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.

Given information:An elliptic curve is represented by the equation , where and are constants.The…

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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