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

I believe you may be referring to exponentiation by squaring . This technique isn't used for multiplying, but for raising to a power x^n, where n may be large. Rather than multiply x times itself N times, one performs a series of squaring and adding operations which can be mapped to the binary representation of N. The number of multiplication operations (which are more expensive than additions for large numbers) is reduced from N to log(N) with respect to the naive exponentiation algorithm.