  The present paper analyses and presents several improvements to the algorithm
for finding the $(a,b)$-pairs of integers used in the $k$-ary reduction of the
right-shift $k$-ary integer GCD algorithm. While the worst-case complexity of
Weber's "Accelerated integer GCD algorithm" is $\cO\l(\log_\phi(k)^2\r)$, we
show that the worst-case number of iterations of the while loop is exactly
$\tfrac 12 \l\lfloor \log_{\phi}(k)\r\rfloor$, where $\phi := \tfrac 12
\l(1+\sqrt{5}\r)$.\par We suggest improvements on the average complexity of the
latter algorithm and also present two new faster residual algorithms: the
sequential and the parallel one. A lower bound on the probability of avoiding
the while loop in our parallel residual algorithm is also given.
