Algorithms and Examples
There is one algorithm that can be used to add two different points or to double; it is given by Joye and Quisquarter. Then, the following result gives the possibility the obtain the doubling operation by the addition:
Proposition. Let P=(X,Y,Z) be a point on a Hessian elliptic curve E(K). Then: 2(X:Y:Z) = (Z:X:Y) + (Y:Z:X) (2). Furthermore, we have (Z:X:Y)≠(Y:Z:X).
Finally, contrary to other parameterizations, there is no subtraction to compute the negation of a point. Hence, this addition algorithm can also be used for subtracting two points and on a Hessian elliptic curve:
( X1:Y1:Z1) - ( X2:Y2:Z2) = ( X1:Y1:Z1) + (Y2:X2:Z2) (3)
To sum up, by adapting the order of the inputs according to equation (2) or (3), the addition algorithm presented above can be used indifferently for: Adding 2 (diff.) points, Doubling a point and Subtracting 2 points with only 12 multiplications and 7 auxiliary variables including the 3 result variables. Before the invention of Edwards curves, these results represent the fastest known method for implementing the elliptic curve scalar multiplication towards resistance against side-channel attacks.
For some algorithms protection against side-channel attacks is not necessary. So, for these doublings can be faster. Since there are many algorithms, only the best for the addition and doubling formulas is given here, with one example for each one:
Read more about this topic: Hessian Form Of An Elliptic Curve
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