The ubiquitous no-signaling constraints state that the probability distributions of outputs of any subset of parties in a Bell experiment are independent of remaining parties' inputs. These constraints are considered to form ultimate limits for physical correlations and led to the fields of post-quantum cryptography, randomness generation besides identifying information-theoretic principles underlying quantum theory. Here we show that while these constraints are sufficient, they are not necessary to enforce relativistic causality in multi-party correlations, i.e., the rule that correlations do not allow casual loops. Depending on the space-time coordinates of the measurement events, causality only imposes a subset of no-signaling conditions. We first consider the n-party Bell experiment (n > 2) and identify all configurations where subsets of the constraints suffice. Secondly, we examine the implications for device-independent cryptography against an eavesdropper constrained only by relativity, detailing among other effects explicit attacks on well-known randomness amplification and key distribution protocols.