Point Pattern of Reduced Elliptic Curves

This section describes elliptic the repeatable pattern of integer points on reduced elliptic curves. If we know the integer points of the curve in one region, we can move them parallelly to any other region.

In the previous tutorial, we demonstrated how to find integer points on elliptic curves reduced by modular arithmetic of prime numbers p in a single region of (x,y) in {0, 1, 2, ..., p-1}

Element Set in a Single Region:
   All P = (x,y), such that:
      y2 = x3 + ax + b (mod p)
   where:
      a and b are integers
      p is a prime number
      4a3 + 27b2 != 0
      x and y are integers in {0, 1, 2, ..., p-1}

Finding integer points the same reduced elliptic curve in other regions is easy. You can just move those points parallelly from the first range to any other region by adding multiples of p to x and y coordinates. You can prove this easily using modular arithmetic properties.

Let's take the same reduced elliptic curve from the previous tutorial of (a,b) = (1,4) and p = 23 as an example:

Reduced Elliptic Curve:
   y2 = x3 + x + 4 (mod 23)

We found 28 integer points on this curve in the first region:

Points (x,y) on the curve and in the region of {0, 1, 2, ..., 22} 
  ( 0, 2)  ( 1,11)  ( 4, 7)  ( 7, 3)  ( 8, 8)  ( 9,11)  (10, 5)
  ( 0,21)  ( 1,12)  ( 4,16)  ( 7,20)  ( 8,15)  ( 9,12)  (10,18)
  (11, 9)  (13,11)  (14, 5)  (15, 6)  (17, 9)  (18, 9)  (22, 5)
  (11,14)  (13,12)  (14,18)  (15,17)  (17,14)  (18,14)  (22,18)

If we want to find integer points in the region of x in {0, 1, 2, ..., 22} and y in {23, 24, 25, ..., 45}, we can just move integer points in the first region by adding 23 to the y coordinates:

Points (x,y) on the curve and in the region
x in {0, 1, 2, ..., 22} and y in {-23, -22, -21, ..., -1}:
  ( 0,25)  ( 1,34)  ( 4,30)  ( 7,26)  ( 8,31)  ( 9,34)  (10,28)
  ( 0,44)  ( 1,35)  ( 4,49)  ( 7,43)  ( 8,38)  ( 9,35)  (10,41)
  (11,32)  (13,34)  (14,28)  (15,29)  (17,32)  (18,32)  (22,28)
  (11,37)  (13,35)  (14,41)  (15,40)  (17,37)  (18,37)  (22,41)

The entire integer 2-dimensional space can be divided into infinite number of such regions. So all integer points on a reduced elliptic curve can be viewed as a single pattern repeats itself in all 4 directions.

Here is a diagram showing this repeatable pattern of integer points on the reduced elliptic curve of (a,b) = (1,4) and p = 23:

Repeatable Pattern of Integer Points on Reduced Elliptic Curve
Repeatable Pattern of Integer Points on Reduced Elliptic Curve

Last update: 2019.

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 = p^n

 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