Scalar Multiplication

Core idea: Scalar multiplication repeats point addition:

$$kP=\underbrace{P+P+\cdots+P}_{k\text{ additions}}.$$

It does not multiply the coordinates: $k(x,y)\ne(kx,ky)$.

Efficient computation

• Build larger multiples from doubling and addition; for example, $$5P=4P+P, \qquad 4P=2P+2P.$$
• Double-and-add reads the binary digits of $k$: double for every bit, then add $P$ when the bit is $1$.
• The work grows roughly as $\log_2 k$, making enormous scalars practical.
• Every intermediate result remains in the same elliptic-curve group.

Next: Replace approximate real-number coordinates with exact modular arithmetic.