Elliptic curve point addition and point doubling operations using Projective and Jacobian coordinates require fewer field multiplication operations when considering $Z$ coordinates of input points equal to one ($Z_1=Z_2=1$). Is there any drawback of this strategy?
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5$\begingroup$ If you add two points with $Z_1=Z_2=1$, the resulting point will (in general) have $Z_3 \ne 1$. Given that we typically do a full addition chain, and not just add two points together, what is your stategy when it comes time to do the second addition? $\endgroup$– ponchoCommented Sep 23, 2015 at 20:25
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2$\begingroup$ In other words, the drawback is that it only works when $Z_1=Z_2=1$. :) And of course you must normalize the resulting $(X_3,Y_3,Z_3)$ if you want the $(x_3,y_3)$ coordinates that point. $\endgroup$– ChrisCommented Sep 23, 2015 at 21:58
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This question was answered in the comments:
If you add two points with $Z_1=Z_2=1$, the resulting point will (in general) have $Z_3\neq1$. Given that we typically do a full addition chain, and not just add two points together, what is your stategy when it comes time to do the second addition? – poncho
In other words, the drawback is that it only works when $Z_1=Z_2=1$. :) And of course you must normalize the resulting $(X_3,Y_3,Z_3)$ if you want the $(x_3,y_3)$ coordinates that point. – Chris