What are elliptic curves? – Elliptic Curves
8.1 What are elliptic curves? Historically, elliptic curves are rooted in so-called Diophantine equations, named after ancient Greek mathematician Diophantus of Alexandria. Diophantine equations are […]
Smoothness – Elliptic Curves
8.1.2 Smoothness Before we can start defining a group operation on the points of an elliptic curve, we need to add one requirement: The curve […]
Projective coordinates – Elliptic Curves
8.1.3 Projective coordinates In order to get to grips with the mysterious point at infinity O, we need to extend the set of points that […]
Elliptic curves as abelian groups – Elliptic Curves
8.2 Elliptic curves as abelian groups Two points P and Q on a smooth elliptic curve E in reduced Weierstrass form can be added to […]
Explicit formulae – Elliptic Curves
8.2.2 Explicit formulae The geometrical viewpoint is nice to get an intuitive picture of point addition, but in order to actually implement things on a […]
Elliptic curves over finite fields – Elliptic Curves
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 […]
Elliptic curves over 𝔽2k – Elliptic Curves
8.3.2 Elliptic curves over 𝔽2k Although not strictly necessary in the context of TLS, for the sake of completeness, we’ll also briefly discuss here how […]
Discrete logarithms and Diffie-Hellman key exchange protocol – Elliptic Curves
8.3.3 Discrete logarithms and Diffie-Hellman key exchange protocol In Section 7.2, Groups, we defined the discrete logarithm problem for generic groups 𝔾. We repeat the […]
Security of elliptic curves – Elliptic Curves
8.4 Security of elliptic curves The security of cryptographic mechanisms based on elliptic curves relies on the following assumptions: In Chapter 7, Public-Key Cryptography, we […]