Semidefinite Programs for Randomness Extractors

Abstract

Randomness extractors are an important building block for classical and quantum cryptography. However, for many applications it is crucial that the extractors are quantum-proof, i.e., that they work even in the presence of quantum adversaries. In general, quantum-proof extractors are poorly understood and we would like to argue that in the same way as Bell inequalities (multiprover games) and communication complexity, the setting of randomness extractors provides a operationally useful framework for studying the power and limitations of a quantum memory compared to a classical one. We start by recalling how to phrase the extractor property as a quadratic program with linear constraints. We then construct a semidefinite programming (SDP) relaxation for this program that is tight for some extractor constructions. Moreover, we show that this SDP relaxation is even sufficient to certify quantum-proof extractors. This gives a unifying approach to understand the stability properties of extractors against quantum adversaries. Finally, we analyze the limitations of this SDP relaxation.