In many articles i have found directly the calculation of prime elliptic curve. How can i calculate this prime $p$ ? For example if I consider NIST P-256, $ p = 2^{256}-2^{224}+2^{192}+2^{96}-1$. Why ? Is there a formula ? Thanks for your support.
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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", you mean elliptic curves with a prime number of points? The number of points is not the prime $p$, and is computed using the Schoof-Elkies-Atkin algorithm.
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$\begingroup$ I understand that $p$ is the size of $\mathbb{F}_{p}$. Ok but why $ p = 2^{256}-2^{224}+2^{192}+2^{96}-1$ and not for example only $p=2^{256}$ ? Why are there $2^{224}+2^{192}+2^{96}$ ? (I don't understand the last calculation. Why $-1$ and not $+1$ ? $\endgroup$ – Ellipticat Sep 20 '14 at 9:43
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$\begingroup$ Because $p$ has to be a prime number. $2^{256}$ is definitely not a prime number (even though there is a field of order $2^{256}$, we don't want to use it). Same answer for +1, if you replace -1 with +1, the result is not a prime number. $\endgroup$ – Calodeon Sep 20 '14 at 10:33
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$\begingroup$ As they said $p$ is chosen to fill some conditions: to be prime and to allow fast computation modulo $p$. A good explaination could be found in this post: imperialviolet.org/2010/12/04/ecc.html $\endgroup$ – ddddavidee Sep 20 '14 at 14:37
