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Based on a generalized chaos synchronization theorem for discrete-time systems, this paper introduces a new 4-dimensional generalized chaos synchronization system. A chaos-based pseudorandom number generator (CPNG) is designed based on the new system. We use the FIPA-140-2 tests issued by the National Institute of Standard and Technology to verify the random properties of the binary number sequences generated by the CPNG. The results showed that all binary number sequences passed the tests. The confidence interval analysis showed that the binary number sequences have sound random properties. Furthermore, the highly sensitivity to the seeds (initial conditions of the CPNG) may make the CPNG be used for “one-time-pad” communication.