# ETOOBUSY ðŸš€ minimal blogging for the impatient

# Stein's algorithm for GCD

**TL;DR**

I read about Steinâ€™s algorithm for calculating the greatest common divisor between two integers. Interesting.

So it seems that finding the greatest common divisor is a hot topic in this blog, considering that I already wrote about it in The extended Euclidâ€™s algorithm. Go figure.

The algorithm is rooted in a few equivalences, which are fine. The most challenging might be considered the last one, i.e.:

$g = gcd(u, v) = gcd(|u âˆ’ v|, min(u, v))$ if $u$ and $v$ are both odd.

When you think about it, anyway, itâ€™s pretty obvious: an integer $g$ that divides both $u$ and $v$ must also divide their difference and its absolute value. The contrary also applies, which accounts for reading the equivalence in the reverse direction.

Hereâ€™s a possible iterative implementation (you know Iâ€™m fond of them):

```
sub stein ($u, $v) {
die 'the greatest common divisor for (0, 0) is undefined)'
unless $u || $v;
# cope with edge cases, insist on using positive integers only
$u = -$u if $u < 0;
return $u unless $v;
$v = -$v if $v < 0;
return $v unless $u;
# we have to go into the rabbit hole here...
my $retval = 1;
# first phase, find out the contributing power of 2, if any
while (($u % 2 == 0) && ($v % 2 == 0)) {
$retval <<= 1;
$v >>= 1;
$u >>= 1;
}
# second phase, from now on either u and v will be odd
while ('necessary') {
$v >>= 1 while $v % 2 == 0; # we don't need evens here
$u >>= 1 while $u % 2 == 0; # ditto
return $retval * $u if $v == $u;
($u, $v) = $u < $v ? ($v - $u, $u) : ($u - $v, $v);
}
}
```

Aside question: why $u$ and $v$ instead of $n$ and $m$? I guess it was a personal preference of Steinâ€¦

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