[dmd]

I don't understand why the x^2 thing is interesting at all.

If I multiply x by 0, the distribution is now just 0.

If I take abs(x) now it is positive.

Are these confusing to anyone?

[StavrosK]

It's confusing to me why the author is confused by these. As you say, he's literally saying "if I know nothing about x, then I should also know nothing about 0x". Wtf? No.

[geofft]

I think the confusion is between the idea that a hash function conveys zero information about the input, and the idea that you will have zero information about the input once seeing the output of the hash.

But if we move it out of math, it's easier to understand: if you have some guesses about x, and I tell you nothing, you have the same guesses about x. You don't stop having guesses.

[acchow]

In code:

i(x) = x // We know i is uniformly distributed just as x is - preserve "unknowability"

f(x) = x^2 // We know f has higher probability between [0,1] than [1,2]

g(x) = 1 // We know g is always 1.

Just because your inputs are random, doesn't mean your output is - the implementation matters.

[dmd]

... which is exactly what I said. What are you adding here?