8.4 Security of elliptic curves
The security of cryptographic mechanisms based on elliptic curves relies on the following assumptions:
The elliptic curve discrete logarithm problem is computationally hard for the chosen elliptic curve
The chosen elliptic curve has no mathematical backdoors
Alice’s and Bob’s public keys are authentic
In the case of ECDH, the Diffie-Hellman problem is computationally hard in the cyclic group 𝔾 generated by the base point G on the elliptic curve
In Chapter 7, Public-Key Cryptography, we already discussed – both for Diffie-Hellman key exchange as well as the RSA algorithm – why it is important for Alice’s and Bob’s public keys to be authentic. We also explained why the Diffie-Hellman problem must be computationally hard for the key agreement to be secure.
In this chapter, we will discuss how to fulfill the first two assumptions. We’ll start by looking at the available algorithms for determining...
