A novel pseudo-random number generator based on z-logistic map is proposed. The observed binary sequence of the chaotic orbit which is realized exactly under finite computing precision mostly retains the statistical characteristics and the randomness of the chaos-based information source which is defined on real domain, so the cryptographic properties of this novel chaos-based pseudo-random number generator (CPRNG) can be supported theoretically. Moreover, the period of this CPRNG is predicable and the weak keys can be excluded using a simple algorithm. This CPRNG overcomes the disadvantage of the traditional CPRNG whose existing weak keys are difficult to be excluded. Theoretical analysis and simulation results demonstrate that the cryptographic properties of the novel CPRNG are good, so it has potential application prospect in many areas including data encryption.