Let be a field and let. Then, the Doubling-oriented Doche–Icart–Kohel curve with parametera in affine coordinates is represented by: Equivalently, in projective coordinates: with and. Notice that, since this curve is a special case of Weierstrass form, transformations to the most common form of elliptic curve are not needed.
Group law
It is interesting to analyze the group law in elliptic curve cryptography, defining the addition and doubling formulas, because these formulas are necessary to compute multiples of points P. In general, the group law is defined in the following way: if three points lies in the same line then they sum up to zero. So, by this property, the group laws are different for every curve shape. In this case, since these curves are special cases of Weierstrass curves, the addition is just the standard addition on Weierstrass curves. On the other hand, to double a point, the standard doubling formula can be used, but it would not be so fast. In this case, the neutral element is , for which. Then, if is a non-trivial element, then the inverse of this point is –P=.
Addition
In this case, affine coordinates will be used to define the addition formula: += where x3 = x12+/ y3 = x13+x12+x1+y1+))/
The fastest addition is the following one, and the cost that it takes is 4 multiplications, 4 squaring and 10 addition. A = Y2-Y1 AA = A2 B = X2-X1 CC = B2 F = X1CC Z3 = 2CC D = X2Z3 ZZ3 = Z32 X3 = 2-aZ3-D Y3 = -Y2ZZ3
Example
Let. Let P, Q and a=1, then A=2 AA=4 B=1 CC=1 F=2 Z3=4 D=4 ZZ3=16 X3=-4 Y3=336 Thus, P+Q=
Doubling
The following algorithm is the fastest one, and the cost that it takes is 1 multiplication, 5 squaring and 7 additions. A = X12 B = A-a16 C = a2A YY = Y12 YY2 = 2YY Z3 = 2YY2 X3 = B2 V = 2-YY-X3 Y3 = V ZZ3 = Z32
Example
Let and a=1. Let P=, then Q=P= is given by: A=1 B=-15 C=2 YY=4 YY2=8 Z3=16 X3=225 V=27 Y3=9693 ZZ3=256 Thus, Q=.
The addition and doubling computations should be as fast as possible, so it is more convenient to use the following representation of the coordinates: are represented by satisfying the following equations: Then, the Doubling-oriented Doche–Icart–Kohel curve is given by the following equation: In this case, is a general point with inverse. Furthermore, the points over the curve satisfy: for all nonzero. Faster doubling formulas for these curves and mixed-addition formulas were introduced by Doche, Icart and Kohel; but nowadays, these formulas are improved by Daniel J. Bernstein and Tanja Lange.
