Randomness Extractor - Formal Definition of Extractors

Formal Definition of Extractors

When defining an extractor, we need a measurement of how well it extracts. In other words, we need to define the level of randomness it produces. A measure of the level of randomness of a distribution defined on is provided by the min-entropy (denoted by ), which measures the amount of randomness in the worst case. This is not to be confused with the average case randomness described by Shannon-entropy. The min-entropy is used to define an extractor as follows:

Definition (Extractor): A (k, ε)-extractor is a function

such that for every distribution on {0, 1}n with the distribution is ε-close (measured by statistical distance) to the uniform distribution on {0, 1}m (denoted by and similarly denotes the distribution of the random seed of dimension d).

Again, the length of the weakly random input must be longer than the length of the output. The aim is to have a short as possible seed (low d), and an output that is as long as possible (high m), while keeping the min-entropy over a certain limit. In short we will have: n > m and we aim for d<< m.