  We propose a protocol for secret sharing, called dual quantum information
splitting (DQIS), that reverses the roles of state and channel in standard
quantum information splitting. In this method, a secret is shared via
teleportation of a fiducial input state over an entangled state that encodes
the secret in a graph state basis. By performing a test of violation of a Bell
inequality on the encoded state, the legitimate parties determine if the
violation is sufficiently high to permit distilling secret bits. Thus, the code
space must be maximally and exclusively nonlocal. To this end, we propose two
ways to obtain code words that are degenerate with respect to a Bell operator.
The security of DQIS comes from monogamy of nonlocal correlations, which we
illustrate by means of a simple single-qubit attack model. The nonlocal basis
of security of our protocol makes it suitable for security in general
monogamous theories and in the more stringent, device-independent cryptographic
scenario.
