**EC Cryptography Tutorials - Herong's Tutorial Examples** - v1.02, by Dr. Herong Yang

Build New Curves with tinyec

This section provides a tutorial example on how to create elliptic curve, actually a reduced elliptic curve group, with tinyec Python library.

If you want to build a new elliptic curve (a reduced elliptic curve group) with tinyec Python library, you must do it in two steps.

1. Create an ec.SubGroup object with the curve modulo and other parameters. For example:

>>> s = ec.SubGroup( p = 97, # the modulo used in the modular arithmetic g = (0,0), # the base point n = 1, # the order of the subgroup h = 1 # the cofactor of the subgroup )

Parameters g, n, and h are all required. But they have no impact on the elliptic curve group. So you can provide any values at the beginning.

2. Create an ec.Curve object with curve coefficients and an ec.SubGroup object. For example:

>>> c = ec.Curve( a = 2, # the 'a' coefficient of the curve b = 3, # the 'b' coefficient of the curve field = s, # a subgroup to provide the modulo name = 'My Curve' # a name given to the curve, optional )

Here is the Python script to create the elliptic curve,
E_{97}(2,3):

>>> 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') C:\Python\Python36-32\lib\site-packages\tinyec\ec.py:119: UserWarning: Point (0, 0) is not on curve ""p97a2b3" => y^2 = x^3 + 2x + 3 (mod 97)" ... >>> print(c) "p97a2b3" => y^2 = x^3 + 2x + 3 (mod 97)

Ok. The elliptic curve, c, is created. But the base point given in the Subgroup, s.g, is not on the curve. This is why we are getting the warning message. However the curve is good.

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)