  When two players achieve a superclassical score at a nonlocal game, their
outputs must contain intrinsic randomness. This fact has many useful
implications for quantum cryptography. Recently it has been observed (C.
Miller, Y. Shi, Quant. Inf. & Comp. 17, pp. 0595-0610, 2017) that such scores
also imply the existence of local randomness -- that is, randomness known to
one player but not to the other. This has potential implications for
cryptographic tasks between two cooperating but mistrustful players. In the
current paper we bring this notion toward practical realization, by offering a
near-optimal bound on local randomness for the CHSH game, and also proving the
security of a cryptographic application of local randomness (single-bit
certified deletion).
