Re: Generating Random Numbers...

[catalyst@netcom.com (Scott Collins)]

Bob Baldwin writes:
  >The english language has an entropy of 1.0 to 1.5 bits per letter, so it
  >is safe to extract one bit per character.  [Shannon's work et al
  >demonstrates] that this bit per character is truely random.

  - entropy -
In information theory, a visceral meaning of the term 'entropy' is 'a
measure of surprise', or unpredictability.  Entropy is the amount of
information in some chunk of data, that could not be deduced.  Equating
entropy with randomness is a tautology.  Entropy (like velocity) is
relative, however, to the observer.  If the observer has a good model of
the data, the entropy will be low (not much information present).  If the
observer has a bad model, the entropy will be high.  One example that
demonstrates a difference in relative entropies is a block of ciphertext. 
The entropy of the block with respect to its intended recipient (who has
the key) is low; with respect to an interloper (no key), it is high.

  - the entropy of english text -
The paper cited by nobody, and alluded to by Bob Baldwin, is a landmark
paper in estimating the entropy of english text.  Note that these estimates
are not lower bounds, but empirical values with respect to the human
'predictors'; and that the (1, 1.5) bits per character are an _average_
across an entire message.  For example, if the message begins "Four score
", then, with respect to a typical American history student, there is very
little entropy in the remainder of the sentence, should it continue " and
seven years ago".

  - turning entropy into random numbers -
From reading Bob Baldwin's post, I am not sure how he intended to extract
this 'one bit per character'.  You would certainly not just pick the n-th
bit from every byte.  A good way to turn entropy into actual bits is to
build a compressor.  A compressor is usually a combination of a predictor
(or model) and an encoder.  The predictor plays the role of the human
'predictors' in Shannon's experiment, and for each new symbol of the
message, generates a probability that represents (inversely) its 'surprise'
at seeing that symbol.  A good encoder uses the probabilities from the
predictor to encode the symbol in the smallest number of bits.  The better
the predictor, the less it is suprised, the larger the probabilities it
returns to the encoder, the fewer number of bits output.  Thus, good
predictors (and competent encoders) lead to efficient compressors.

See Bell, Cleary, & Witten, "Text Compression" Prentice Hall, 1990.  ISBN
0-13-911991-4.  Especially Ch. 5 'From Probabilities to Bits'.

  - entropy and transformations -
Both Bob Baldwin and nobody discuss the use of hash functions on the text
to get random numbers, and wonder if redundancy in the underlying message
will 'show through'.  Cryptographically secure deterministic hash function
are designed with this criteria in mind.  That is, one goal is that the
entropy in the bits of the hash itself, relative to an observer who knows
the input stream, is 0; to an observer who does not know the input stream,
it is maximal (i.e., has a lower bound related to your predictability in
selecting an input).

nobody asks:
  >could someone here describe how this compares to other pseudo-random
  >sources?  how well would this function as a seed to some other
  >pseudo-random number generation process?

Using hashes as random number generators is as secure as the hash which
(for a cyptographically poor hash) may be dependent on the entropy of the
input, and your use of it (e.g., don't always hash the 'Gettysburg Address'
to get your random numbers).  Using compression to convert the entropy in a
message (with respect to the compression model) into a random number is
dependent on the quality of the model.  If you use a poor model, someone
else will be able to find, and perhaps capitalize on, redundancy in your
output.

Related to this, in a conversation on this topic with Ron Rivest he noted
that good enough compression might be secure in itself (with the model and
its initial state as the key), and that he had students who were
researching this.

Hope this helps,


Scott Collins         | "Few people realize what tremendous power there
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