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

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SUMMARY

This discussion focuses on solving elliptic curve equations, specifically y² = x³ + x + 1 mod 17 and y² = x³ + 3x + 1 mod 13. Participants suggest evaluating x values from 0 to 16 to identify all relevant points on the curve. This method allows for the determination of inverses as well. The conversation emphasizes the importance of understanding the periodic nature of the results when plotting these curves.

PREREQUISITES
  • Understanding of elliptic curve equations
  • Familiarity with modular arithmetic
  • Basic calculus skills for curve plotting
  • Knowledge of inverse functions in mathematics
NEXT STEPS
  • Learn how to compute elliptic curve points using Python libraries like 'ecdsa'
  • Research modular arithmetic applications in cryptography
  • Explore graphical representation of elliptic curves using tools like Desmos
  • Study the mathematical properties of elliptic curves in cryptographic systems
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Mathematicians, cryptographers, and students interested in elliptic curve cryptography and its applications in secure communications.

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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