Question

Let \(C\) be a nonsingular cubic given by the equation \(Y^{2} Z=X^{3}+a X^{2} Z+b X Z^{2}+\) \(c Z^{3}, O=[0: 1: 0]\). Let \(P_{i}=\left[x_{i}: y_{i}: 1\right], i=1,2,3\), and suppose \(P_{1} \oplus P_{2}=P_{3}\). If \(x_{1} \neq x_{2}\), let \(\lambda=\left(y_{1}-y_{2}\right) /\left(x_{1}-x_{2}\right) ;\) if \(P_{1}=P_{2}\) and \(y_{1} \neq 0\), let \(\lambda=\left(3 x_{1}^{2}+2 a x_{1}+b\right) /\left(2 y_{1}\right)\). Let \(\mu=y_{i}-\lambda x_{i}, i=1,2\). Show that \(x_{3}=\lambda^{2}-a-x_{1}-x_{2}\), and \(y_{3}=-\lambda x_{3}-\mu\). This gives an explicit method for calculating in the group.

Short answer

In this exercise, we are given a nonsingular cubic curve C and three points P1, P2, P3 on the curve such that P1 ⊕ P2 = P3. We are asked to show some expressions for the x3 and y3 coordinates of P3. By following the steps outlined in the solution, we showed the expressions for x3 and y3 as: x3 = λ² - a - x1 - x2 y3 = -λx3 - μ Where λ depends on the conditions stated in step 1 and μ is given in the problem.

Step-by-step solution

Define \(\lambda\)

Depending on the given conditions, find the appropriate value for \(\lambda\): - If \(x_1 \neq x_2\), then \(\lambda = \frac{y_1 - y_2}{x_1 - x_2}\). - If \(P_1 = P_2\) and \(y_1 \neq 0\), then \(\lambda = \frac{3x_1^2 + 2ax_1 + b}{2y_1}\).

Find the line passing through \(P_1\) and \(P_2\)

Using the given value for \(\mu\), we can express the line passing through \(P_1\) and \(P_2\) as \(y = \lambda x + \mu\).

Find the third point of intersection of the line with the curve

Substitute \(y = \lambda x + \mu\) into the curve equation: \((\lambda x+\mu)^2Z = X^3 + aX^2Z + bXZ^2 + cZ^3\). Rearrange the equation to make it a cubic equation in terms of \(x\). Let \(x_3\) be the root of this cubic equation that is different from \(x_1\) and \(x_2\). Then, find the corresponding \(y_3\) using the line equation: \(y_3 = \lambda x_3 + \mu\).

Prove the required expressions for \(x_3\) and \(y_3\)

To prove the given expressions for \(x_3\) and \(y_3\), we'll use the properties of the group law on elliptic curves. Since \(P_1 \oplus P_2 = P_3\), we know that \(P_1 \oplus P_2 \oplus P_3 = O\). So, now consider \(P_1 \oplus P'_1 \oplus P'_2 \oplus P'_3 = O\). Because \(P_1 \oplus P'_1 \oplus P'_2 = O\), it follows that \(P_3 = P'_3\). Lastly, using line and curve equations, obtain the expressions: - \(x_3 = \lambda^2 - a - x_1 - x_2\) - \(y_3 = -\lambda x_3 - \mu\)

Key concepts

Elliptic Curves

Elliptic curves are fascinating mathematical structures with applications in number theory, cryptography, and complex analysis. An elliptic curve over the real numbers is commonly represented by the equation \[y^2 = x^3 + ax + b,\] where the discriminant \[4a^3 + 27b^2 eq 0\] ensures that the curve is nonsingular (without self-intersections or cusps). These curves are graphical representations in the xy-plane featuring a distinct 'doughnut shape'.

Interestingly, elliptic curves possess a group structure, where the points on the curve can be added together following specific rules. The point at infinity, often denoted as O, serves as the identity element in this group. The exercise explores an important case of the elliptic curve, set in projective space to avoid issues with points at infinity, showcasing that the sum of two points on the curve, say \(P_1\) and \(P_2\), can be determined analytically. In projective coordinates, the curve equation expands to \[Y^2Z = X^3 + aX^2Z + bXZ^2 + cZ^3,\] incorporating the homogenizing variable \(Z\).

An understanding of this group law is crucial for applications such as elliptic curve cryptography (ECC), where the security of encryption relies on the computational difficulty of solving problems related to group operations on elliptic curves.

Group Law in Algebraic Geometry

The group law in algebraic geometry is a rule for combining elements (points) on an algebraic curve to get another point on the same curve. For elliptic curves, this group law defines how to 'add' two points together and get a third point, following the curve's symmetry and intersection properties.

When two points, say \(P_1\) and \(P_2\), on an elliptic curve are 'added' (\(P_1 oplus P_2 = P_3\)), the line intersecting \(P_1\) and \(P_2\) will also intersect the curve at exactly one other point. This third intersection point is then reflected over the x-axis to find the resultant point \(P_3\), completing the 'addition.' If \(P_1\) equals \(P_2\), the line is the tangent at that point, and a similar intersection and reflection process follow.

Working through \(P_1 oplus P_2 = P_3\)

The exercise illustrates this addition by providing the equation for the secant (or tangent) line and the curve equation in projective space. Calculation of the sum of the points involves algebraic manipulation to find the coordinates of the third point \(P_3\). The outlined steps guide us through finding a common factor (\(lambda\)) when the x-coordinates of \(P_1\) and \(P_2\) differ or when they are the same point, the definition of a line equation using this common factor, and finally the intersection points—which in this context are the group elements themselves.

Nonsingular Cubic Curves

Nonsingular cubic curves are those without any points where the curve crosses itself (no double points) or has sharp corners (no cusps). It's essential for the curves in our context because the structure of the group law would not hold if there were singular points. The exercise given in the problem ensures that the cubic in question is nonsingular by using a homogeneous polynomial equation in projective space. This form guarantees smoothness across the entire curve, including points at infinity.

The given cubic equation in projective coordinates, \[Y^2Z = X^3 + aX^2Z + bXZ^2 + cZ^3,\] describes such a nonsingular curve. This equation can be converted into Weierstrass form, which is more familiar in the context of elliptic curves. The nonsingularity ensures the existence of well-defined tangent lines at every point on the curve, which is a property used in calculating the group law. When we follow the steps in the solution to find a third point on the curve, using secants or tangents, we rely on the fact that these lines intersect the curve at exactly one additional point that is not a singularity.

Importance of Nonsingularity

The nonsingular nature of an elliptic curve is vital for its use in cryptography and complex analysis as it guarantees the smoothness necessary for defining complex operations and ensuring security in cryptographic systems.