Crypto Algorithm Question - Are prime numbers still important to cryptography?

Hi,

I understand RSA and some older public key algorithms quite well but have fallen a little out of touch with new advances. I have a few high level questions and and wonder would appreciate hearing from anyone who has an interest in primes or knows the new algorithms intimately.

(1) Are prime numbers and pseudo primes central to the strength of popular modern ciphers like AES?

(2) There are a few internet communities constantly finding new large prime numbers. I understand these projects probably advance various computer science fields in some small way - but do the actual discoveries of new large primes have any bearing on crypto security? (e.g. are there code-breakers adding the new primes to their lists?!).

(3) If some government happened find significant factorisation shortcuts and could built a very large list of consecutive prime numbers (well exceeding the large known mersenne primes) - what could they do with this, in context with the most commonly used cryptographic systems used today (SSL, AES, WPA2 etc)?

Thanks for reading!

1) Prime numbers still play a crucial role in asymmetric ciphers like RSA, but as far as I know they are not very significant to symmetric ciphers like AES.

2) Finding large new primes wouldn't really have any bearing on the security of these ciphers.

3) New, fast factorization techniques would essentially completely destroy the security of many ciphers. Just having a large list of primes is not a major security concern.