We study a gcd algorithm directed by Least Significant Bits, the so--called LSB algorithm, and provide a precise average--case 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 2--adic 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.