2005 International Conference on Analysis of Algorithms
Conrado Martínez (ed.)
DMTCS Conference Volume AD (2005), pp. 7194
author:  Benoît Daireaux, Véronique MaumeDeschamps and Brigitte Vallée 

title:  The Lyapunov tortoise and the dyadic hare 
keywords:  
abstract: 
We study a gcd algorithm directed by Least Significant
Bits, the socalled LSB algorithm, and provide a precise
averagecase analysis of its main parameters [number of
iterations, number of shifts, etc…]. This analysis
is based on a precise study of the dynamical systems which
provide a continuous extension of the algorithm, and, here,
it is proved convenient to use both a 2adic extension and
a real one. This leads to the framework of products of
random matrices, and our results thus involve a constant
γ
which is the Lyapunov exponent of the set of matrices
relative to the algorithm. The algorithm can be viewed as a
race between a dyadic hare with a speed of 2 bits by step
and a ``real'' tortoise with a speed equal to
γ/log2 ∼0.05
bits by step. Even if the tortoise starts before the
hare, the hare easily catches up with the tortoise [unlike
in Aesop's fable [Ae]…], and the algorithm
terminates.

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reference:  Benoît Daireaux and Véronique MaumeDeschamps and Brigitte Vallée (2005), The Lyapunov tortoise and the dyadic hare, in 2005 International Conference on Analysis of Algorithms, Conrado Martínez (ed.), Discrete Mathematics and Theoretical Computer Science Proceedings AD, pp. 7194 
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