
TLS Cryptography In-Depth
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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: 𝔽p = {0,1,2,…,p − 1}, where p is a prime number, and 𝔽p[X]∕M, where p is a prime number and M is an irreducible polynomial of degree n with coeffcients ai ∈𝔽p. The essential difference between the two is that 𝔽p has p elements, whereas 𝔽p[X]∕M has pn elements. For this reason, 𝔽p[X]∕M is often called 𝔽pn without explicitly stating the polynomial M.
We focus on the case p > 3, so that char(𝔽p) > 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 𝔽p
Let p > 3 be a prime number. An elliptic curve over 𝔽p is the set of points (x,y) satisfying...
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