Abstract: The effects of what values the a, b, x take in the Weierstrass function ∑aⁱ cos(bⁱπx) on the randomness of the sequence {xn} are studied. It is proved that it will generate an unpredictable good random sequence {xn} when b>1. Then, a random number generator is designed for data encryption. Compared with other generators, it has several merits such as the unpredictability of the sequence, bigger secret key space, smaller storage space, simpler algorithm and easier to implement in addition to its long period, quick running speed, high compatibility and reproducibility of good randomness. The security of random number generator is thus enhanced, to a certain extent, thus meeting the requirements of a high-performance generator.