Existing double random phase encryption (DRPE) schemes in the DFT, discrete fractional Fourier transform (DFRFT), and multiple-parameter DFRFT domains are all linear and vulnerable to the known-plaintext attack. In this brief, we first propose a random nonlinearizing block, which is then added to and nonlinearizes the encryption process of the conventional linear DRPE scheme. Besides, from computer experiments, we show the resulting random nonlinearized DRPE schemes are very sensitive to decryption key errors. Consequently, the proposed random nonlinearized DRPE schemes can resist the known-plaintext attack with enhanced security.