Given that memristor with memory property is an ideal electronic component for implementing the artificial neural synaptic function, a brand-new tristable locally active memristor model is first proposed in this paper. On this basis, a novel four dimensional fractional-order memristive cellular neural network (FO-MCNN) model with hidden attractors is constructed to enhance the engineering feasibility of original CNN model and its performances. And then, its hardware circuit implementation and complicated dynamic properties are investigated in multi-simulation platforms. Subsequently, it is used toward the secure communication application scenarios. Taking it as the pseudo-random number generator (PRNG), a new privacy image security scheme is designed based on the adaptive sampling rate compressive sensing (ASR-CS) model. Eventually, the simulation analysis and comparative experiments manifest that the proposed data encryption scheme possesses strong immunity against the various security attack models and satisfactory compression performance.