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In the context of RSA, when we say an "N bit prime", we mean that it's a prime in the range $[2^{n-1}, 2^n)$. In addition, when we say an RSA key is an "N bit key", we mean that it's in the range $[2^{n-1}, 2^n)$. What this means that if you pick two random $N/2$ bit primes, and multiply them together, you'll get an $N-1$ bit modulus about half the time. ...

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This prime $p$ corresponds to the size on the base field $\mathbb F_p$ on which is defined the elliptic curve. It has not been computed, it has been chosen. It is chosen to satisfy a few properties, like, of course, being prime, but also being large enough (here, 256 bits), and allowing fast computations modulo $p$. Maybe when you say "prime elliptic curve", ...

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In your question, you already pointed out, that the necessary condition is More generally largest prime minus one does not consists of smallest prime as a prime factor therefore, it is sufficient to just check if $p$ divides $q-1$ by computing division. You can just verify this condition during the key generation. I don't know of a more efficient ...

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