## On the Periods of Generalized Fibonacci Recurrences (1992)

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@MISC{Brent92onthe,

author = {Richard P. Brent},

title = {On the Periods of Generalized Fibonacci Recurrences},

year = {1992}

}

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### 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 dened 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. 1.

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Citation Context ... is sometimes called the principal period [19] of the linear recurrence, sometimes simply the period [4]. For brevity we de ne = 1. An irreducible polynomial in Z2[t] is a factor of t2r , t (see e.g. =-=[18]-=-), so j2r , 1. We saythat Q(t) is primitive (mod 2) if =2r , 1. Note that primitivity is a stronger condition than irreducibility1 , i.e. Q(t) primitive implies that Q(t) is irreducible, but the conve... |

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Citation Context ... means that Q(t) mod 2 is irreducible (resp. primitive) in Z2[t]. 2 For example, the polynomial 1 + t + t 2 + t 4 + t 6 is irreducible, but not primitive, since it has =21< 2 6 , 1. 2 (8)sIt is known =-=[7, 12, 19]-=- that pw 2 w,1 with equality holding for all w>0 i it holds for w = 3. The main aim of this paper is to give a simple necessary and su cient condition for pw =2 w,1 : (9) The result is stated in Theor... |

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Citation Context ...umbers satisfy a linear recurrence Generalized Fibonacci recurrences of the form Fn = Fn,1 + Fn,2: xn = xn,s xn,r mod 2 w are of interest because they are often used to generate pseudo-random numbers =-=[1, 5, 6, 11, 13, 17]-=-. We assume throughout that x0;:::;xr,1 are given and not all even, and w>0 is a xed exponent. Usually w is close to the wordlength of the (binary) computer used. Apart from computational convenience,... |

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Citation Context ...an irreducibility1 , i.e. Q(t) primitive implies that Q(t) is irreducible, but the converse is not generally true unless 2r , 1 is prime2 .Tables of irreducible and primitive trinomials are available =-=[4, 10, 14, 16, 20, 22, 23, 24, 25]-=-. In the following we usually assume that Q(t) is irreducible. Our assumption that q0 and qr are odd excludes the trivial case Q(t) =t, and implies that ~Q(t) is irreducible (or primitive) of degree r... |