How is scalar multiplication performed on a point in Elliptic Curve Cryptography

Question

can someone please explain how the multiplication(386(0,376)) is performed in the given example in the image. ECC example

Answer 1

You have to know the doubling formula for a point on the curve, here for $(0,376)$, and also the addition formula (usually these have different forms). See page 21 of this introduction or any good book on ellitic curves over finite fields.

To compute $386\cdot(0,376)$ we do a repeated doubling and addition.

$386 = 2 + 256 + 128$ so we keep doubling $(0,376)$ $8$ times and get intermediate results $2\cdot(0,376)$, $128\cdot(0,376)$ and $256\cdot(0,376)$. Adding these 3 points will give the desired $386 \cdot(0,376)$; see also double-and-add on Wikipedia.

The formulae in the quoted document: the curve is of the form $y=x^3+ax+b$ modulo $p$ and then $2\cdot(x_1, y_1)$ has $x$-coordinate $x_3 = \lambda^2 - 2x_1 \pmod{p}$ and $y$-coordinate $y_3 = \lambda(x_1-x_3)-y_1$, where $\lambda = \frac{3x_1^2 + a}{2y_1}$; all divisions to be replaced by multiplications with inverse elements modulo $p$ of course. Do the computations by hand or with help of a program like Sage..