Group Law

Learn about group law and basic group operations in this lesson.

In this section, we present the basic group operations on elliptic curves, namely point addition and point doubling. We’ll show that the points on an elliptic curve EE have an algebraic structure, i.e., ( Bezout’s theoremconcept_4), we’ll show that adding two points PP and QQ in E(K)E(\mathbb{K}) gives a third point in E(K)E(\mathbb{K}), whereas the set of E(K)E(\mathbb{K}) together with the addition operation forms an abelian group with O\mathcal{O} as its identity. The best way to explain this operation is to consider the elliptic curves over the reals R\mathbb{R} and to explain the rule geometrically. As already mentioned, we only consider fields with characteristic char(K)>3char(\mathbb{K})>3.

Point addition

Let E(K)E(\mathbb{K}) be an elliptic curve

y2=x3+Ax+By^{2}=x^{3}+A x+B

over a field K\mathbb{K} with char(K)2,3char(\mathbb{K}) \neq 2,3. Furthermore, let P=(x1,y1)P=\left(x_{1}, y_{1}\right) and Q=(x2,y2)Q=\left(x_{2}, y_{2}\right) be two distinct points on the elliptic curve E(K).E(\mathbb{K}). Let Pˉ\bar{P} denote the conjugate point of PP, i.e., if P=(x,y)P=(x, y), then Pˉ=(x,y)\bar{P}=(x,-y) (i.e., the point PP is reflected across the xx-axis by changing the sign of the yy-coordinate). Then, the sum RR of the addition P+QP+Q is defined as follows (Darrel Hankerson et al. (2006)Darrel Hankerson, Alfred J. Menezes, and Scott Vanstone. Guide to Elliptic Curve Cryptography. Springer Professional Computing. New York, 2006. Springer., Lawrence C. Washington (2008)Lawrence C. Washington. Elliptic Curves: Number Theory and Cryptography, Second Edition. Discrete Mathematics and Its Applications. New York, 2008. CRC Press.):

We draw a line through the two points PP and QQ as depicted in the figure given below. This line intersects EE at a unique third point RR^{\prime}, which we denote by R=PQR^{\prime}=P * Q. Then, we obtain the sum RR by the reflection of RR^{\prime} across the xx-axis.

Figure 1

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