The effects of reduced "free will" on Bell-based randomness expansion

With the advent of quantum information, the violation of a Bell inequality is used as evidence of the absence of an eavesdropper in cryptographic scenarios such as key distribution and randomness expansion. One of the key assumptions of Bell's Theorem is the existence of experimental "free will", meaning that measurement settings can be chosen at random and independently by each party. The relaxation of this assumption potentially shifts the balance of power towards an eavesdropper. We consider a no-signalling model with reduced "free will" and bound the adversary's capabilities in the task of randomness expansion.