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

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: y^{2}= x^{3}+ ax + b (mod p) where: a and b are integers p is a prime number 4a^{3}+ 27b^{2}!= 0 x and y are integers in {0, 1, 2, ..., p-1}

Finding integer points on 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: y^{2}= x^{3}+ 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:

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)