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To answer your question about the size of primes: The current recommendation for RSA keys recommends using two 1024-bit keys. That means they're somewhat larger than $10^{300}$; Diffie-Hellman on the integers modulo $p$ is normally used with a single prime of at least 2048 bits (so, larger than $10^{600}$!) Even worse, these numbers tend to grow over time; ...

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This question is really broad. I'll try to answer in a few sentences. Of course, $\mathbb Z$ in its widely accepted definition has infinitely many primes. This means: the properties people usually expect from something we may rightly call "the integers" already imply that this thing contains infinitely many primes. Hence it is impossible to keep everything ...

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I think the confusion lies in mixing two distinct hard problems. RSA is related to the difficulty of factoring large numbers. The scheme you link to in the paper is based on the difficulty of solving discrete logarithms. Since they are very different hard problems, you can't really compare the two in terms of what must be kept secret. For RSA, ...

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