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\begin{itemize}  \item \textbf{Existence of the Unit}  $$\forall P \in \mathcal{E} \ \ \exists \mathcal{O} \ s.\ t. \ \ \ P \oplus \mathcal{O} = P $$  We discussed before about the property of the point $\mathcal{O}$  \item \textbf{Existence of the Inverse}  $$\forall P \in \mathcal{E} \ \ \exists \mathcal{\ominus P} \ s.\ t. \ \ \ P \oplus (\ominus P) = \mathcal{O} $$  We could consider as $\ominus P$ the symmetric point respects to the x-axis  \item \textbf{Commutativity}  $$\forall P,Q \in \mathcal{E} \ \ \ P \oplus Q=Q \oplus P $$  This is obvious.   \end{itemize}\ \\ \ \\  From now we could simply call $\oplus = + $ and $\ominus = - $. \\  We could define the quantity  $$nP=P+P+ \ldots + P \ \ \ \ \ n \ times $$   \ \\  \ \\  To define completely an elliptic curve, we have to add the following condition:  $$\mathcal{E} : \ Y^{2} = X^{3} +AX+B$$  represents an Elliprici Curve iff   $$4A^{3}+27B^{2} \neq 0 $$  This condition tells us that $\\vartriangle_{\mathcal{E}} \neq 0$ and the curve doesn't have singular points (the curve passes at least two times by these points); if the curve has singular points the addition law defined on the curve doesn't work.  Now we present in practice the \textbf{Elliptic Curve Addition Algorithm}  \begin{ppt}[\textbf{Elliptic Curve Addition Algorithm}]  Let $$\mathcal{E} : \ Y^{2} = X^{3} +AX+B$$ be an elliptic curve and $P_{1}(x_{1},y_{1}),P_{2}(x_{2},y_{2}) \in \mathcal{E} $ and say $P_{1} + P_{2}=(x_{3},y_{3})$. \\  Define $\lambda$ by  $$\lambda = \begin{cases}   \dfrac{y_{2}-y_{1}}{x_{2}-x_{1}} \ \ \ if P_{1} \neq P_{2} \\  \dfrac{3x_{1}^{2}+A}{2y_{1}} \ \ \ \ if P_{1}=P_{2}  \end{cases}$$  $$THEN$$  $$x_{3}=\lambda^{2}-x_{1}-x_{2} \ \ \ \ \ and \ \ \ \ y_{3}= \lambda (x_{1}-x_{3})-y_{1}.$$  \end{ppt}   \ \\