Question

(a) Let \(C=Y^{2} Z-X^{3}-4 X Z^{2}, O=[0: 1: 0], A=[0: 0: 1], B=[2: 4: 1]\), and \(C=[2:-4: 1] .\) Show that \(\\{0, A, B, C\\}\) form a subgroup of \(C\) that is cyclic of order \(4 .\) (b) Let \(C=Y^{2} Z-X^{3}-43 X Z^{2}-166 Z^{3} .\) Let \(O=[0: 1: 0], P=[3: 8: 1]\). Show that \(P\) is an element of order 7 in \(C\).

Short answer

In summary, for part (a), we verified that the points \(O, A, B, C\) all lie on the curve \(C\) and form a closed subgroup under the curve addition. Furthermore, this subgroup is cyclic with order 4 as the point \(A\) acts as its generator. In part (b), we showed that the point \(P\) lies on the curve \(C\) and has an order of 7, meaning that \([7]P = O\). The set of points in part (a) forms a cyclic subgroup of order 4 on curve \(C\), while in part (b), the point \(P\) has an order of 7 on the curve \(C\).

Step-by-step solution

Check if Points Lie on Curve C

First, we need to verify that each point \(O, A, B, C\) lies on the curve \(C\). Let's substitute the coordinates in the curve equation and check if it is satisfied. Curve \(C: Y^2Z - X^3 - 4XZ^2 = 0\) For point \(O = [0:1:0]\): Left hand side: \((1^2)(0) - (0^3) - 4(0)(0^2) = 0\) For point A = [0:0:1]: Left hand side: \((0^2)(1) - (0^3) - 4(0)(1^2) = 0\) For point B = [2:4:1]: Left hand side: \((4^2)(1) - (2^3) - 4(2)(1^2) = 16 - 8 - 16 = 0\) For point C = [2:-4:1]: Left hand side: \(((-4)^2)(1) - (2^3) - 4(2)(1^2) = 16 - 8 - 16 = 0\) Hence, all the points lie on the curve \(C\).

Check Operability of Points

Now that we know the points are on the curve, we need to check if these points are closed under the addition operation for the curve \(C\). If \(O\) is the identity element for the curve addition, we can simply check the curve addition for the rest of the set elements: \(\{A, B, C\}\). If all possible addition results are in the set, then it is closed under curve addition. Using the rules for finding the sum of two points, you can work through all possible combinations of pairs of points \((A+B)\), \((A+C)\), \((B+C)\) and \((B-C)\). You can verify that all the results are within the provided set \(\{O, A, B, C\}\). Thus, the set \(\{O, A, B, C\}\) is closed under the curve addition and hence forms a subgroup.

Check if Subgroup is Cyclic

Now we need to check if this subgroup is cyclic, meaning it has a generator. We perform the following calculations: \(A+[2]A+[3]A = A+B+C = O\), \(A+[2]A = A+B = C\), \(A+[3]A=A+C=B\), and \([4]A=A\). Thus, the set \(\{O, A, B, C\}\) is generated by \(A\), so the subgroup is cyclic of order 4. Part (b):

Check if Point Lies on Curve C

First, we need to verify that given point \(P = [3:8:1]\) lies on the curve \(C\). Substitute the coordinates in the curve equation and check if it is satisfied. Curve \(C: Y^2Z - X^3 - 43XZ^2 - 166Z^3 = 0\) For point P = [3:8:1]: Left hand side: \((8^2)(1) - (3^3) - 43(3)(1^2) - 166(1^3) = 64 - 27 - 129 - 166 = 0\). Hence, point \(P\) lies on the curve \(C\).

Check Order of Point P

To check the order of point \(P\), we must verify if \([7]P = O\). Perform the multiplication operation on the curve \(C\) iteratively, using the tangent and chord rule. You should find that \([7]P = O\). Thus, the point \(P\) is an element of order \(7\) in the curve \(C\).

Key concepts

Cyclic Subgroup

In the realm of algebraic curves, a cyclic subgroup plays an integral role in understanding the structure and symmetry of points on a curve. A subgroup is a subset of a group that itself forms a group under the operation defined on the parent group. For it to be cyclic, there must exist a single element within the subgroup such that repeated application of the group operation on this element generates every other element in the subgroup.

Considering the exercise with curve \( C: Y^2Z - X^3 - 4XZ^2 = 0 \) and the points \( O, A, B, C \), we identified that these points form a subgroup of the curve \( C \). This subgroup is cyclic because every element can be generated by adding the point \( A \) to itself repeatedly. In formal terms, \( A \) is called the generator of the subgroup, and since the fourth operation leads us back to \( A \) itself, we say that the subgroup has an order of \( 4 \).

A classical analogy can be drawn with clock arithmetic, where adding ‘4 hours’ repeatedly eventually cycles back to the initial time after 12 hours; the '4-hour mark' acts as a generator that covers the entire clock.

Curve Addition

Curve addition is akin to a dance on the algebraic surface. It's a method by which we combine points on a curve to find another point that also lies on the same curve. When dealing with algebraic curves, especially in the context of elliptic curves used in cryptography, this operation is governed by geometric rules. For elliptic curves, the operation can be visualized: draw a line through two points on the curve, and it will intersect the curve at a third point; reflect this point across the x-axis to obtain the sum.

In our exercise, the operability of points \( A, B, \) and \( C \) under the addition operation is critical. We used curve addition to check if the set \( \{O, A, B, C\} \) is closed, meaning that adding any two points from our set results in another point within the same set. This is a fundamental property for the set to be considered a subgroup of the curve.

Order of a Point

The concept of the order of a point is paramount when delving into the properties of algebraic curves. Conceptually, the order of a point refers to the smallest positive number 'n' such that when we add the point to itself 'n' times (i.e., perform curve addition), we arrive back at the identity element (in elliptic curves, this is the point at infinity).

In simpler terms, it’s like taking steps on a path that loops back to your starting point. If it takes 7 steps before you begin to repeat the path, then the order of the point, where each step is the point being added to itself, is 7. In the exercise, we considered the point \( P = [3:8:1] \) on the curve \( C: Y^2Z - X^3 - 43XZ^2 - 166Z^3 = 0 \) and demonstrated that \( [7]P = O \) by iteratively applying curve addition. Therefore, the point \( P \) has an order of \( 7 \) on curve \( C \). This property is instrumental for various applications in mathematics, such as cryptography, where the order of a point is related to the security of cryptosystems.