On the periods of generalized Fibonacci recurrences

Author:

Brent Richard P.

Abstract

We give a simple condition for a linear recurrence ( mod 2 w ) \pmod {2^w} of degree r to have the maximal possible period 2 w 1 ( 2 r 1 ) {2^{w - 1}}({2^r} - 1) . It follows that the period is maximal in the cases of interest for pseudorandom number generation, i.e., for three-term linear recurrences defined by trinomials which are primitive ( mod 2 ) \pmod 2 and of degree r > 2 r > 2 . We consider the enumeration of certain exceptional polynomials which do not give maximal period, and list all such polynomials of degree less than 15.

Publisher

American Mathematical Society (AMS)

Subject

Applied Mathematics,Computational Mathematics,Algebra and Number Theory

Reference25 articles.

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