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2

As already mentioned in a previous comment, ECIES (a hybrid encryption scheme) is typically the way to go when implementing asymmetric encryption on elliptic curves, as it is standardized. It provides chosen ciphertext security (IND-CCA). But as you are looking for "pure" public key encryption schemes, here we go: ElGamal can not only be implemented in ...

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Let's suppose that you work in field $\mathbb{F}_p$ for a prime $p$; i.e. the $x$, $y$, $a$ and $b$ values are integers modulo $p$. The curve order $n$ is the number of points on the curve. By Hasse's theorem we know that $n$ is relatively close to $p$; namely, the theorem states that: $$|n-(p+1)| \leq 2\sqrt{p}$$ So you get a range of possible values for ...

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Actually you have a bug in your step of key generation, it should be $Q_A=d_A \times G$ and you want to have a point $G$ of large prime order $n$ such that the ECDLP is hard on the group. The last check ensures that the public key $Q_A$ is a point of order $n$. If this is the case, then \$Q_A\times n = (d_A \times G)\times n = d_A\times(n\times G)=d_A\times ...

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