  A device-independent randomness expansion protocol aims to take an initial
random string and generate a longer one, where the security of the protocol
does not rely on knowing the inner workings of the devices used to run it. In
order to do so, the protocol tests that the devices violate a Bell inequality
and one then needs to bound the amount of extractable randomness in terms of
the observed violation. The entropy accumulation theorem lower bounds the
extractable randomness of a protocol with many rounds in terms of the
single-round von Neumann entropy of any strategy achieving the observed score.
Tight bounds on the von Neumann entropy are known for the one-sided randomness
(i.e., where the randomness from only one party is used) when using the
Clauser-Horne-Shimony-Holt (CHSH) game. Here we investigate the possible
improvement that could be gained using the two-sided randomness. We generate
upper bounds on this randomness by attempting to find the optimal eavesdropping
strategy, providing analytic formulae in two cases. We additionally compute
lower bounds that outperform previous ones and can be made arbitrarily tight
(at the expense of more computation time). These bounds get close to our upper
bounds, and hence we conjecture that our upper bounds are tight. We also
consider a modified protocol in which the input randomness is recycled. This
modified protocol shows the possibility of rate gains of several orders of
magnitude based on recent experimental parameters, making device-independent
randomness expansion significantly more practical. It also enables the locality
loophole to be closed while expanding randomness in a way that typical
spot-checking protocols do not.
