A Fast Algorithm for Computing Large Fibonacci Numbers

Information Processing Letters 75 (2000) 243–246

A fast algorithm for computing large Fibonacci numbers
Daisuke Takahashi
Department of Information and Computer Sciences, Saitama University, 255 Shimo-Okubo, Urawa-shi, Saitama 338-8570, Japan Received 13 March 2000; received in revised form 19 June 2000 Communicated by K. Iwama

Abstract We present a fast algorithm for computing large Fibonacci numbers. It is known that the product of Lucas numbers algorithm uses the fewest bit operations to compute the Fibonacci number Fn . We show that the number of bit operations in the conventional product of Lucas numbers algorithm can be reduced by replacing multiplication with the square operation. © 2000 Elsevier Science B.V. All rights reserved.
Keywords: Program derivation; Fibonacci numbers

1. Introduction

1.1. Fibonacci and Lucas numbers We define the Fibonacci numbers as F0 = 0, F1 = 1, n 0. (1)

Many algorithms for computing Fibonacci numbers have been well studied [11,12,5,8,4,3,7,9,2]. It is known that the product of Lucas numbers algorithm uses the fewest bit operations to compute Fn [2]. In this paper, we present a fast algorithm for computing large Fibonacci numbers. This algorithm is based on the product of Lucas numbers algorithm [9, 2]. The conventional product of Lucas numbers algorithm uses multiplication and square operation. In general, it is known that the number of bit operations used to square an n-bit number is less than that used to multiply two n-bit numbers. Thus, the number of bit operations in the conventional product of Lucas numbers algorithm can be reduced by replacing multiplication with the square operation.

Fn+2 = Fn+1 + Fn ,

The Lucas numbers are defined as L0 = 2, L1 = 1, n 0. (2)

Ln+2 = Ln+1 + Ln ,

We also have the formulas αn − β n , α−β Ln = α n + β n , Fn = (3)

E-mail address: [e-mail] (D. Takahashi).

(4) √ √ where α = (1 + 5)/2 and β = (1 − 5)/2. We will use γ n to represent the

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