A Fast Jump Ahead Algorithm for Linear Recurrences in a Polynomial Space

Linear recurring sequences with very large periods are widely used as the basic building block of pseudorandom number generators. In many simulation applications, multiple streams of random numbers are needed, and these multiple streams are normally provided by jumping ahead in the sequence to obtain starting points that are far apart. For maximal-period generators having a large state space, this jumping ahead can be costly in both time and memory usage. We propose a new jump ahead method for this kind of situation. It requires much less memory than the fastest algorithms proposed earlier, while being approximately as fast (or faster) for generators with a large state space such as the Mersenne twister.