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**1 - 2**of**2**### Efficient computation of full Lucas sequences

, 1996

"... . Recently, Yen and Laih [1] proposed a algorithm to quickly compute LUC digital signatures. This signature is based on a special type of the Lucas sequence V k . In this paper, we shall generalize their method to any type of Lucas sequences, and we shall extend it to the 'sister' Lucas sequence U ..."

Abstract
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. Recently, Yen and Laih [1] proposed a algorithm to quickly compute LUC digital signatures. This signature is based on a special type of the Lucas sequence V k . In this paper, we shall generalize their method to any type of Lucas sequences, and we shall extend it to the 'sister' Lucas sequence U k . As application, we shall quickly compute the order of an elliptic curve over GF(2 m ). 1 Basic facts In this section, we shall only include the minimal amount of background necessary to understand the article. For a systematic treatment, see the references [2, 3]. Let P and Q be two rational integers, and let ff be a root of x 2 \GammaP x+Q = 0 in the field Q( p D), where D = P 2 \Gamma 4Q is a non-square. Let fi be the conjugate of ff, i.e. fi = ¯ ff. The Lucas sequences fU k g k0 and fV k g k0 with parameters P and Q are given by U k (P; Q) = ff k \Gamma fi k ff \Gamma fi ; (1) V k (P; Q) = ff k + fi k : (2) It can easily be shown that the numbers U i and V i satisf...