In this paper a spectral characterization of the synchronization property of Boolean dynamical systems is provided. Conditions on the spectrum of the next-state function are derived for two systems coupled in a unidirectional way - also called master-slave configuration - to guarantee self-synchronization. Two kinds of self-synchronization are discussed: the statistical one and the finite one. Next, some conditions are stated for a specific input sequence to allow the system to be self-synchronizing. Some of the results are based on the notion of influence of variables, a notion that is extended to vectorial Boolean functions for the purpose of the paper. A potential application to cryptography is finally given.