How does a public/private key encrypt and decrypt each other?

Question

I understand the logic that when a communication takes place both parties pass their public keys to each other which is used to encrypt all messages. Once the party receives the messages the private key is used to decrypt them however I'm wondering how a private key is generated from a public key. If the private key is based on an algorithm wouldn't each party be able to generate what the other person's private key would be based on the public? Wouldn't a third party?

Answer 1

How the public and private keys are generated depends on what public key cryptosystem is being used, but in general the private key cannot be derived from the public with a computationally feasable algorithm, while the public key can be derived from the private key very quickly. Two examples:
RSA: private keys are 2 primes, p and q, and an encryption exponent d. Public key is the product p*q, and an encryption exponent e. How does the attacker get p and q, or d, from n and e? The best attack known against this (for properly chosen p, q, and d) is factoring. Factoring can be made infeasable by choosing the primes to be large enough.
Diffie-Hellman: a prime modulus p and a base g < p is known by everyone (including the attacker). The private key is an integer x chosen randomly, 2 <= x < p-1 (there are better ways to choose x). The public key is g^x mod p. How does the attacker get x from g^x mod p? Again, the best known attack is one that is computationally roughly equal to factoring a composite number of about the size of p.