Question

Check that the curve \(E=\left\\{(x, y) \in \mathbb{F}_{7}^{2}: y^{2}=x^{3}+x+3\right\\}\) over E7 is nonsingular. Compute all points on it, and verify that it is cyclic and generated by \((4.1)\).

Short answer

The curve is nonsingular, and all points on it form a cyclic group generated by (4,1).

Step-by-step solution

Understanding the Elliptic Curve

An elliptic curve over a finite field is defined by an equation of the form \(y^2 = x^3 + ax + b\). In this problem, the curve is given by \(y^2 = x^3 + x + 3\) over \(\mathbb{F}_7\). Our task is to first check if this curve is nonsingular.

Checking for Nonsingularity

A curve is nonsingular if it has no points where the partial derivatives of the defining equation both vanish. For the equation \(y^2 = x^3 + x + 3\), we calculate the partial derivatives: \(dy/dx = 0\) and \(dy/dy = 0\). Compute and simplify \(f_x(x, y) = 3x^2 + 1\) and \(f_y(x, y) = 2y\) in \(\mathbb{F}_7\). The curve is nonsingular if there is no \(x, y\) in \(\mathbb{F}_7\) such that both partial derivatives are zero. Evaluate around \(\mathbb{F}_7\) to confirm this.

Finding Points on the Curve

Enumerate all \(x\) values in \(\mathbb{F}_7\), i.e., \(0, 1, 2, 3, 4, 5, 6\). For each \(x\) value, compute \(x^3 + x + 3\) mod 7. Check if the result is a quadratic residue in \(\mathbb{F}_7\). For each valid \(x\), solve \(y^2 \equiv x^3 + x + 3 \pmod{7}\) to find \(y\) values. This will enumerate all points on the curve including the point at infinity.

Verification of Cyclic Property

With all points found, check if these form a cyclic group. By computation or point addition properties, see if there exists a generator for the group. Specifically, see if repeated addition of the point \((4,1)\) returns to the starting point (identity). This involves performing elliptic curve addition for the points, acknowledging the modular arithmetic over \(\mathbb{F}_7\).

Finding a Generator

Using the candidate generator \((4,1)\), perform point addition on the curve to generate all points. Verify that cycling through multiples of \((4,1)\) generates all point previously found. Using the addition formulae for elliptic curves, calculate multiples of \((4,1)\) to verify cyclic nature, checking at each step that operation remains within mod 7.

Key concepts

Nonsingular Curves

When studying elliptic curves, it's important to determine whether a curve is nonsingular. This ensures that the curve forms a well-behaved geometric object suitable for mathematical study, such as forming groups.
A curve is described as nonsingular if it doesn't have any point where both partial derivatives of its defining equation are zero. To achieve this, we evaluate the partial derivatives and verify that no solution vanishes both.

• For the given curve: \(y^2 = x^3 + x + 3\), we need to compute the derivatives with respect to \(x\) and \(y\).
• The partial derivative with respect to \(x\) is \(f_x(x,y) = 3x^2 + 1\), and with respect to \(y\) is \(f_y(x,y) = 2y\).

Checking these equations over a finite field, such as \(\mathbb{F}_7\), means evaluating them at all possible points to ensure that neither simultaneously equals zero. This process confirms the curve's nonsingularity, which allows it to have no cusps or self-intersections and ensures the integrity of the mathematical behavior expected of elliptic curves.

Cyclic Groups

A cyclic group is a fundamental concept in algebra, defined as a group that can be generated by repeatedly applying the group operation to a particular element.
In the context of elliptic curves, we must verify whether the points on the curve form such a group. The task involves using a specific generator point to reproduce all other points on the curve through the group operation.
We use the point \((4,1)\) as a candidate generator to explore this property.

• The operation here is the addition of points on the curve, related to the algebraic structure defined by the curve's equation.
• If starting from \((4,1)\) and repeatedly adding it using elliptic curve addition gets back to the infinity point, and we cover all distinct points on the curve, then it's a cyclic group.

This would imply that each point can be described as a multiple of \((4,1)\), verifying the cyclic nature. Such properties are crucial, especially in fields like cryptography, where the predictability and generation of elements are vital for constructing secure systems.

Finite Fields

Finite fields, often denoted as \(\mathbb{F}_q\), where \(q\) is a prime power, play a significant role in the operations on elliptic curves. In our case, \(\mathbb{F}_7\) represents a finite field with exactly seven elements.
Working with finite fields entails performing arithmetic operations modulo \(q\), where the results are always within the set \(\{0, 1, 2, ..., q-1\}\).

• When checking points on an elliptic curve in a finite field, we perform calculations like \(x^3 + x + 3\) modulo 7.
• Then, determine if the result is a quadratic residue, which involves checking if there's a \(y\) such that \(y^2 \equiv x^3 + x + 3 \pmod{7}\).

The finite field encapsulates the environments where these computations occur. It ensures the calculations are manageable while preserving the key properties necessary for applications such as cryptographic schemes and computations on elliptic curves.