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 . 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.

## 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)

## References

**^**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)

## External links