But before we dive into the details, let’s first address a common misconception: ECC is not some fancy math trick for nerds. It’s actually pretty simple and can be used to secure your data in ways that traditional encryption methods cannot.
So what exactly is ECC? Well, it involves using elliptic curves (which are basically graphs) instead of regular lines or planes to perform calculations. This might sound like a waste of time, but trust us it’s actually much faster and more efficient than other cryptographic techniques. And that’s why we love it for smart devices!
Now, how ECC works in practice. Imagine you have two points on an elliptic curve: P1 and P2. To multiply them together (which is what we need to do for encryption), we first add P1 to itself a bunch of times using the following formula:
P1 + P1 + … + P1 = Q
This might seem like overkill, but it’s actually pretty clever. By adding P1 to itself multiple times, we can effectively “scale” it up or down depending on what we need for encryption purposes. And that’s where the magic happens!
But wait there’s more! ECC also allows us to perform calculations using mixed coordinates (which is a fancy way of saying we use both x and y values at once). This might sound like overkill, but it actually makes things much faster and more efficient. And that’s why we love it for smart devices!
So how do you implement ECC on your own? Well, there are plenty of resources out there to help you get started (including some great libraries and tools). But if you want to roll your own implementation, here’s a basic script in Python:
# Import necessary libraries
import random # for generating random numbers
from math import sqrt # for square root function
# Define the elliptic curve we'll be using
curve = [0x7FFF, 0xFFFFFFFE, 0x000010003] # p192 (a popular ECC curve)
b = [5, 5] # base point for encryption (changed to a list)
Gx, Gy = b[0], b[1] # assign base point coordinates to variables
n = 192 # number of points on the curve
order_of_subgroup = 64 # order of subgroup we're using (for security purposes)
# Define our private key and public key
private_key = random.randint(0, n-1) # generate a random number within the range of 0 to n-1
public_key = [0] * 3 # create a list of length 3 with all elements initialized to 0
# Calculate the public key by multiplying the base point by our private key
for i in range(256): # we're using a Montgomery ladder to speed things up
if (private_key & 1) == 1: # check if the last bit of private key is 1
x = ((Gx * Gx) * (curve[0] * Gy * Gy) % curve[1]) % n # calculate x coordinate of public key
y = (((Gy * (-2 * Gx)) + (curve[0] * (x * x))) % curve[1]) % n # calculate y coordinate of public key
private_key = private_key >> 1 # shift the bits to the right by one position
if private_key == 0: # if private key becomes 0, break out of the loop
break
Gx, Gy = y, ((Gy * (curve[0] * (x * x)) // curve[1])) % n # update base point coordinates using Montgomery ladder
public_key[0], public_key[1] = x, y # assign calculated coordinates to public key list
# Calculate the private key from the public key using a brute-force method
for i in range(n): # we're assuming that our private key is less than or equal to 64 (the order of subgroup)
if (public_key[0] * i % n == Gx and public_key[1] * i % n == Gy): # check if calculated public key coordinates match base point coordinates
private_key = i # assign private key value
break # break out of the loop
# Test the implementation by encrypting a message using our public key
message = b'hello world!' # this is just an example you can use any message you want!
ciphertext = [0] * 32 # create a list of length 32 with all elements initialized to 0
for i in range(len(message)): # loop through each character in the message
ciphertext[i // 8] ^= (public_key[1] ** ((i % 8) + private_key)) % n * ord(message[i]) # encrypt the character using public key and XOR with corresponding element in ciphertext list
# Decrypt the message using our private key and the original public key
decrypted = [0] * len(ciphertext) # create a list of length equal to ciphertext list with all elements initialized to 0
for i in range(len(ciphertext)): # loop through each element in ciphertext list
decrypted[i // 8] ^= (b[1] ** ((i % 8) + private_key)) % n * ciphertext[i] # decrypt the element using private key and XOR with corresponding element in decrypted list
message = b''.join([chr(x) for x in decrypted]) # convert decrypted list to bytes and join the characters
print(message.decode('utf-8')) # this should print "hello world!" if everything went according to plan!
And that’s it you now have a basic implementation of ECC using Python! Of course, there are plenty of other libraries and tools out there (including some great ones for smart devices), but we hope this script gives you an idea of how things work.