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

Universal security for randomness expansion from the spot-checking protocol