The inverse of a non-linear dynamical system is introduced and its synchronization with the original system is discussed. If the original system is chaotic and if despite this the inverse system synchronizes, the two systems can be used respectively as a modulator and a demodulator for a chaotic carrier signal. The transmitted signal is then hidden in chaos. We show that the inverse system can have a different order from the original system. The difference corresponds to the relative degree of the original system. In an inverse system with reduced order it is easier to achieve synchronization; on the other hand, such a system may distort a noisy input signal considerably. Examples from the literature are discussed from the point of view of order reduction. New circuit examples are given where the input and the output are not variables at the same port, as is the case for all published examples in the context of communication with chaotic signals. Finally, a quite general system structure together with its inverse that can be designed for synchronization is given.