I have gone through one example where i saw a curve defined over some prime number containing non-prime order.
Do the elliptic curves over prime fields must always contain prime number of elements (prime order)?
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$\begingroup$ You've just seen an example of a curve over a prime field which has non-prime order, and your question is whether they exist? How could you have found one if they don't? $\endgroup$ – CurveEnthusiast Jan 25 '17 at 8:01
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$\begingroup$ Don't you think you just answered your own question? $\endgroup$ – fkraiem Jan 25 '17 at 11:00
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$\begingroup$ Montgomery and Edwards curves can't have a prime order. $\endgroup$ – CodesInChaos Jan 25 '17 at 13:51
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In general, an elliptic curve over $\mathbb{F}_p$ can have a nonprime order. Much more is known.
As mentioned in Stinson's book Cryptography:Theory and Practice, 2nd Edition, Theorem 6.1, the additive group of an Elliptic curve $E$ over $\mathbb{F}_p$ with $p>3,$ and prime, is isomorphic to $Z_{n_1}\times Z_{n_2}$ with $$n_2|n_1,\quad n_2|(p-1).$$
