An elliptic curve is a curve of the form y^2 = x^3 + ax + b for some constants a, b. These curves are extremely special in that they admit an addition given by the following picture: P R P+Q P Q R=0 Addition of distinct points T P TOT=0 R ΡΘΡ Adding a point to itself (a) Verify that the point (-4, -3) lies on the curve y^2 = x^3 + 73. (b) Compute the sum (-4, -3) + (-4, -3) using the description and picture above. Verify that this point is also on the curve.
An elliptic curve is a curve of the form y^2 = x^3 + ax + b for some constants a, b. These curves are extremely special in that they admit an addition given by the following picture: P R P+Q P Q R=0 Addition of distinct points T P TOT=0 R ΡΘΡ Adding a point to itself (a) Verify that the point (-4, -3) lies on the curve y^2 = x^3 + 73. (b) Compute the sum (-4, -3) + (-4, -3) using the description and picture above. Verify that this point is also on the curve.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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How do I find P+P in part B? All I have gotten up to is solving for the necessary equation 8x+y+35=0 but from there I do not know how to find P+P. BTW this is not part of a graded assignment; this is just a practice problem.
![**Elliptic Curves and Their Special Properties**
An elliptic curve is a curve of the form:
\[ y^2 = x^3 + ax + b \]
for some constants \( a, b \). These curves are extremely special in that they admit an addition given by the following picture:
**Graph Explanation:**
The image contains two diagrams illustrating the addition of points on an elliptic curve.
1. **Addition of Distinct Points:**
- Points \( P \) and \( Q \) are distinct points on the curve.
- A line is drawn through \( P \) and \( Q \) and it intersects the curve at a third point \( R \).
- The point \( P + Q \) is determined by reflecting \( R \) over the x-axis.
2. **Adding a Point to Itself:**
- The point \( P \) is considered and a tangent line at \( P \) is drawn.
- This tangent line intersects the curve at another point \( T \).
- The point \( P + P \) (or \( T \oplus T \)) is found by reflecting \( T \) over the x-axis.
In both cases, the operation results in a point denoted as \( O \), showing the closure property under addition.
**Exercises:**
(a) **Verify that the point \((-4, -3)\) lies on the curve \( y^2 = x^3 + 73 \).**
This involves substituting \((-4, -3)\) into the equation and confirming if the equality holds.
(b) **Compute the sum \((-4, -3) \oplus (-4, -3)\) using the description and picture above. Verify that this point is also on the curve.**
This requires using the rules of point addition on an elliptic curve as described, and then verifying that the resulting point satisfies the equation of the curve.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6e5ea172-2838-4041-9e7f-74e8cb5bb4e3%2F3b33257d-f533-4dd4-9900-ca5fe75484b3%2F20d8vp8_processed.png&w=3840&q=75)
Transcribed Image Text:**Elliptic Curves and Their Special Properties**
An elliptic curve is a curve of the form:
\[ y^2 = x^3 + ax + b \]
for some constants \( a, b \). These curves are extremely special in that they admit an addition given by the following picture:
**Graph Explanation:**
The image contains two diagrams illustrating the addition of points on an elliptic curve.
1. **Addition of Distinct Points:**
- Points \( P \) and \( Q \) are distinct points on the curve.
- A line is drawn through \( P \) and \( Q \) and it intersects the curve at a third point \( R \).
- The point \( P + Q \) is determined by reflecting \( R \) over the x-axis.
2. **Adding a Point to Itself:**
- The point \( P \) is considered and a tangent line at \( P \) is drawn.
- This tangent line intersects the curve at another point \( T \).
- The point \( P + P \) (or \( T \oplus T \)) is found by reflecting \( T \) over the x-axis.
In both cases, the operation results in a point denoted as \( O \), showing the closure property under addition.
**Exercises:**
(a) **Verify that the point \((-4, -3)\) lies on the curve \( y^2 = x^3 + 73 \).**
This involves substituting \((-4, -3)\) into the equation and confirming if the equality holds.
(b) **Compute the sum \((-4, -3) \oplus (-4, -3)\) using the description and picture above. Verify that this point is also on the curve.**
This requires using the rules of point addition on an elliptic curve as described, and then verifying that the resulting point satisfies the equation of the curve.
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