EUCLID'S GCD ALGORITHM IS NOT OPTIMAL PART I. UPPER BOUND

Abstract

Using linear arithmetic operations, the floor operation, and indirect addressing, we sequentially compute $GCD(x,y)$ for $0~<=~x,y~<=~N$, and find two integers $-N~<=~a,b~<=~N$ such that $ax~+~by~=~GCD(x,y)$ with operation complexity 0 left ( {log~N} over {log~log~N} right ) and space complexity $0(($log$~N) sup epsilon )$ for any constant $0~<~epsilon~<~1$. The intermediate numbers obtained during the execution of these algorithms are all less than $max(x,y)$. In the second part of this paper (Part II. Lower Bound), we prove that, using the given operations, this bound is tight. We also study the direct sum complexity of computing the GCD and also the complexity of computing the GCD using tables of bounded size.