On Sep 3, 2013, at 3:16 PM, Faré <fah...@gmail.com> wrote:
> Can't you trivially transform a hash into a PRNG, a PRNG into a
> cypher, and vice versa?

> hash->PRNG: append blocks that are digest (seed ++ counter ++ seed)
Let H(X) = SHA-512(X) || SHA-512(X)
where '||' is concatenation.  Assuming SHA-512 is a cryptographically secure 
hash H trivially is as well.  (Nothing in the definition of a cryptographic 
hash function says anything about minimality.)  But H(X) is clearly not useful 
for producing a PRNG.

If you think this is "obviously" wrong, consider instead:

H1(X) = SHA-512(X) || SHA-512(SHA-512(X))

Could you determine, just from black-box access to H1, that it's equally bad as 
a PRNG?  (You could certainly do it with about 2^256 calls to H1 with distinct 
inputs - by then you have a .5 chance of a duplicated top half of the output, 
almost certainly with a distinct bottom half.  But that's a pretty serious bit 
of testing....)

I don't actually know if there exists a construction of a PRNG from a 
cryptographically secure hash function.  (You can build a MAC, but even that's 
not trivial; people tried all kinds of things that failed until the HMAC 
construction was proven correct.)
                                                        -- Jerry

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