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

The square root of 2ⁿ is 2^{n / 2} or 2^{n >> 1}, so if your number is a power of two everything is totally simple once you know the power. To multiply is even simplier: 2⁴ * 2⁸ is 2⁴⁺⁸. There's no sense in this statements you've done.