Abstract
The binary algorithm is a variant of the Euclidean algorithm that performs well in practice. We present a quasi-linear time recursive algorithm that computes the greatest common divisor of two integers by simulating a slightly modified version of the binary algorithm. The structure of our algorithm is very close to the one of the well-known Knuth-Schönhage fast gcd algorithm; although it does not improve on its O(M(n) log n) complexity, the description and the proof of correctness are significantly simpler in our case. This leads to a simplification of the implementation and to better running times.
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References
Brent, R.P.: Twenty years’ analysis of the binary Euclidean algorithm. In: Roscoe, A.W., Davies, J., Woodcock, J. (eds.) Millenial Perspectives in Computer Science: Proceedings of the 1999 Oxford-Microsoft Symposium in honour of Professor Sir Antony Hoare, Palgrave, New York, pp. 41–53 (2000)
Brent, R.P., Kung, H.T.: A systolic VLSI array for integer GCD computation. In: Hwang, K. (ed.) Proceedings of the 7th Symposium on Computer Arithmetic (ARITH-7), IEEE CS Press, Los Alamitos (1985)
Granlund, T.: GNU MP: The GNU Multiple Precision Arithmetic Library, 4.1.2 edn. (2002), http://www.swox.se/gmp/#DOC
Knuth, D.: The analysis of algorithms. Actes du Congrès International des Mathématiciens de 3, 269–274 (1970); Gauthiers-Villars, Paris ( 1971)
Montgomery, P.L.: Modular multiplication without trial division. Math. Comp. 44(170), 519–521 (1985)
Schönhage, A.: Schnelle Berechnung von Kettenbruchentwicklungen. Acta Informatica 1, 139–144 (1971)
Schönhage, A., Strassen, V.: Schnelle Multiplikation grosser Zahlen. Computing 7, 281–292 (1971)
Shand, M., Vuillemin, J.E.: Fast implementations of RSA cryptography. In: Swartzlander, E.E., Irwin, M.J., Jullien, J. (eds.) Proceedings of the 11th IEEE Symposium on Computer Arithmetic (ARITH-11), pp. 252–259. IEEE Computer Society Press, Los Alamitos (1993)
Vallée, B.: Gauss’ algorithm revisited. Journal of Algorithms 12, 556–572 (1991)
Vallée, B.: Dynamical analysis of a class of Euclidean algorithms. Th. Computer Science 297(1-3), 447–486 (2003)
von zur Gathen, J., Gerhard, J.: Modern Computer Algebra, 2nd edn. Cambridge University Press, Cambridge (2003)
Yap, C.K.: Fundamental Problems in Algorithmic Algebra. Oxford University Press, Oxford (2000)
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Stehlé, D., Zimmermann, P. (2004). A Binary Recursive Gcd Algorithm. In: Buell, D. (eds) Algorithmic Number Theory. ANTS 2004. Lecture Notes in Computer Science, vol 3076. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-24847-7_31
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DOI: https://doi.org/10.1007/978-3-540-24847-7_31
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-22156-2
Online ISBN: 978-3-540-24847-7
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