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Suppose one had a complete list of primes up to $2^{n+1}-1$. Then wouldn't one be able to crack an $n$-bit RSA public key in time $O(\pi(2^{n+1}-1))$, making RSA insecure?



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up vote 5 down vote accepted

Actually, one would be able to crack a $2n$-bit RSA public key in $O(\pi(2^{n+1})-1)$ time. However, $O(\pi(2^{n+1})-1) = O(2^n / n)$, and we already know how to factor $2n$-bit numbers faster than that.

Hence, even if someone could come up with such a list (and find some place to store it), it wouldn't actually affect the security of RSA.

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