Editing Linear congruential generator (section)

==LCG derivatives==
There are several generators which are linear congruential generators in a different form, and thus the techniques used to analyze LCGs can be applied to them.

One method of producing a longer period is to sum the outputs of several LCGs of different periods having a large [[least common multiple]]; the [[Wichmann–Hill]] generator is an example of this form. (We would prefer them to be completely [[coprime]], but a prime modulus implies an even period, so there must be a common factor of 2, at least.) This can be shown to be equivalent to a single LCG with a modulus equal to the product of the component LCG moduli.

[[George Marsaglia|Marsaglia]]'s add-with-carry and [[Subtract with carry|subtract-with-borrow]] PRNGs with a word size of ''b''=2<sup>''w''</sup> and lags ''r'' and ''s'' (''r''&nbsp;>&nbsp;''s'') are equivalent to LCGs with a modulus of ''b<sup>r</sup>''&nbsp;±&nbsp;''b<sup>s</sup>''&nbsp;±&nbsp;1.<ref>{{cite conference
 |title=On the Lattice Structure of the Add-with-Carry and Subtract-with-Borrow Random Number Generators
 |first1=Shu |last1=Tezuka |first2=Pierre |last2=L'Ecuyer
 |url=https://core.ac.uk/download/pdf/39215926.pdf
 |date=October 1993
 |publisher=Kyoto University
 |conference=Workshop on Stochastic Numerics
}}</ref><ref>{{cite conference 
 |url=http://www.informs-sim.org/wsc92papers/1992_0059.pdf
 |first1=Shi |last1=Tezuka |first2=Pierre |last2=L'Ecuyer
 |title=Analysis of Add-with-Carry and Subtract-with-Borrow Generators
 |conference=Proceedings of the 1992 Winter Simulation Conference
 |date=December 1992 |pages=443–447
}}</ref>

[[Multiply-with-carry]] PRNGs with a multiplier of ''a'' are equivalent to LCGs with a large prime modulus of ''ab<sup>r</sup>''−1 and a power-of-2 multiplier ''b''.

A [[permuted congruential generator]] begins with a power-of-2-modulus LCG and applies an output transformation to eliminate the short period problem in the low-order bits.