Cyclostationary features of communication signals are known to compromise the security of transmissions against eavesdropping attacks. They can be used for signal detection, modulation recognition or for blind estimation of PHY layer parameters. In the context of underwater acoustic communications, this work presents a method that voluntarily distorts the transmitted signal to hide the cyclostationary patterns. This distortion is obtained with a pseudo-random time-varying filter that combines time warping and dispersive filtering. The proposed method acts as a plugin that is applicable to most existing transmission scheme. It is shown that this distortion can be easily reversed by the cooperative receiver using a simple matched filter combined with resampling. Numerical results with replay simulations of acoustic channels measured at sea illustrate the benefits of the proposed method. For both a coherent and a noncoherent modem, the induced distortion is shown to be robust to existing cyclostationary attacks, at the cost of a slight reduction in data rate (~7%). Furthermore, no performance degradation in terms of packet error ratio is observed for cooperative transmissions.