Random Walks Arising in Random Number Generation

Abstract

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. Random number generators often work by recursively computing Xn+ 1 aXn + b (mod p). Various schemes exist for combining these random number generators. In one scheme, a and b are themselves chosen each time from another generator. Assuming that this second source is truly random, we investigate how long it takes for Xn to become random. For example, if a = 1 and b = O 1, or -1 each with probability 1 , then cp2 steps are required to achieve randomness. On the other hand, if a = 2 and b = 0, 1, or -1, each with probability , then c log p log log p steps always suffice to guarantee randomness, and for infinitely many p, are necessary as well, although, in fact, for almost all odd p, 1.02 log2p steps are enough.