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

Find Subgroup with Point Addition

This section provides a tutorial example on how to find the subgroup of a given point on an elliptic curve using a loop of point additions with tinyec Python library.

Once you have elliptic curve created with tinyec Python library, you can find the subgroup of a given point on the curve with a loop of point additions:

>>> import tinyec.ec as ec >>> s = ec.SubGroup(p=97,g=(0,0),n=1,h=1) >>> c = ec.Curve(a=2,b=3,field=s,name='p97a2b3') >>> p = ec.Point(curve=c,x=3,y=6) >>> print(p) (3, 6) on "p97a2b3" => y^2 = x^3 + 2x + 3 (mod 97) >>> z = ec.Inf(c) # represents the infinite point on the curve >>> r = p >>> for i in range(0,97): ... print(r) ... if (r == z): ... break ... r = r + p # point addition operation ... ^Z (3, 6) on "p97a2b3" => y^2 = x^3 + 2x + 3 (mod 97) (80, 10) on "p97a2b3" => y^2 = x^3 + 2x + 3 (mod 97) (80, 87) on "p97a2b3" => y^2 = x^3 + 2x + 3 (mod 97) (3, 91) on "p97a2b3" => y^2 = x^3 + 2x + 3 (mod 97) Inf on "p97a2b3" => y^2 = x^3 + 2x + 3 (mod 97)

Ok. We found the subgroup of base point (or generator) of p = (3,6). The group order is 5.

Note that we created an ec.Inf object to represent the infinite point on the curve, and used it as the condition to stop the loop.

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

►tinyec - Python Library for ECC

Perform Point Addition with tinyec

►Find Subgroup with Point Addition

Set Subgroup Order to Higher Value

ECDH (Elliptic Curve Diffie-Hellman) Key Exchange

ECDSA (Elliptic Curve Digital Signature Algorithm)