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On Sequences of Pseudo-Random Numbers of Maximal Length

On Sequences of Pseudo-Random Numbers of Maximal Length On S e q u e n c e s of P s e u d o - R a n d o m N u m b e r s of Maximal Length* J. C ERTAINE Nuclear Development Corporation of America, White Plains, N. Y. Introduction The method of obtaining sequences of pseudo-random numbers b y calculating successive powers (modulo m) 1 of a number a has been described in [1, 2]. When such sequences are generated on digital computers, the module number m is usually taken to be ,~ power of the base of the system which is used to represent numbers to the computer. In general, m will be the largest power of 2 or 10 which will fit into the machine. The question naturally arises as to what number a will generate ~ non-repeating sequence of maximal length. I t is a consequence of Euler's theorem (cf. next section) t h a t there exists a number O(m) such t h a t no non-repeating sequence generated by a number relatively prime to m m a y have length longer t h a n O(m). Indeed, the length of any such sequence must be http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of the ACM (JACM) Association for Computing Machinery

On Sequences of Pseudo-Random Numbers of Maximal Length

Certaine, J.
Journal of the ACM (JACM) , Volume 5 (4) – Oct 1, 1958
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Publisher
Association for Computing Machinery
Copyright
Copyright © 1958 by ACM Inc.
ISSN
0004-5411
DOI
10.1145/320941.320949
Publisher site
See Article on Publisher Site

Abstract

On S e q u e n c e s of P s e u d o - R a n d o m N u m b e r s of Maximal Length* J. C ERTAINE Nuclear Development Corporation of America, White Plains, N. Y. Introduction The method of obtaining sequences of pseudo-random numbers b y calculating successive powers (modulo m) 1 of a number a has been described in [1, 2]. When such sequences are generated on digital computers, the module number m is usually taken to be ,~ power of the base of the system which is used to represent numbers to the computer. In general, m will be the largest power of 2 or 10 which will fit into the machine. The question naturally arises as to what number a will generate ~ non-repeating sequence of maximal length. I t is a consequence of Euler's theorem (cf. next section) t h a t there exists a number O(m) such t h a t no non-repeating sequence generated by a number relatively prime to m m a y have length longer t h a n O(m). Indeed, the length of any such sequence must be

Journal

Journal of the ACM (JACM) – Association for Computing Machinery

Published: Oct 1, 1958

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