8.3 Elliptic curves over finite fields

Now let’s see what elliptic curves over finite fields look like. As we established in the last chapter, there are only two kinds of finite fields: 𝔽_p = {0,1,2,…,p − 1}, where p is a prime number, and 𝔽_p[X]∕M, where p is a prime number and M is an irreducible polynomial of degree n with coeffcients a_i ∈𝔽_p. The essential difference between the two is that 𝔽_p has p elements, whereas 𝔽_p[X]∕M has pⁿ elements. For this reason, 𝔽_p[X]∕M is often called 𝔽_pⁿ without explicitly stating the polynomial M.

8.3.1 Elliptic curves over 𝔽_p

We focus on the case p > 3, so that char(𝔽_p) > 3. Then it is always possible to generate the reduced Weierstrass form of the curve, and we can use the following definition.

Elliptic curve over 𝔽_p

Let p > 3 be a prime number. An elliptic curve over 𝔽_p is the set of points (x,y) satisfying...