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

Suppose you want to expand out the multiplication (a+b)×(c+d). It splits up into four individual multiplications: a×c + a×d + b×c + b×d.

But if you want to expand out (a+b)², then it only needs three multiplications (and a doubling): a² + 2ab + b².

(Note also that two of the multiplications are themselves squares.)

Hopefully this just begins to give an insight into some of the speedups that are possible when performing a square over a regular multiplication.