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 (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 ${\textstyle x:=x\oplus (x\gg k)}$ 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.

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.

Implementations

• Implementations of WELL512a, WELL1024a, WELL19937a, WELL19937c, WELL44497a, WELL44497b in C (Free for non-commercial use)
• Implementations of same algorithms in Scala
• Implementations in C++
• Implementations of WELL512, WELL1024, WELL607 in Java
• Implementations of WELL512, WELL1024 in BBC BASIC
• Modified "maximally equidistributed" implementations of WELL19937, WELL44497 in C (Free for non-commercial use)
• Implementation of WELL512 in C (Public Domain)