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Topic: Karatsuba multiplication

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	MathTrek: Divide-and-Conquer Multiplication
		One particularly intriguing and efficient multiplication algorithm was developed in the late 1950s by Anatolii Alexeevich Karatsuba, now at the Steklov Institute of Mathematics in Moscow.
		Karatsuba's "divide-and-conquer" multiplication algorithm has its roots in a method that Carl Friedrich Gauss (1777-1855) introduced involving the multiplication of complex numbers.
		Karatsuba's insight was to apply Gauss optimization to this divide-conquer-and-glue approach, replacing some multiplications with extra additions.
	blog.sciencenews.org /mathtrek/2007/02/divideandconquer_multiplicatio.html (983 words)

	math lessons - Multiplication algorithm
		An old method for multiplication, that does not require multiplication tables, is the Peasant multiplication algorithm; this is actually a method of multiplication using base 2.
		Karatsuba is a special case of Toom-Cook using two parts.
		multiplication for the cost of five size-N multiplications, improvement by a factor of 9/5 compared to the Karatsuba method's improvement by a factor of 4/3.
	www.mathdaily.com /lessons/Multiplication_algorithm (917 words)

	GNU MP 4.1.2
		The Karatsuba multiplication algorithm is described in Knuth section 4.3.3 part A, and various other textbooks.
		The inputs x and y are treated as each split into two parts of equal length (or the most significant part one limb shorter if N is odd).
		Karatsuba multiplication is asymptotically an O(N^1.585) algorithm, the exponent being log(3)/log(2), representing 3 multiplies each 1/2 the size of the inputs.
	www.delorie.com /gnu/docs/gmp/gmp_86.html (430 words)

	Karatsuba multiplication (Site not responding. Last check: 2007-10-24)
		Karatsuba multiplication is a technique for multiplying integers together which is very different from the traditional grade-school technique.
		Karatsuba multiplication is often taught in computer science classes and analyzed theoretically.
		I have written a relatively efficient implementation of both grade-school and Karatsuba multiplication using C++.
	ozark.hendrix.edu /~burch/proj/karat (78 words)

	Karatsuba Multiplication - GNU MP 4.2.1
		The basecase algorithm will take a time of the form M(N) = aN^2 + bN + c and the Karatsuba algorithm K(N) = 3M(N/2) + dN + e, which expands to K(N) = 3/4aN^2 + 3/2bN + 3c + dN + e.
		The factor 3/4 for a means per-crossproduct speedups in the basecase code will increase the threshold since they benefit M(N) more than K(N).
		Of course all speedups reduce total time, and in that sense the algorithm thresholds are merely of academic interest.
	www.swox.com /gmp/manual/Karatsuba-Multiplication.html (402 words)

	Za-Zm
		The ZEN library is written to perform efficient arithmetic operations for general polynomial extensions of finite rings.
		Zephyr meets the high-throughput, high fan-out communications requirements of large-scale workstation environments and is designed as a suite of layered services based on a reliable, authenticated notice protocol, with the authentication usually provided by Kerberos.
		Multiple, redundant Zephyr servers provide basic routing, queueing, and dispatching services to clients which communicate via the Zephyr Client Library, and more advanced services are built upon this base.
	stommel.tamu.edu /~baum/linuxlist/linuxlist/node57.html (1889 words)

	Publications of the SPACES team
		Multiplication by an integer constant: Lower bounds on the code length.
		The multiplication by a constant problem consists in generating code to perform a multiplication by an integer constant, using elementary operations, such as left shifts (multiplications by powers of two), additions and subtractions.
		An algorithm is given and proved which reduces the computations of these zeros to the resolution of a single univariate equation whose degree is the number of common zeros.
	www-calfor.lip6.fr /~safey/Spaces/publications.html (13078 words)

	Karatsuba multiplication (Site not responding. Last check: 2007-10-24)
		I thought it would be fun to program up Karatsuba's multiplication to see how it really performs compared to the grade-school method.
		Sometimes these techniques to optimize big-O bounds don't work well for reasonably sized problems; the best way to check is to run some experiments.
		What we see is that Karatsuba, properly implemented, beats grade-school multiplication even for 16-digit numbers.
	www.cs.cmu.edu /~cburch/251/karat/index.html (178 words)

	ConcurrentContext (Javolution v5.0 API)
		This class represents a concurrent context; it is used to accelerate execution of concurrent algorithms on multi-processors systems.
		When a thread enters a concurrent context, it may execute multiple concurrent
		Here is an example of concurrent/recursive implementation of the Karatsuba multiplication for large integers:
	javolution.org /api/javolution/context/ConcurrentContext.html (865 words)

	Publikationen
		Weimerskirch, C. Paar, "Generalizations of the Karatsuba Algorithm for Efficient Implementations", Ruhr-Universität-Bochum, Germany, 2003.
		Amanor, V. Bunimov, C. Paar, J. Pelzl, M. Schimmler, "Efficient Hardware Architectures for Modular Multiplication on FPGAs", International Conference on Field Programmable Logic, Reconfigurable Computing, and Applications, Tampere, Finland, August 24-28, 2005.
		T. Blum, C. Paar "Montgomery Modular Multiplication on Reconfigurable Hardware", 14th IEEE Symposium on Computer Arithmetic (ARITH-14), Adelaide, Australia, April 14-16, 1999.
	www.crypto.rub.de /publications.html (3235 words)

	Math Forum Discussions
		Replies: 1 Last Post: Mar 8, 2004 7:35 AM
		> I have created a preliminary bottom-up Karatsuba multiplication algorithm.
		> rearranging the base case multiplications in such a way that factors are
	www.mathforum.org /kb/thread.jspa?threadID=107953&messageID=556539 (170 words)

	Properties of 17 (seventeen)
		Each time these points reach a multiple of 17, the reader receives a pin.
		For my training period of the D.E.A., I have to implement some algorithms of multiple precision multiplication.
		With my program, the classical algorithm in n² is better than Karatsuba's algorithm up to a size of 17 words, where a word is a 50-bit integer.
	www.vinc17.org /d17_eng.html (9811 words)

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