Is there a 3-D alternative for elliptic curves? 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?
So, i've just started to figure out how encryption works. 


From what you've said, I fear that's not true. 


  As i understand the most widely used method to exchange encrypted data over untrusted network is Diffie-Hellman protocol. 


Partially true, as DH is used as key agreement protocol to further derive session encryption &amp; MAC key. DH is asymmetric, where as encryption and MAC (message authentication code) are symmetric. 


  So, original idea was to use big prime numbers (1 number, 1-D) ... After a while elliptic curves (x and y coordinates, 2-D) was discovered, so now everyone using them. 


The original realization of the idea was to use a big prime number, but those numbers that can be represented in binary system have a exploitable structure make possible attacks faster than brutal-force search. 

DH is still realized using scalars, except this time, the scalars are points on elliptic curves over finite fields. 


  Question is - do elliptic curves have more cryptographic strength than classical approach because of 2 dimensions?


No, they are stronger because they don't have the same exploitable structure as (big) integers. 


  Is there researches related to 3-rd (or even more) order polynomes that potentially can provide even more cryptographic strength than elliptic curves?


Yes, one of my questions here attracted this comment, but I haven't thoroughly comprehended it, I advise you read it with care if you have time.