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Well equidistributed long-period linear

The Well Equidistributed Long-period Linear (WELL) is a family of pseudorandom number generators developed in 2006 by François Panneton, Pierre L'Ecuyer, and Makoto Matsumoto [ja] (松本 眞).[1] It is a form of linear-feedback shift register optimized for software implementation on a 32-bit machine.

Operational design

The structure is similar to the Mersenne Twister, a large state made up of previous output words (32 bits each), from which a new output word is generated using linear recurrences modulo 2 over a finite binary field ${\displaystyle F_{2}}$. However, a more complex recurrence produces a denser generator polynomial, producing better statistical properties.

Each step of the generator reads five words of state: the oldest 32 bits (which may straddle a word boundary if the state size is not a multiple of 32), the newest 32 bits, and three other words in between.

Then a series of eight single-word transformations (mostly of the form `x := x ⊕ (x >> k)`) and six exclusive-or operations combine those into two words, which become the newest two words of state, one of which will be the output.

Variants

Specific parameters are provided for the following generators:

• WELL512a
• WELL521a, WELL521b
• WELL607a, WELL607b
• WELL800a, WELL800b
• WELL1024a, WELL1024b
• WELL19937a, WELL19937b, WELL19937c
• WELL21701a
• WELL23209a, WELL23209b
• WELL44497a, WELL44497b.

Numbers give the state size in bits; letter suffixes denote variants of the same size.

References

1. ^ Panneton, François O.; l'Ecuyer, Pierre; Matsumoto, Pierre (March 2006). "Improved long-period generators based on linear recurrences modulo 2" (PDF). ACM Transactions on Mathematical Software. 32 (1): 1–16. CiteSeerX 10.1.1.73.5499. doi:10.1145/1132973.1132974.CS1 maint: ref=harv (link)
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