On the periods of generalized Fibonacci recurrences

TL;DR

This paper establishes a simple criterion for linear recurrences modulo powers of two to achieve maximal periods, with implications for pseudo-random number generation and enumeration of special polynomials.

Contribution

It provides a straightforward condition for maximal period in linear recurrences modulo 2^w and enumerates exceptional polynomials with non-maximal periods below degree 15.

Findings

Maximal period condition for linear recurrences mod 2^w

Identification of primitive trinomials for pseudo-random generators

Complete list of non-maximal polynomials of degree < 15

Abstract

We give a simple condition for a linear recurrence (mod 2^w) of degree r to have the maximal possible period 2^(w-1).(2^r-1). It follows that the period is maximal in the cases of interest for pseudo-random number generation, i.e. for 3-term linear recurrences defined by trinomials which are primitive (mod 2) and of degree 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.