Question

I’m using .NET 4’s System.Numerics.BigInteger structure.

I need to calculate the square (x²) of very large numbers – millions of decimal digits.

If x is a BigInteger, What is the time complexity of:

x*x;

or

BigInteger.Pow(x,2);

?

How can multiply such big numbers in the fastest way using .NET 4 BigInteger? Is there an implementation for Schönhage–Strassen algorithm?

Answer 1

That depends on how large your numbers are. As the Wikipedia page tells you:

In practice the Schönhage–Strassen algorithm starts to outperform older methods such as Karatsuba and Toom–Cook multiplication for numbers beyond 2²¹⁵ to 2²¹⁷ (10,000 to 40,000 decimal digits).

System.Numerics.BigInteger uses the Karatsuba algorithm or standard schoolbook multiplication, depending on the size of the numbers. Karatsuba has a time complexity of O(n ^log₂ 3). But if your numbers are smaller than above quoted figures, then you likely won’t see much speedup from implementing Schönhage–Strassen.

As for Pow() this itself squares the number at least once during its calculations (and it does that by simply doing num * num – so I think this won’t be more efficient, either.