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Have a look to Shanks-Tonelli algorithms about modular square root. Here is one link ( http://en.wikipedia.org/wiki/Tonelli%E2%80%93Shanks_algorithm ) On binary curves y^2 + (x)y - (x^3 + bx^2 + b) = 0, you can rewrite it as y^2 + Ay + B = 0 you need to solve a quadratic equation in F(2^m) ( ...


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Let $x\in\mathbb Z/p\mathbb Z$ be the point's first coordinate, and define $z := x^3+ax+b$. We know that there exists a square root $y\in\mathbb Z/p\mathbb Z$ of $z$, i.e. $y^2=z$. Let's assume we have already found such an $y$. Since the order of $(\mathbb Z/p\mathbb Z)^\ast$ is $p-1$, Lagrange's theorem implies $y^p=y\text,$ hence ...



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