The next two exercises deal with the game of Euclid. Two players begin with a pair of positive integers and take turns making moves of the following type. A player can move from the pair of positive integers {x, y} with x ≥ y, to any of the pairs {x − ty, y}, where t is a positive integer and x − ty ≥ 0. A winning move consists of moving to a pair with one element equal to 0.
Show that the number of bit operations needed to use the Euclidean algorithm to find the greatest common divisor of two positive integers a and b with a > b is 0((log2 a)2). (Hint: First show lhat the complexity of division of the positive integer q by the positive integer d is O (log d log q).)
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