B.12 Examples for Chapter 21

Example 40

All of the elliptic curves we work with in this chapter are elliptic curves mod n. However, it is helpful use the graphs of elliptic curves with real numbers in order to visualize what is happening with the addition law, for example, even though such pictures do not exist mod n.

Let’s graph the elliptic curve y2=x(x1)(x+1). We’ll specify that 1x3 and 5y5,  and make sure that x and y are cleared of previous values.

> x:=’x’;y:=’y’;implicitplot(yaˆ2=x*(x-1)*(x+1), x=-1..3,y=-5..5)
A graph plots an oval shaped curve at the left along with an elliptic curve pointing towards right.

Example 41

Add the points (1, 3) and (3, 5) on the elliptic curve y2x3+24x+13(mod ...

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