ECCImplementation

This is a Python Implementation of Elliptical Curve Cryptography created as my Final Project for CSC471: Cryptography.

UML Diagram

Elliptic Curve Implementation (ec.elliptic_curve.py)

Defines an EllipticCurve object with the following properties and functions:

self.PTATINF: Point at Infinity

__init__(self, a, b, p): Initialize an Elliptic Curve of the form $y^2 = x^3 + ax + b\mod{p}$

add_points(self, p1, p2): Returns $P_1+P_1$ where $P_1,P_2\in E$

scalar_product(self, t, p): Returns $t\cdot P$ using the Double-and-Add algorithm seen here Elliptic Curve Paper (UChicago)

neg(self, point): Returns $-P$

get_y_from_x(self, x, pos=False): Returns $y$ coordinate corresponding to the $x$ coordinate given. If pos is True, it returns the negative value of $y\mod{p}$

random_point(self): Returns a securely random point along the Elliptic Curve

decompress_public_key(self, pk): "Decompresses" a point using the algorithm found here BitCoin Stack Exchange

is_point(self, p): Returns True if the point is on the curve, False otherwise

assert_point(self, p): assert self.ispoint(p)

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Elliptic Curve Diffie Helmann Implementation (ecdh/ecdh.py)

Defines an ECDH object that has the following functions:

__init__(self, p, curve): Defines an object capable of participating in Diffie-Helmann key exchanges based on Elliptic Curves.

rand_private(self, bytes=124): Produces a random private key of length bytes

get_shared(self, public): Returns shared key from others public key. Performs scalar multiplication $a\cdot B$ where $a$ is alice's private key and $B$ is Bob's public key.

get_public(self): Returns $a\cdot P$

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Elliptic Curve ElGamal Cryptosystem Implementation (ec_elgamal/ec_actor.py)

Defines a class ECActor to encrypt and decrypt points along an Elliptic Curve

__init__(self, p, ord_p, curve): Creates an actor that uses point $P$ with order $ord_P$ to move along the curve given by curve. Also defines an object self.ecdh=ECDH(p, curve)

get_public(self): Gets the public key of the embedded Elliptic Curve Diffie Helman object

encrypt(self, point, receiver_pub): Encrypts a point using the other actor's public key

decrypt(self, encrypted_output): Decrypts a pair of points containing other actor's public key and their encrypted result