%0 Journal Article %J SIAM Journal on Computing %D 2017 %T Universal Security for Randomness Expansion from the Spot-Checking Protocol %A Carl Miller %A Yaoyun Shi %X

Colbeck (Thesis, 2006) proposed using Bell inequality violations to generate certified random numbers. While full quantum-security proofs have been given, it remains a major open problem to identify the broadest class of Bell inequalities and lowest performance requirements to achieve such security. In this paper, working within the broad class of spot-checking protocols, we prove exactly which Bell inequality violations can be used to achieve full security. Our result greatly improves the known noise tolerance for secure randomness expansion: for the commonly used CHSH game, full security was only known with a noise tolerance of 1.5%, and we improve this to 10.3%. We also generalize our results beyond Bell inequalities and give the first security proof for randomness expansion based on Kochen-Specker inequalities. The central technical contribution of the paper is a new uncertainty principle for the Schatten norm, which is based on the uniform convexity inequality of Ball, Carlen, and Lieb (Inventiones mathematicae, 115:463-482, 1994).

%B SIAM Journal on Computing %V 46 %8 2017/08/01 %G eng %U http://epubs.siam.org/doi/10.1137/15M1044333 %N 4 %R 10.1137/15M1044333