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What's the standard way to express "$x$ knows about $y$", or "$x$ has no knowledge of $y$" in cryptographic notation?

Example (PRNG predictor):

$\exists f : P(f(G(k)|_{0..n}) = G(k)|_{n+1}) \geq 0.5 + \epsilon$, for non negligable $\epsilon$, where $f$ has no knowledge of $k$.

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2 Answers 2

up vote 4 down vote accepted

The notation is "", i.e., the empty string. $\;\;$ Since $k$ is not an input of $f$, $f$ has no knowledge of $k$.

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So you're saying there's no such notation? –  Polynomial May 25 '12 at 9:44
@Polynomial He's saying that in your example such a notation is not necessary, because it's implicit in the parameters of $f$. A function by definition only knows what it sees in its parameters. –  CodesInChaos May 25 '12 at 9:56
But your example is problematic, since you did not restrict the cost of calculating $f$. If you make no such restriction, you need entropy exceeding the output size, i.e. a true RNG, and not a PRNG. –  CodesInChaos May 25 '12 at 9:57
@CodeInChaos I was just using it as an example. If I were really writing that in a document, I'd put something like "where $f(x)$ runs in polynomial time or better". –  Polynomial May 25 '12 at 10:10

If you really wanted a symbol for that, I suppose you could borrow a notation from probability theory and write $f \perp k$ for "$f$ is independent of $k$". But that's definitely not standard usage, so you're going to have to define it explicitly.

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