Random Walks Arising in Random Number Generation

Abstract

Random number generators often work by recursively computing $X_{n+1} \equiv aX_n + b (\mod p)$. 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 $X_n$ to become random. For example, if $a = 1$ and $b = 0, 1$, or $-1$ each with probability $\frac{1}{3}$, then $cp^2$ steps are required to achieve randomness. On the other hand, if $a = 2$ and $b = 0, 1$, or $-1$, each with probability $\frac{1}{3}$, 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 $p, 1.02 \log_2 p$ steps are enough.