### Abstract

Marsaglia and Zaman recently proposed new classes of random number generators, called add-with-carry1993 and subtract-with-borrow(SWB), which are capable of quickly generating very long-period (pseudo)-random number sequences using very little memory. We show that these sequences are essentially equivalent to linear congruential sequences with very large prime moduli. So, the AWC/SWB generators can be viewed as efficient ways of implementing such large linear congruential generators. As a consequence, the theoretical properties of such generators can be studied in the same way as for linear congruential generators, namely, via the spectral and lattice tests. We also show how the equivalence can be exploited to implement efficient jumping-ahead facilities for the AWC and SWB sequences. Our numerical examples illustrate the fact that AWC/SWB generators have extremely bad lattice structure in high dimensions.

Original language | English |
---|---|

Pages (from-to) | 315-331 |

Number of pages | 17 |

Journal | ACM Transactions on Modeling and Computer Simulation (TOMACS) |

Volume | 3 |

Issue number | 4 |

DOIs | |

Publication status | Published - Jan 10 1993 |

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### All Science Journal Classification (ASJC) codes

- Modelling and Simulation
- Computer Science Applications

### Cite this

*ACM Transactions on Modeling and Computer Simulation (TOMACS)*,

*3*(4), 315-331. https://doi.org/10.1145/159737.159749

**On the Lattice Structure of the Add-With-Carry and Subtract-With-Borrow Random Number Generators.** / Tezuka, Shu; L'Ecuyer, Pierre; Couture, Raymond.

Research output: Contribution to journal › Article

*ACM Transactions on Modeling and Computer Simulation (TOMACS)*, vol. 3, no. 4, pp. 315-331. https://doi.org/10.1145/159737.159749

}

TY - JOUR

T1 - On the Lattice Structure of the Add-With-Carry and Subtract-With-Borrow Random Number Generators

AU - Tezuka, Shu

AU - L'Ecuyer, Pierre

AU - Couture, Raymond

PY - 1993/1/10

Y1 - 1993/1/10

N2 - Marsaglia and Zaman recently proposed new classes of random number generators, called add-with-carry1993 and subtract-with-borrow(SWB), which are capable of quickly generating very long-period (pseudo)-random number sequences using very little memory. We show that these sequences are essentially equivalent to linear congruential sequences with very large prime moduli. So, the AWC/SWB generators can be viewed as efficient ways of implementing such large linear congruential generators. As a consequence, the theoretical properties of such generators can be studied in the same way as for linear congruential generators, namely, via the spectral and lattice tests. We also show how the equivalence can be exploited to implement efficient jumping-ahead facilities for the AWC and SWB sequences. Our numerical examples illustrate the fact that AWC/SWB generators have extremely bad lattice structure in high dimensions.

AB - Marsaglia and Zaman recently proposed new classes of random number generators, called add-with-carry1993 and subtract-with-borrow(SWB), which are capable of quickly generating very long-period (pseudo)-random number sequences using very little memory. We show that these sequences are essentially equivalent to linear congruential sequences with very large prime moduli. So, the AWC/SWB generators can be viewed as efficient ways of implementing such large linear congruential generators. As a consequence, the theoretical properties of such generators can be studied in the same way as for linear congruential generators, namely, via the spectral and lattice tests. We also show how the equivalence can be exploited to implement efficient jumping-ahead facilities for the AWC and SWB sequences. Our numerical examples illustrate the fact that AWC/SWB generators have extremely bad lattice structure in high dimensions.

UR - http://www.scopus.com/inward/record.url?scp=0027677150&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0027677150&partnerID=8YFLogxK

U2 - 10.1145/159737.159749

DO - 10.1145/159737.159749

M3 - Article

AN - SCOPUS:0027677150

VL - 3

SP - 315

EP - 331

JO - ACM Transactions on Modeling and Computer Simulation

JF - ACM Transactions on Modeling and Computer Simulation

SN - 1049-3301

IS - 4

ER -