XORshift

Xorshift is a family of pseudorandom number generators based on the repeated application of XOR and bitwise shift operations. Designed by George Marsaglia in 2003, xorshift generators are valued for their speed and simplicity on common CPUs, requiring only integer arithmetic and fast bitwise operators.

A typical xorshift generator maintains a small internal state, updates it with a fixed sequence of shifts and XORs, and uses the resulting state as the next output. A widely cited 32-bit configuration performs three operations in sequence: x ^= x << 13; x ^= x >> 17; x ^= x << 5.

There are many parameterizations for 64-bit and larger states. The 128-bit version uses two 64-bit words to form the state and yields a long period, often described as 2^128 - 1 for nonzero seeds. The 1024-bit variant uses an array of sixteen 64-bit values, with a recurrence that produces a period of 2^1024 - 1. Some designs include an output scrambler, such as a multiplicative step, to improve statistical properties of the produced numbers.

Properties and limitations: xorshift generators are fast and simple, but their linear structure can introduce correlations in outputs and they may fail certain statistical tests. They are not cryptographically secure and should not be used where strong security is required. They are commonly employed for non-cryptographic tasks such as simulations and graphics, and are sometimes combined with additional transforms to enhance statistical quality.