negligible function

We say that two probability ensembles and are statistically close if is a negligible function in .

This leads to the definitions of negligible functions given at the top of this article.

That this extractor fulfills the criteria of the lemma is trivially true as is a negligible function.

Since the constants can be expressed as with a constant polynomial this shows that negligible functions are a subset of the infinitesimal functions.

Statistical zero-knowledge means that the distributions are not necessarily exactly the same, but they are statistically close, meaning that their statistical difference is a negligible function.

Then for all non-uniform probabilistic polynomial time algorithms that output and of increasing length k, the probability that and is a negligible function in k.

Definition (negligible function): In the proof of this theorem, we need a definition of a negligible function.

Definition (k-ERF): An adaptive k-ERF is a function where, for a random input , when a computationally unbounded adversary can adaptively read all of except for bits, for some negligible function (defined below).

An adversary is said to have a negligible "advantage" if it wins the above game with probability , where is a negligible function in the security parameter k, that is for every (nonzero) polynomial function there exists such that for all .

In computational complexity, if and are two distribution ensembles indexed by a security parameter n (which usually refers to the length of the input), then we say they are computationally indistinguishable if for any non-uniform probabilistic polynomial time algorithm A, the following quantity is a negligible function in n:

Definition (k-APRF): A APRF is a function where, for any setting of bits of the input to any fixed values, the probability vector of the output over the random choices for the remaining bits satisfies for all and for some negligible function .