I have a point in affine coordinates: $(x,y)$.

What should I do when I want to convert to $(X,Y,Z)$ in Jacobian coordinates? I need it for calculating ECC in a prime field.


2 Answers 2


When you go from Affine to Jacobian, $X$ and $Y$ stay the same, and $Z$ is equal to $1$

Affine -> Jacobian:

$(X',Y',Z') = (X,Y,1)$

Jacobian -> Affine:

$(X',Y') = (\frac{X}{Z^2}, \frac{Y}{Z^3} )$

  • 4
    $\begingroup$ Note that the division in the Jacobian -> Affine transformation is based on the Modular multiplicative inverse. You can't just use the integer division operation offered by common programming languages. $\endgroup$ Oct 13, 2014 at 10:20
  • $\begingroup$ and what if $Z=0$? $\endgroup$ Jun 1, 2021 at 23:48
  • $\begingroup$ Z = 0 means something went wrong in your algorithm. It should never occur if you add or multiply points on the curve. $\endgroup$
    – Nayuki
    Jul 31, 2021 at 2:55

If you have a point $(x,y)$ to get its Jacobian coordinates $(X,Y,Z)$, take a $\lambda \in K^*$ and $(X,Y,Z) = (x\lambda^2, y\lambda^3, \lambda)$. The special/identity point is mapped to $(1,1,0)$.

Where $K$ is the field (of the definition of the Elliptic curve)


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