# How to convert projective to jacobian co-ordinate in ECC?

I am doing a small project using elliptic curve in cryptography. My doubt is, can I directly convert a projective to a Jacobian coordinate system without using the affine conversion in elliptic curve cryptography? Actually, I am stuck at this point…

## 1 Answer

If the question is how to take a point in projective coordinates (a triplet $(A,B,C)$, where the corresponding affine coordinate is $(AC^{-1}, BC^{-1})$), and you want to convert it into the same point in Jacobian coordinates $(D, E, F)$ (where the affine coordinate is $(DF^{-2}, EF^{-3})$), I believe this works:

$$D = AC$$

$$E = BC^2$$

$$F = C$$

We can easily see that $DF^{-2} = AC^{-1}$ and $EF^{-3} = BC^{-1}$, hence $(D,E,F)$ refers to the same point as the original, and this conversion takes only three multiplies.

• Thank you @poncho. To reverse the operation will the below equations work? $$A = DF$$ $$B = E$$ $$C = F^{3}$$ – vijita Jun 30 '14 at 8:39
• @vijita: that reversed operation also preserves $DF^{-2}=AC^{-1}$ and $EF^{-3} = BC^{-1}$, and so, yes, those equations also "works". – poncho Jun 30 '14 at 12:42