### Abstract

A necessary and sufficient condition is established for the generalized feedback shift register (GFSR) sequence introduced by Lewis and Payne to be k-distributed. Based upon the theorem, a theoretical test for k-distributivity is proposed and performed in a reasonable amount of computer time, even for k = 16 and a high degree of resolution (for which statistical tests are impossible because of the astronomical amount of computer time required). For the special class of GFSR generators considered by Arvillias and Maritsas based on the primitive trinomial D_{p} + D_{q} + 1 with q = an integral power of 2, it is shown that the sequence is k-distributed if and only if the lengths of all subregisters are at least k. The theorem also leads to a simple and efficient method of initializing the GFSR generator so that the sequence to be generated is k-distributed.

Original language | English |
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Pages (from-to) | 516-523 |

Number of pages | 8 |

Journal | Communications of the ACM |

Volume | 26 |

Issue number | 7 |

DOIs | |

Publication status | Published - Jul 1 1983 |

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### All Science Journal Classification (ASJC) codes

- Computer Science(all)

### Cite this

*Communications of the ACM*,

*26*(7), 516-523. https://doi.org/10.1145/358150.358159

**The k-distribution of generalized feedback shift register pseudorandom numbers.** / Fushimi, M.; Tezuka, S.

Research output: Contribution to journal › Article

*Communications of the ACM*, vol. 26, no. 7, pp. 516-523. https://doi.org/10.1145/358150.358159

}

TY - JOUR

T1 - The k-distribution of generalized feedback shift register pseudorandom numbers

AU - Fushimi, M.

AU - Tezuka, S.

PY - 1983/7/1

Y1 - 1983/7/1

N2 - A necessary and sufficient condition is established for the generalized feedback shift register (GFSR) sequence introduced by Lewis and Payne to be k-distributed. Based upon the theorem, a theoretical test for k-distributivity is proposed and performed in a reasonable amount of computer time, even for k = 16 and a high degree of resolution (for which statistical tests are impossible because of the astronomical amount of computer time required). For the special class of GFSR generators considered by Arvillias and Maritsas based on the primitive trinomial Dp + Dq + 1 with q = an integral power of 2, it is shown that the sequence is k-distributed if and only if the lengths of all subregisters are at least k. The theorem also leads to a simple and efficient method of initializing the GFSR generator so that the sequence to be generated is k-distributed.

AB - A necessary and sufficient condition is established for the generalized feedback shift register (GFSR) sequence introduced by Lewis and Payne to be k-distributed. Based upon the theorem, a theoretical test for k-distributivity is proposed and performed in a reasonable amount of computer time, even for k = 16 and a high degree of resolution (for which statistical tests are impossible because of the astronomical amount of computer time required). For the special class of GFSR generators considered by Arvillias and Maritsas based on the primitive trinomial Dp + Dq + 1 with q = an integral power of 2, it is shown that the sequence is k-distributed if and only if the lengths of all subregisters are at least k. The theorem also leads to a simple and efficient method of initializing the GFSR generator so that the sequence to be generated is k-distributed.

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UR - http://www.scopus.com/inward/citedby.url?scp=0020793402&partnerID=8YFLogxK

U2 - 10.1145/358150.358159

DO - 10.1145/358150.358159

M3 - Article

AN - SCOPUS:0020793402

VL - 26

SP - 516

EP - 523

JO - Communications of the ACM

JF - Communications of the ACM

SN - 0001-0782

IS - 7

ER -