**EC Cryptography Tutorials - Herong's Tutorial Examples** - v1.03, by Herong Yang

_{1931}(443,1045)

This section provides an example of a reduced Elliptic Curve group E_{1931}(443,1045).

Let's take a look at another reduced elliptic curve group, E_{1931}(433,1045),
as discussed in "2018 Math Summer Camp -
Explicit construction of elliptic curves with prescribed order over finite fields"
at mathduc.com/summercamp/.

Here is the elliptic curve in the real number space:

y^{2}= x^{3}- 443x + 1045

Here is the reduced elliptic curve group using modular arithmetic of prime number 1931,
E_{1931}(443,1045):

y^{2}= x^{3}- 443x + 1045 (mod 1931)

As you can see, there are a lots of elements in a reduced elliptic curve group, when a large prime number is used. Is there any formula to calculate the number of elements (also called the order) of an a reduced elliptic curve group? We will discuss it in the next tutorial.

Table of Contents

Geometric Introduction to Elliptic Curves

Algebraic Introduction to Elliptic Curves

Abelian Group and Elliptic Curves

Discrete Logarithm Problem (DLP)

Generators and Cyclic Subgroups

►Reduced Elliptic Curve Groups

Converting Elliptic Curve Groups

Elliptic Curves in Integer Space

Python Program for Integer Elliptic Curves

Elliptic Curves Reduced by Modular Arithmetic

Python Program for Reduced Elliptic Curves

Point Pattern of Reduced Elliptic Curves

Integer Points of First Region as Element Set

Reduced Point Additive Operation

Modular Arithmetic Reduction on Rational Numbers

Reduced Point Additive Operation Improved

What Is Reduced Elliptic Curve Group

Reduced Elliptic Curve Group - E_{23}(1,4)

Reduced Elliptic Curve Group - E_{97}(-1,1)

Reduced Elliptic Curve Group - E_{127}(-1,3)

►Reduced Elliptic Curve Group - E_{1931}(443,1045)

Finite Elliptic Curve Group, E_{q}(a,b), q = p^{n}

tinyec - Python Library for ECC

ECDH (Elliptic Curve Diffie-Hellman) Key Exchange

ECDSA (Elliptic Curve Digital Signature Algorithm)