More Efficient Privacy Amplification With Less Random Seeds via Dual Universal Hash Function

TLDR: This work explicitly construct random hash functions for privacy amplification (extractors) that require smaller random seed lengths than the previous literature, and still allow efficient implementations with complexity $O(n\log n)$ for input length $n$ .

Abstract: We explicitly construct random hash functions for privacy amplification (extractors) that require smaller random seed lengths than the previous literature, and still allow efficient implementations with complexity $O(n\log n)$ for input length $n$ . The key idea is the concept of dual universal2 hash function introduced recently. We also use a new method for constructing extractors by concatenating $\delta $ -almost dual universal2 hash functions with other extractors. Besides minimizing seed lengths, we also introduce methods that allow one to use non-uniform random seeds for extractors. These methods can be applied to a wide class of extractors, including dual universal2 hash function, as well as to the conventional universal2 hash functions.