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[[File:Karatsuba_multiplication.svg|thumb|300px|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]] for [[Integer|integers]]. It was discovered by [[Anatoly Karatsuba]] in 1960 and published in 1962.<ref name="kara1962">
{{cite journal
| author = A. Karatsuba and Yu. Ofman
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: ''z''<sub>1</sub> = ('''12''' + '''345''') '''×''' ('''6''' + '''789''') − ''z''<sub>2</sub> − ''z''<sub>0</sub> = 357 '''×''' 795 − 72 − 272205 = 283815 − 72 − 272205 = 11538
We get the result by just adding these three partial results, shifted accordingly (and then taking carries into account by decomposing these three inputs in base ''1000''
: result = ''z''<sub>2</sub> · (''B''<sup>''m''</sup>)<sup>''2''</sup> + ''z''<sub>1</sub> · (''B''<sup>''m''</sup>)<sup>''1''</sup> + ''z''<sub>0</sub> · (''B''<sup>''m''</sup>)<sup>''0''</sup>, i.e.
: result = 72 · ''1000''<sup>2</sup> + 11538 · ''1000'' + 272205 = '''83810205'''.
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