Recursive cheating strategies for the relativistic $F_Q$ bit commitment protocol

In this paper, we study relativistic bit commitment, which uses timing and location constraints to achieve information theoretic security. We consider the $F_Q$ multi-round bit commitment scheme introduced by Lunghi et al. [LKB+15]. This protocol was shown secure against classical adversaries as long as the number of rounds $m$ is small compared to $\sqrt{Q}$ where $Q$ is the size of the used field in the protocol [CCL15,FF16]. In this work, we study classical attacks on this scheme. We use classical strategies for the $CHSH_Q$ game described in [BS15] to derive cheating strategies for this protocol. In particular, our cheating strategy shows that if $Q$ is an even power of any prime, then the protocol is not secure when the number of rounds $m$ is of the order of $\sqrt{Q}$. For those values of $Q$, this means that the upper bound of [CCL15,FF16] is essentially optimal.