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

What Is Reduced Elliptic Curve Group

This section describes Reduced Elliptic Curve Groups or Elliptic Curve over Prime Field GF(p), denoted as E_{p}, which uses elliptic curve equations reduced by modular arithmetic of prime number p to define the group element set, and uses point addition operation based on the rule of chord reduced by the same modular arithmetic as the group operation.

After applying modular arithmetic reduction of prime number p to the elliptic equation and the point addition operation, I think we have found the complete definition of a new type of elliptic curve Abelian groups, we can call them reduced elliptic curve groups for now.

1. Group Element Set - All integer points P = (x, y) that satisfy the elliptic equation of coefficients a and b, reduced by modular arithmetic of prime number p as expressed below:

Element Set: 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}

2. Group Operation - The elliptic curve point addition operation based on the rule of chord reduced by modular arithmetic of prime number p as expressed below:

Additive operation: For any two given points on the curve: P = (x_{P}, y_{P}) Q = (x_{Q}, y_{Q}) R = P + Q is a third point on the curve: R = (x_{R}, y_{R}) Where: x_{R}= m^{2}- x_{P}- x_{Q}(mod p) (11) y_{R}= m(x_{P}- x_{R}) - y_{P}(mod p) (12) If P != Q, m is determined by: m(x_{P}- x_{Q}) = y_{P}- y_{Q}(mod p) (18) If P = Q, m is determined by: 2m(y_{P}) = 3(x_{P})^{2}+ a (mod p) (19)

3. Identity Element - The infinite point of 0 = (∞, ∞).

I don't have any reference resource that proves the above definition produces Abelian group. But I will provide some verifications of some specific cases of p, a and b.

Reduced elliptic curve groups are called with different names in different reference sources like: Elliptic curves over prime filed GF(p) as in "Elliptic Curve Cryptography: finite fields and discrete logarithms" at andrea.corbellini.name/2015/05/23 /elliptic-curve-cryptography-finite-fields-and-discrete-logarithms/, and in "Elliptic Curve Cryptography" at https://slideplayer.com/slide/6610565/.

A reduced elliptic curve group can be denoted as
E_{p}(a,b) for a specific set of p, a and b.
It is also called an elliptic curve over a finite field.

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)