  Bit commitment and coin flipping occupy a unique place in the
device-independent landscape, as the only device-independent protocols thus far
suggested for these tasks are reliant on tripartite GHZ correlations. Indeed,
we know of no other bipartite tasks, which admit a device-independent
formulation, but which are not known to be implementable using only bipartite
nonlocality. Another interesting feature of these protocols is that the
pseudo-telepathic nature of GHZ correlations -- in contrast to the generally
statistical character of nonlocal correlations, such as those arising in the
violation of the CHSH inequality -- is essential to their formulation and
analysis. In this work, we present a device-independent bit commitment protocol
based on CHSH testing, which achieves the same security as the optimal
GHZ-based protocol. The protocol is analyzed in the most general settings,
where the devices are used repeatedly and may have long-term quantum memory. We
also recast the protocol in a post-quantum setting where both honest and
dishonest parties are restricted only by the impossibility of signaling, and
find that overall the supra-quantum structure allows for greater security.
