[Twisol]

If we have any probability distribution, that tells us something about x. If someone tells us that license plates numbers are uniformly distributed, we can pretend to be a license plate maker by sampling from a uniform distribution, and nobody else could tell the difference by looking at the license plate numbers we make.

Zero information is more like not knowing what probability distribution a variable comes from. Rather than hiding "what value does x take", we can hide "what probability distribution does x come from". That's one level up. (Then you could ask what the likelihood of x having a given probability distribution is, and so on -- zero information is having none of this information, all the way up.)

[klodolph]

Uniform distribution gives minimum information, assuming the range is known. Any other distribution will give more information. Talking about information capacity of coded messages, here.

I think what you are saying when you use the phrase "zero information" is really "zero knowledge".

[Twisol]

"Minimum information" is much more accurate than "zero information". A necessary condition for a good cryptographic hash function F is that F(X) is a uniform distribution if X is a uniform distribution. This is a property of F, not of X, which is where the article is a bit muddy.

The article seems to want to say that "zero information" ought to be a probability distribution such that for all functions f, f(X) is the same zero-information distribution, i.e. from nothing we get nothing. The point is that no such X exists, because every probability distribution encodes some amount of information. What we want is some function F such that for every X in a large class of probability distributions, F(X) is uniform. Which is exactly why x^2 is a terrible hash function.