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According to wiki, regarding elliptic curve points addition, the following formula for the slope holds

$$s = \frac{y_Q-y_P}{x_Q-x_P}$$ where $P(x_P,y_P)$ and $Q(x_Q,y_Q)$ are distinct points on the curve $y^2 = x^3 + ax + b \pmod q$, $q$ odd prime.

I do know that $$s = \frac{y_Q-y_P}{x_Q-x_P} = (y_Q-y_P)\cdot(x_Q-x_P)^{-1} \pmod q$$

At my university, my teacher said that we can avoid the computation of the multiplicative inverse $(x_Q-x_P)^{-1}$ using projective coordinates (i.e. avoiding using the Extended Euclidean Algorithm). For example each affine point $M(x_M,y_M)$ in the projective coordinates has the form $M(x_M,y_M,1)$.

My question : Could you provide me an example of how to do this?

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penguina
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    The use of projective coordinates in elliptic curve crypto primitives does offer some computational savings. The idea is to take advantages of the fact that in projective coordinates (aka homogeneous coordinates) the points $[x:y:1]$ and $[\lambda x:\lambda y:\lambda]$ are the same point. The savings come from writing down a point addition formula in projective coordinates and using that throughout a double-and-add algorithm when calculating $[n]P$ for a point $P\in E$ and an integer $n$. Only at the end you may choose to convert back to affine coordinates, getting away with a single division. – Jyrki Lahtonen May 24 '17 at 09:44
  • "According to Wiki"? That's the slope's formula we know from junior high school. Of course, it must be $;x_Q\neq x_P;$ . – DonAntonio May 24 '17 at 09:45
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    See here for a bit more. IIRC there are on-line resources doing cost analysis in both affine and projective coordinates. May be you can find a link to such material? Here are the formulas from our sister site specializing in crypto. – Jyrki Lahtonen May 24 '17 at 09:48

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