MHB How to Use Elliptic Curve Cryptography to Find Inverses and Points on the Curve?

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To solve the elliptic curve equations y² = x³ + x + 1 mod 17 and y² = x³ + 3x + 1 mod 13, one approach is to evaluate the equations for all integer values of x within the specified modulus. This method allows for the identification of all points on the curve. Once the points are determined, inverses can also be calculated based on the results. Additionally, visualizing the curves by plotting the points can enhance understanding of the relationships between them. This process aids in improving calculus skills related to elliptic curve cryptography.
vokoyo
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May I know how to solve the equation as below:

(1) y2 = x3 + x + 1 mod 17

Finding Inverses
Finding Points on the Curve

(2) y2 = x3 + 3x + 1 mod 13

Finding Inverses
Finding Points on the Curve
 
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vokoyo said:
May I know how to solve the equation as below:

(1) y2 = x3 + x + 1 mod 17

Finding Inverses
Finding Points on the Curve

Hi vokoyo,

How about filling in $x=0, ..., 16$.
Those are all the relevant possibilities for $x$.
After that the same results will appear periodically.

That way we find all the points on the curve.
And from the results we can also find all inverses, can't we?
 
Thank you very much for your advice and suggestion

Please show me your sample solution draft
so that I can improve my calculus skills

I fact I would like to draw the curve line or point by point
 
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