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Answering on the subquestion: Why $|E_d|+|E'_d|=2\cdot p+2$ ? It follows from the definition of quadratic twist. In fact, let's consider all possible $\tilde{x}$ coordinates for points, that is all the values in $\mathbb{F_p}$, and an elliptic curve $E$ with equation $y^2=x^3+ax+b$, then: Case $\tilde{x}^3+a\tilde{x}+b\neq0$: So either $\tilde{x}^3+a\tilde{...


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Similarly, just send the identity element $\mathcal{O}$ as the public key. Alice must be a fool to not validate the public key to fall into this trap. The first rule is to check that $P \neq \mathcal{O}$ Update per user's comment How can i find the identity element of curve 25519? The identity of some curves like Curve25519 is the point of infinity ( ...


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