To calculate from elliptic curves we will use a simple approximation. For example,consider the curve y² = x3 + 17 over the field ℚ and the points P = (-1,4) andQ = (2,5).Check that P, Q lie on the curve. Write an equation y = ax + b for the line throughP, Q and plug it into the equation of the curve. You already know that x₁ = -1 andx2 = 2 belong to points on the curve, so you can factor the right side of the equationas (x-x1)(x − x2)(x − x3) where x3 is still unknown. Use this to calculate thethirdintersection R of the line with the curve and show that P⊕ Q = (-(8/9), -(109/27)). Note thatthe coordinates of this are again in ℚ. How generally applicable is this method tocalculate P⊕ Q?

Consider, the curve is y² = x3 + 17. The points are given by P=-1,4 and Q=2,5.

Answered: To calculate from elliptic curves we…

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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Question

To calculate from elliptic curves we will use a simple approximation. For example,
consider the curve y² = x³ + 17 over the field ℚ and the points P = (-1,4) and
Q = (2,5).

Check that P, Q lie on the curve. Write an equation y = ax + b for the line through
P, Q and plug it into the equation of the curve. You already know that x₁ = -1 and
x₂ = 2 belong to points on the curve, so you can factor the right side of the equation
as (x-x₁)(x − x₂)(x − x₃) where x₃ is still unknown. Use this to calculate the
third
intersection R of the line with the curve and show that P⊕ Q = (-(8/9), -(109/27)). Note that
the coordinates of this are again in ℚ. How generally applicable is this method to
calculate P⊕ Q?