Karatsuba multiplication of az+b and cz+d (boxed), and 1234 and 567 with z=100. Magenta arrows denote multiplication, amber denotes addition, silver denotes subtraction and cyan denotes left shift. (A), (B) and (C) show recursion with z=10 to obtain intermediate values.
The Karatsuba algorithm is a fast multiplication algorithm. It was discovered by Anatoly Karatsuba in 1960 and published in 1962.[1][2][3] It is a divide-and-conquer algorithm that reduces the multiplication of two n-digit numbers to three multiplications of n/2-digit numbers and, by repeating this reduction, to at most single-digit multiplications. It is therefore asymptotically faster than the traditional algorithm, which performs single-digit products.
The Karatsuba algorithm was the first multiplication algorithm asymptotically faster than the quadratic "grade school" algorithm.
The Toom–Cook algorithm (1963) is a faster generalization of Karatsuba's method, and the Schönhage–Strassen algorithm (1971) is even faster, for sufficiently large n.
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A. Karatsuba and Yu. Ofman (1962). "Multiplication of Many-Digital Numbers by Automatic Computers". Proceedings of the USSR Academy of Sciences. 145: 293–294. Translation in the academic journal Physics-Doklady, 7 (1963), pp. 595–596{{cite journal}}: CS1 maint: postscript (link)
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A. A. Karatsuba (1995). "The Complexity of Computations" (PDF). Proceedings of the Steklov Institute of Mathematics. 211: 169–183. Translation from Trudy Mat. Inst. Steklova, 211, 186–202 (1995){{cite journal}}: CS1 maint: postscript (link)
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Knuth D.E. (1969) The Art of Computer Programming. v.2. Addison-Wesley Publ.Co., 724 pp.
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