Following the response to my previous question, I would like to know if you could give me some information or give me a link on how to perform arithmetic operations once I changed a point from the field $\mathbb{F}_p$ to $\mathbb{F}_{p^2}$ ?
If I have an elliptic curve $Y^2 = X^3 + 1$ over $\mathbb{F}_p$ and I want to make a distortion map $\phi(x,y) \rightarrow (\beta x,y)$ to have a point of the form $\{a+bi : a,b \in \mathbb{F_p}\}$ in $\mathbb{F}_{p^2}$.
How can I compute the addition of two points in $\mathbb{F}_{p^2}$ for example since the field arithmetic is not the same ?
Thank you for your answers
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$\begingroup$ Instead of significantly modifying a question, especially after it has received answers, just ask a new question. It's okay, there's nothing wrong with asking many questions. $\endgroup$– fkraiemCommented Jul 31, 2018 at 9:02
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$\begingroup$ And if an answer satisfactorily answers the question as originally asked, please accept it. Questions with no accepted answer are periodically bumped to the front page, which is annoying if it happens too often. $\endgroup$– fkraiemCommented Jul 31, 2018 at 9:20
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As you note, the elements of $\mathbf F_{p^2}$ can be represented as $a+bi$ with $a,b \in \mathbf F_p$ and $i^2 = -1$; however note that this only works if $p\equiv 3 \pmod 4$.
Then you can add and multiply two elements of $\mathbf F_{p^2}$ in basically the same way you add and multiply complex numbers. Namely, $(a+bi)+(c+di) = (a+c)+(b+d)i$ and $(a+bi)(c+di) = (ac-bd)+(ad+bc)i$.
And the formula for point addition is the same, but you need to keep in mind that all your "numbers" are elements of $\mathbf F_{p^2}$ and must be handled as above.
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$\begingroup$ Thank you, I edited my question to show my computations. Is it correct ? I speak french if it helps to communicate more easily. $\endgroup$ Commented Jul 31, 2018 at 9:01