A Lower Bound for Integer Greatest Common Divisor Computations

It is proved that no finite computation tree with operations { +, -, *, /, mod, < } can decide whether the greatest common divisor 1991 of a and b is one, for all pairs of integers a and b. This settles a problem posed by Gro¨tschel et al. Moreover, if the constants explicitly involved in any operation performed in the tree are restricted to be “0” and “1” (and any other constant must be computed), then we prove an Ω(log log n) lower bound on the depth of any computation tree with operations { +, -, *, /, mod, < } that decides whether the gcd of a and b is one, for all pairs of n-bit integers a and b. A novel technique for handling the truncation operation is implicit in the proof of this lower bound. In a companion paper, other lower bounds for a large class of problems are proved using a similar technique.