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

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 universal$_2$ hash function introduced recently. We also use a new method for constructing extractors by concatenating $\delta$-almost dual universal$_2$ 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 universal$_2$ hash function, as well as to conventional universal$_2$ hash functions.