binary GCD

(algorithm)

Definition: Compute the greatest common divisor of two integers, u and v, expressed in binary. The run time complexity is O((log₂ u v)²) bit operations.

See also Euclid's algorithm.

Note: Discovered by J. Stein in 1967. Another source says discovered by R. Silver and J. Tersian in 1962 and published by G. Stein in 1967.

The algorithm uses the following observations.

If u and v are both even, gcd(u,v) = 2 gcd(u/2, v/2).
If u is even and v is odd, gcd(u,v) = gcd(u/2, v).
Otherwise both are odd, and gcd(u,v) = gcd(|u-v|/2, v). (Euclid's algorithm with a division by 2 since the difference of two odd numbers is even.)

Here is the algorithm. It is especially efficient for operations on binary representations.

g = 1
while u is even and v is even

u = u/2 (right shift)
v = v/2
g = 2*g (left shift)
now u or v (or both) are odd
while u > 0

if u is even, u = u/2
else if v is even, v = v/2
else

t = |u-v|/2
if u < v, then v = t else u = t

return v*g

Knuth calls this algorithm 4.5.2B in [Knuth97, 2:338]. Torsten Eichstädt suggested incrementing g in step 2.3 (g++) and returning v left shifted g times in step 4 (return v<<g). 4 April 2006. Alex Peyser pointed out that g must start at 0. 10 July 2008.

Author: PEB

More information

Binary Euclid's Algorithm including examples and references.

Go to the Dictionary of Algorithms and Data Structures home page.

If you have suggestions, corrections, or comments, please get in touch with Paul Black.

Entry modified 4 September 2009.
HTML page formatted Mon Feb 2 13:10:39 2015.

Cite this as:
Paul E. Black, "binary GCD", in Dictionary of Algorithms and Data Structures [online], Vreda Pieterse and Paul E. Black, eds. 4 September 2009. (accessed TODAY) Available from: http://www.nist.gov/dads/HTML/binaryGCD.html