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Recent randomness expansion protocols have been proposed which are able togenerate an unbounded amount of randomness from a finite amount of truly randominitial seed. One such protocol, given by Miller and Shi, uses a pair ofnon-signaling untrusted quantum mechanical devices. These play XOR games withinputs given by the user in order to generate an output. Here we present ananalysis of the required seed size, giving explicit upper bounds for the numberof initial random bits needed to jump-start the protocol. The bits output fromsuch a protocol are $\varepsilon$-close to uniform even against quantumadversaries. Our analysis yields that for a statistical distance of$\varepsilon=10^{-1}$ and $\varepsilon=10^{-6}$ from uniformity, the number ofrequired bits is smaller than 225,000 and 715,000, respectively; in general itgrows as $O(\log\frac{1}{\varepsilon})$.

Bounding the seed length of Miller and Shi's unbounded randomness expansion protocol