EUCLIDEAN GCD ALGORITHM IS NOT OPTIMAL

Abstract

Using the operations {$+,~-$}, multiplication and division by constants {$x sub z ,~/z$}, floor operation, {$(lf~(rf$} and indirect addressing, we compute $GCD(x,y),~x,y~(mo~[0,N]$ and find $a,b~(mo~[0,N]$ 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 numbers that are produced in the algorithms are less than $ max(x,y)$. We also prove that using these operations our bound is tight. In the boolean model we prove that to obtain this upper bound we must use $OMEGA ((log N) sup epsilon )$ space for some constant $ epsilon $. We also study the direct sum complexity of GCD and prove that GCD function does not satisfy the direct sum conjecture and we study the operation complexity of computing GCD and LCM of $n$ numbers and find tight bounds for these problems.