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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 (松本眞).^[1] It is a form of linear-feedback shift register optimized for software implementation on a 32-bit machine.

Operational design

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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 👁 {\textstyle x:=x\oplus (x\gg 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.