Re: Not that many primes
Not really a problem. We know roughly how many primes there would be to deal with :
For N=2^1024, only one number in ln(N) = 1420 (that's a natural base-e log) will be prime. Call it about N=2^1013 = 10^304 (roughly) primes. Store those on, say, ten-terabyte (10^16 byte) drives, with each prime consuming 1024/8 = 128 bytes, and you're looking at 10^(304-16)/10^2 = about 10^286 hard drives.
If each hard drive weighs a kilogram (being optimistic here), we can use the fact that the sun masses about 2x10^30 kilograms to determine that we need 5x10^255 solar masses worth of hard drives. We'd have to turn a lot of observable universes worth of matter into hard drives to get this to work.
On the other hand, we could rely on Moore's Law. If those hard drives double in capacity every 18 months, then in about 1400 years, you'll be able to fit all those primes on a single hard drive (or whatever equivalent storage medium our descendants are using at that point.) At that point, they'll have to move to 2048-bit primes, which will get them about another 1400 years.
(Which is not to say prime-number cryptography should be assumed to be "ultimately secure". I think it has a better theoretical justification than anything else out there, unless you're counting one-time pads and similar exotica.)