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Geometric Introduction to Elliptic Curves

This chapter provides a geometric introduction of elliptic curves and the associated addition operation. Topics includes what is an elliptic curve and its geometric properties; geometric algorithm defining an addition operation; infinity point or identity element; commutativity and associativity of the addition operation.

What Is an Elliptic Curve?

Elliptic Curve Geometric Properties

Addition Operation on an Elliptic Curve

Prove of Elliptic Curve Addition Operation

Same Point Addition on an Elliptic Curve

Infinity Point on an Elliptic Curve

Negation Operation on an Elliptic Curve

Subtraction Operation on an Elliptic Curve

Identity Element on an Elliptic Curve

Commutativity of Elliptic Curve Operations

Associativity of Elliptic Curve Operations

Elliptic Curve Operation Summary

Conclusion:

• An elliptic curve is symmetric about the x-axis.
• A straight line can only intersect with an elliptic curve at up to 3 locations.
• An operation, called "addition", can be defined on an elliptic curve using a geometric algorithm called "rule of the chord".
• The infinity point can be considered as part of an elliptic curve and used as the "identity element" of the "addition" operation.
• The "addition" operation is commutative and associative.

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