Why is squaring a number faster than multiplying two random numbers?

Question

Multiplying two binary numbers takes n^2 time, yet squaring a number can be done more efficiently somehow. (with n being the number of bits) How could that be?

Or is i

Answer 1

Squaring an n digit number may be faster than multiplying two random n digit numbers. Googling I found this article. It is about arbitrary precision arithmetic but it may be relevant to what your asking. In it the authors say this:

In squaring a large integer, i.e. X^2 = (xn-1, xn-2, ... , x1, x0)^2 many cross-product terms of the form xi * xj and xj * xi are equivalent. They need to be computed only once and then left shifted in order to be doubled. An n-digit squaring operation is performed using only (n^2 + n)/2 single-precision multiplications.