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

Integer Points of First Region as Element Set

This section describes how to use all integer points from the first region on a reduced elliptic curve as the element set to try to construct an Abelian group.

Once we know how find all integer points on an elliptic curve reduced by modular arithmetic of a prime number. We need to consider a way to define an Abelian group out of those integer points.

Let's consider all integer points from the first region on a reduced elliptic curve as the element set:

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}

Can we still use the rule of chord operation on this set as the group operation?

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

Element Set in a Single Region: All P = (x,y), such that: y^{2}= x^{3}+ x + 4 (mod 23) where: x and y are integers in {0, 1, 2, ..., 22}

If we want perform the point doubling operation of P = (0,2), obviously we can not do it geometrically using the rule of chord. But we can do it using our algebraic equations developed earlier in the book:

Calculation of 2P = P +P: 2P = P + P = R = (x_{R}, y_{R}) P = (x_{P}, y_{P}) = (0, 2) x_{R}= m^{2}- 2x_{P}(4) y_{R}= m(x_{P}- x_{R}) - y_{P}(5) 3(x_{P})^{2}+ a m = --------- (6) 2(y_{P}) m = (3*0*0 + 1)/(2*2) = 1/4 x_{R}= (1/4)*(1/4) - 2*0 = 1/16 y_{R}= (1/4)*(0 - 1/16) - 2 = -1/64 - 2 = -129/64 Result: 2P = R = (x_{R}, y_{R}) = (1/16, -129/64)

As you can see, the resulting point R is not an integer point and it is not in the first region! In other words, (0,2) + (0,2) using the original rule of chord operation as the addition operation does not satisfy the "Closure" condition of Abelian group.

See the next tutorial on how to reduce the point addition operation.

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)