to repeat?
by JavaFan (Canon) on Mar 09, 2012 at 03:21 UTC
    But the (non-repeating) sequences it could produce are any permutation of the following 24 permutations of the 4 basic values it can produce:
    Can you explain how it does that? Given just four different values for the seed, how can you pick from 24, with each element having a chance to be selected?
    Hence, the 32-bit, Mersenne Twister MT19937 can produce 219937 - 1 values (from any given starting point) before it repeats itself exactly.
    Sure. But how many different such sequences can it make? Looking at the pseudo code implementation on Wikipedia, it's all derived from a single, 32-bit seed. Which would limit the number of possible sequences to 232.
[reply]
      Given just four different values for the seed, how can you pick from 24,

      I didn't say it could generate all those sequences. Only that from any given starting point, the non-repeating sequence could be any permutation of those 24 permutations.

      Sure. But how many different such sequences can it make?

      That's the wrong question. When generating the OPs 25-char sequences, you don't re-seed before starting each new sequence. You seed (implicitely) once and then follow that sequence until you have enough.

      Therefore the upper bound is the length of the non-repeating sequence (the period) the prng can generate. (4.31e+6001 in the Mersenne Twister).

      Of course, that is further constrained because of the modulo operation to bring the generated random values into the 0 .. 61 range. hence 6.45e44.

      For the 15-bit RCPRNG built-in to perl on win32, the period (at least when seeded(1), seems (by experiment) to be 214741815.

      Which looks suspiciously close to 2^31, but not quite.

      With the rise and rise of 'Social' network sites: 'Computers are making people easier to use everyday'
      Examine what is said, not who speaks -- Silence betokens consent -- Love the truth but pardon error.
      "Science is about questioning the status quo. Questioning authority".
      In the absence of evidence, opinion is indistinguishable from prejudice.

      The start of some sanity?
[reply]
        Only that from any given starting point, the non-repeating sequence could be any permutation of those 24 permutations.
        No. With a seed/state of 2 bits, for any given implementation, and any given starting point, only 4 out of the 24 permutation are possible. Think about it. You're working on a deterministic machine. You have only 4 different begin states. How can you have more than 4 end states?

        If you think I'm wrong, show an algorithm that proves otherwise. Given a 2-bit state, that shouldn't be overly complicated.

        That's the wrong question. When generating the OPs 25-char sequences, you don't re-seed before starting each new sequence. You seed (implicitely) once and then follow that sequence until you have enough.
        Eh, no, it's the right question. As you immediately say after stating "it's the wrong question", a sequence is produced. One that isn't reseeded. So, the question is indeed, "how many different sequences can be produced".
        Therefore the upper bound is the length of the non-repeating sequence (the period) the prng can generate. (4.31e+6001 in the Mersenne Twister).
        No, it's not.

        Here's another generator, with the same period as the Mersenne Twister, in pseudo code: 

        use bigint;
my $state = 0;
sub rand {
    $state = ($state + 1) % (2 ** 19937 - 1);
    $state & 0xFFFFFFFF;
}

        [download]
        It's a simple generator, but produces numbers in the range 0 .. 232-1, and has a sequence length of 219937-1 before it repeats itself. It requires an internal state of about 19937 bits, but has no seed (0 bits).

        So, I claim, on each run of the program that uses the above implementation of random, you get one of 20 == 1 different sequences.

        Now, you may have a point if the OP was generating all the passwords he may ever require in his life, in a single run of the program. Then the number of different produced strings depends on the size of the state that the generator keeps. Which, for a typical PRNG is 32, 48 or 64 bits. For MT19937, the internal state is 19968 bits (smallest multiple of 32 greater than 19937). You'd need a seed of that size if you want to carry this information over to different runs of the program.

[reply]
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