It is possible to perform Multiplication of Large Numbers in (many) fewer operations than the usual brute-force technique of ``long multiplication.'' As discovered by Karatsuba and Ofman (1962), Multiplication of two -Digit numbers can be done with a Bit Complexity of less than using identities of the form

(1) |

As a concrete example, consider Multiplication of two numbers each just two ``digits'' long in base ,

(2) | |||

(3) |

then their Product is

(4) |

Instead of evaluating products of individual digits, now write

(5) | |||

(6) | |||

(7) |

The key term is , which can be expanded, regrouped, and written in terms of the as

(8) |

(9) | |||

(10) | |||

(11) |

so the three ``digits'' of have been evaluated using three multiplications rather than four. The technique can be generalized to multidigit numbers, with the trade-off being that more additions and subtractions are required.

Now consider four-``digit'' numbers

(12) |

(13) |

(14) | |||

(15) |

and the Karatsuba algorithm can be applied to and in this form. Therefore, the Karatsuba algorithm is not restricted to multiplying two-digit numbers, but more generally expresses the multiplication of two numbers in terms of multiplications of numbers of half the size. The asymptotic speed the algorithm obtains by recursive application to the smaller required subproducts is (Knuth 1981).

When this technique is recursively applied to multidigit numbers, a point is reached in the recursion when the overhead of additions and subtractions makes it more efficient to use the usual Multiplication algorithm to evaluate the partial products. The most efficient overall method therefore relies on a combination of Karatsuba and conventional multiplication.

**References**

Borodin, A. and Munro, I. *The Computational Complexity of Algebraic and Numeric Problems.* New York:
American Elsevier, 1975.

Borwein, J. M.; Borwein, P. B.; and Bailey, D. H. ``Ramanujan, Modular Equations, and Approximations
to Pi, or How to Compute One Billion Digits of Pi.'' *Amer. Math. Monthly* **96**, 201-219, 1989.

Brigham, E. O. *The Fast Fourier Transform.* Englewood Cliffs, NJ: Prentice-Hall, 1974.

Brigham, E. O. *Fast Fourier Transform and Applications.* Englewood Cliffs, NJ: Prentice-Hall, 1988.

Cook, S. A. *On the Minimum Computation Time of Functions.* Ph.D. Thesis. Cambridge, MA: Harvard University, pp. 51-77, 1966.

Hollerbach, U. ``Fast Multiplication & Division of Very Large Numbers.'' `sci.math.research` posting, Jan. 23, 1996.

Karatsuba, A. and Ofman, Yu. ``Multiplication of Many-Digital Numbers by Automatic Computers.''
*Doklady Akad. Nauk SSSR* **145**, 293-294, 1962. Translation in *Physics-Doklady* **7**, 595-596, 1963.

Knuth, D. E. *The Art of Computing, Vol. 2: Seminumerical Algorithms, 2nd ed.*
Reading, MA: Addison-Wesley, pp. 278-286, 1981.

Schönhage, A. and Strassen, V. ``Schnelle Multiplikation Grosser Zahlen.'' *Computing* **7**, 281-292, 1971.

Toom, A. L. ``The Complexity of a Scheme of Functional Elements Simulating the Multiplication of Integers.''
*Dokl. Akad. Nauk SSSR* **150**, 496-498, 1963. English translation in *Soviet Mathematics* **3**, 714-716, 1963.

Zuras, D. ``More on Squaring and Multiplying Large Integers.'' *IEEE Trans. Comput.* **43**, 899-908, 1994.

© 1996-9

1999-05-26