# Multiply point by scalar in elliptic curve group

I'm trying to understand how to multiply a point by a scalar to get a point in elliptic curve cryptography. Here's an example from my textbook.

The group is E257(0, -4). That's shorthand for y2 = x3 + 0 x - 4 (mod 257)

According to the book,

101(2, 2) = (197, 167)


How can I get from 101(2, 2) to (197, 167)? How can I get from cP to Q where c is a constant scalar and P and Q are points?

What's multiplication? It's simply repeated addition. Instead of thinking of the problem as $101 \cdot (2, \, 2)$, where $101$ is meaningless within the context of the elliptic curve, think of the operation as $(2, \, 2) + (2, \, 2) + \underbrace{\ldots}_{98} + (2, \, 2)$.
• If this notation seems unusual, just recall that when our group operation is $\cdot$, we often use exponention by a scalar: $g^n = g \cdot g \cdot \underbrace{\cdots}_{n-3} \cdot g$. So for groups where $+$ is the operation, we use multiplication by a scalar instead. – Henry Swanson Nov 3 '14 at 19:57