As I understand the most widely used method to exchange encrypted data over untrusted network is Diffie-Hellman protocol. Protocol itself doesn't define how to encrypt data, but it defines a usage pattern of public/private keys and implementation can vary.
So, original idea was to use big prime numbers (1 number, 1-D) and their combinations as public/private keys. After a while elliptic curves (x and y coordinates, 2-D) was discovered, so now everyone using them.
Do elliptic curves have more cryptographic strength than classical approach because of 2 dimensions? Is there any research related to 3rd (or even more) order polynomial that potentially can provide even more cryptographic strength than elliptic curves?