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Some new algorithms are presented for efficiently generating pseudorandom noise both in hardware and software. In software, a new word of pseudorandom bits can be generated about every 12 machine cycles, and hardware implementations can generate pseudo Gaussian noise with bandwith of 20 MHz or more. The algorithms generate binary maximal-length linear recursive sequencies of high degree and with many nonzero terms. The ability to efficiently implement high-degree recursions is important because the number of consecutive bits which can be guaranteed to be both linearly and statistically independent is equal to the degree of the recursion. The implementations are by interconnection of several short shift registers in a linear manner in such a way that different widely spaced phase shifts of the same p-n sequence appear in the stages of the several registers. Some specific algorithms have been subjected to extensive statistical evaluation, with no evidence found to distinguish the sequences from purely random binary sequences.