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The RSA cryptosystem makes use of $n=pq$ where $p, q$ are large prime numbers.

With quantum computing, factorization might become easier, so it will probably be useful to use much much bigger $p$, $q$ in the future.

If I remember correctly, the source of large prime numbers $p$, $q$ in today's most commons RSA implementations is: probabilistic primalty tests.

In a concrete manner, what could be a possible source of much much larger primer numbers (several order of magnitudes bigger than the $p$, $q$ used today)?

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    $\begingroup$ RSA has died years ago. Long live ECC. Several magnitude order, no one is going to use that power consumer and slow operation. $\endgroup$
    – kelalaka
    Oct 5, 2020 at 10:26
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    $\begingroup$ You mean pqRSA? $\endgroup$
    – DannyNiu
    Oct 5, 2020 at 10:47
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    $\begingroup$ the numbers we use right now are chosen at random and then incremented till prime, I see no reason why we cannot still do that, it will just be slower $\endgroup$ Oct 5, 2020 at 10:51
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    $\begingroup$ however, the actual use of such large primes would be... undesirable from a performance standpoint $\endgroup$ Oct 5, 2020 at 10:52
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    $\begingroup$ @kelalaka: if the threat we're worried about are Quantum Computers, ECC (except for isogeny) isn't the answer - it's just as vulnerable... $\endgroup$
    – poncho
    Oct 5, 2020 at 13:42

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