# The accelerated integer GCD algorithm

@article{Weber1995TheAI,
title={The accelerated integer GCD algorithm},
author={Kenneth Weber},
journal={ACM Trans. Math. Softw.},
year={1995},
volume={21},
pages={111-122}
}
• Kenneth Weber
• Published 1 March 1995
• Computer Science
• ACM Trans. Math. Softw.
Since the greatest common divisor (GCD) of two integers is a basic arithmetic operation used in many mathematical software systems, new algorithms for its computation are of widespread interest. The accelerated integer GCD algorithm discussed here is based on a reduction step proposed by Sorenson (k-ary reduction), coupled with the dmod operation similar to Norton's smod. Some practical limitations of Sorenson's reduction have been eliminated. Worst-case complexity is still O(n2) for n-bit…
59 Citations

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## References

SHOWING 1-10 OF 19 REFERENCES
Parallel integer gcd algorithms and their application to polynomial gcd
• Computer Science
• 1994
Two new integer greatest common divisor (GCD) algorithms, the accelerated GCD and the modular GCD, are presented and a new version of the GCDHEU algorithm, GCDCOLL--the collusive polynomial GCD algorithm, that simultaneously performs more than one search for the result is designed and experimented.
Comparing several GCD algorithms
• T. Jebelean
• Computer Science
Proceedings of IEEE 11th Symposium on Computer Arithmetic
• 1993
The execution times of several algorithms for computing the GCD of arbitrary precision integers are compared, and an improved Lehmer algorithm using two digits in partial consequence computation, and a generation of the binary algorithm using a new concept of modular conjugates are introduced.
An Algorithm for Exact Division
An algorithm which computes the quotient of two long integers in this particular situation, starting from the least-significant digits of the operands, which is better suited for systolic parallelization in a "least-significant digit first" pipelined manner.
Two Fast GCD Algorithms
It is shown that sequential versions of both algorithms take Θ( n 2 /log n ) bit operations in the worst case to compute the GCD of two n -bit integers, which compares favorably to the Euclidean and both binary algorithms, which takes Θ ( n 2 ) time.
A Shift-Remainder GCD Algorithm
An integer greatest common divisor algorithm which uses a "shift-divide" instruction to compute the gcd of two integers u, v, and for uniformly distributed integers in the range [0,u-1] , the average run-time is experimentally 0.555 ln u.
A generalization of the binary GCD algorithm
A generalization of the binary algorithm for operation at ‘word level” by using a new concept of ‘modular conjugates” computes the GCD of multiprecision integers two times faster than Lehmer–Euclid
A p-adic algorithm for univariate partial fractions
Partial fractions is an important algebraic operation with many applications in applied mathematics, physics and engineering. It is also an important operation in any computer symbolic and algebraic
The Art of Computer Programming
The arrangement of this invention provides a strong vibration free hold-down mechanism while avoiding a large pressure drop to the flow of coolant fluid.
The Art of Computer Programmmg