Pseudo-random generator explanation by Bruce Schneier I am reading about Pseudo-Random Sequences in Applied Cryptography book by Bruce Schneier. Description is provided below.


  A pseudo-random sequence is one that looks random. The sequence's
  period should be long enough so that finite sequence of reasonable
  length —that is, one that is actually used— is not periodic. If you
  need a billion random bits, don't choose a sequence generator that
  repeats after only sixteen thousand bits. These relatively short
  nonperiodic subsequences should be as indistinguishable as possible
  from random sequences. For example, they should  have about the same
  number of ones and zeros, about half the runs (sequences of the same
  bit) should be of length one, one quarter of length two, one eights of
  length three, and so on. They should not be compressible. The
  distribution of run lengths for zeros and ones should be the
  same [643,863,99,1357].


My questions on above text


What does statement "or example, they should  have about the same number of ones and zeros, about half the runs(sequences of the same bit) should be of length one, one quarter of length two, one eights of length three, and so on. " mean?
What does statement "Theere should not be compressible"?
What is number 643,863,99,1357? 


Thanks for your time and help
This is a about properties of a random sequence in general.


  What does statement "or example, they should have about the same number of ones and zeros, about half the runs(sequences of the same bit) should be of length one, one quarter of length two, one eights of length three, and so on. " mean?


Consider a coin toss with a perfect coin (i.e. one that doesn't land on the side and lands on head or tail with 0.5 chance). Then about half of the time it will be a head or a tail. If you would toss it infinite times then you would have an equal amounts of heads and tails. However, if you toss it some other amount you'd expect about the same number of heads and tails.

If you would toss the coin an odd number of times you would certainly not have the same amount of ones and zeros, even with a perfect coin.


  What does statement "Theere should not be compressible"?


Probably this should read "they should not be compressible". Things are compressible when a pattern can be found (think .zip files). If a pattern can be found then the sequence is not random. Note that small patterns can exist by chance, so this requirement is a rather heuristic (unless there is a different definition of compressible around).

Then again, you should not be able to always compress the random number sequences generated.


  What is number 643,863,99,1357? 


That looks like a set of references to me, nothing to do with the random numbers in itself :)