Negation Operation on an Elliptic Curve

This section describes the Negation operation on an elliptic curve. If the resulting point the negation operation of an given point P is R, then P + R is the infinity point.

With the introduction of the infinity point, we can now define another operation on an elliptic curve, the negation operation, which is a unary operation.

If we use the minus sign "-" as the negation operator, then the resulting point R of the nagation operation of a point P on an elliptic curve can be expressed as the following:

R = -P, if P + R is the infinity point. 

If we use letter O to represent the infinity point, the above expression can be shortened as the following:

R = -P, if P + R = O

Last update: 2019.

Table of Contents

 About This Book

Geometric Introduction to Elliptic Curves

 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

 Algebraic Introduction to Elliptic Curves

 Abelian Group and Elliptic Curves

 Discrete Logarithm Problem (DLP)

 Finite Fields

 Generators and Cyclic Subgroups

 Reduced Elliptic Curve Groups

 Elliptic Curve Subgroups

 tinyec - Python Library for ECC

 EC (Elliptic Curve) Key Pair

 ECDH (Elliptic Curve Diffie-Hellman) Key Exchange

 ECDSA (Elliptic Curve Digital Signature Algorithm)

 ECES (Elliptic Curve Encryption Scheme)

 Terminology

 References

 Full Version in PDF/EPUB