FAQ: What is the fastest way to multiply two integers?

Question:
    I would like to know what is currently the fastest algorthim used to multiply two arbitary long integers. I would also like to know if the function is available in C/C++.

Yves Gallot replies:

It depends on the size of the numbers:
  • up to about 100 digits, the grammar-school method is the fastest
  • between 100-1,000 digits, the Karatsuba method is the fastest (a recursive formula that replace 4 multiplications by 3).
  • between 1,000-10,000,000 digits, convolution based on FFT using some floating-point numbers is the fastest
  • for number having more than 10,000,000 digits, multiple Number Theoretic Transforms and Chinese Remainder Theorem should be used, because accuracy of floating-point numbers of available processors is not large enough.
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