# binary GCD

(algorithm)

**Definition:**
Compute the *greatest common divisor* of two integers, u and v, expressed in binary. The run time complexity is *O((log*_{2} 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.

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with Paul Black.

Entry modified 4 September 2009.

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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: https://www.nist.gov/dads/HTML/binaryGCD.html