Reduced Elliptic Curve Group - E1931(443,1045)

This section provides an example of a reduced Elliptic Curve group E1931(443,1045).

Let's take a look at another reduced elliptic curve group, E1931(433,1045), as discussed in "2018 Math Summer Camp - Explicit construction of elliptic curves with prescribed order over finite fields" at

Here is the elliptic curve in the real number space:

   y2 = x3 - 443x + 1045
Elliptic Curve E(443,1045) in Real Number Space
Elliptic Curve E(443,1045) in Real Number Space

Here is the reduced elliptic curve group using modular arithmetic of prime number 1931, E1931(443,1045):

   y2 = x3 - 443x + 1045 (mod 1931)
Elliptic Curve Group E1931(443,1045)
Elliptic Curve Group E1931(443,1045)

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

 About This Book

 Geometric Introduction to Elliptic Curves

 Algebraic Introduction to Elliptic Curves

 Abelian Group and Elliptic Curves

 Discrete Logarithm Problem (DLP)

 Finite Fields

 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 - E23(1,4)

 Reduced Elliptic Curve Group - E97(-1,1)

 Reduced Elliptic Curve Group - E127(-1,3)

Reduced Elliptic Curve Group - E1931(443,1045)

 What Is Hasse's Theorem

 Finite Elliptic Curve Group, Eq(a,b), q = pn

 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)

 EC Cryptography in Java

 Standard Elliptic Curves



 Full Version in PDF/EPUB