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!