The principles behind this algorithm were discovered by R. Silver and J. Tersian and independently by Stein [8]. The algorithm computes the greatest common divisor and is based on the following observations:
If u and v are both even, then \(\gcd(u,v) = 2 \gcd(u/2, v/2)\);
If u is even and v is odd, then \(\gcd(u,v) = \gcd(u/2, v)\);
Otherwise both are odd, and \(\gcd(u,v) = \gcd(|u-v|/2, v)\).
The three conditions cover all possible cases for u and v. The algorithm systematically reduces u and v by repeatedly testing the conditions and accordingly applying the reductions. Note that the first condition, i.e., u and v both being even, applies only in the very beginning of the procedure. Thus, the algorithm first factors out the highest common power of 2 from u and v and stores it in g. In the remainder of the computation only the other two conditions are tested. The computation terminates when one of the operands becomes zero. The algorithm is given as follows.
The Binary GCD Algorithm
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References
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Sunar, B. (2005). Binary Euclidean Algorithm. In: van Tilborg, H.C.A. (eds) Encyclopedia of Cryptography and Security. Springer, Boston, MA . https://doi.org/10.1007/0-387-23483-7_26
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