First off, let’s start with some background information. Elliptic curves are mathematical objects that have been used for centuries in various fields such as cryptography, number theory, and physics. They can be represented by a set of points on a plane or surface, which makes them perfect for encoding data in a compact way.
But here’s the catch these elliptic curve points can take up quite a bit of space if they are not compressed properly. That’s where NumMaster comes in! This library provides efficient algorithms for compressing and decompressing elliptic curve points using various compression schemes such as Point Compression (P-128), Point Compression with Affine Coordinates (P-192), and Point Compression with Projective Coordinates (P-256).
Now, Let’s begin exploring with the details. To compress an elliptic curve point using NumMaster, you first need to import the library:
# Import the nummaster library
import nummaster as nm
# Define a function to compress an elliptic curve point
def compress_point(point):
# Check if the point is a valid elliptic curve point
if not nm.is_valid_point(point):
# If not valid, raise an error
raise ValueError("Invalid point")
# Get the x and y coordinates of the point
x, y = point
# Check if the point is at infinity
if nm.is_infinity(point):
# If so, return the point as is
return point
# Check if the point is on the curve
if not nm.is_on_curve(point):
# If not, raise an error
raise ValueError("Point is not on the curve")
# Get the curve parameters
a, b, p = nm.get_curve_params()
# Check if the point is the identity point
if nm.is_identity(point):
# If so, return the identity point
return nm.identity_point()
# Check if the point is a generator point
if nm.is_generator(point):
# If so, return the generator point
return nm.generator_point()
# Check if the point is a quadratic residue
if nm.is_quadratic_residue(y):
# If so, return the compressed point with the first bit set to 0
return (x, 0)
# If not a quadratic residue, return the compressed point with the first bit set to 1
return (x, 1)
# Example usage
# Define an elliptic curve point
point = (5, 7)
# Compress the point using the compress_point function
compressed_point = compress_point(point)
# Print the compressed point
print(compressed_point)
# Output: (5, 1)
Next, create a new instance of the `EllipticCurvePointCompression` class and pass it your desired compression scheme:
# Create an instance of the EllipticCurvePointCompression class and pass it the desired compression scheme
compression = EllipticCurvePointCompression(P128) # P-128 compression scheme
Now, let’s say you have an elliptic curve point that looks like this:
# Define a tuple variable named "point" to store an elliptic curve point
# The point has two coordinates, x and y, in big-endian format
point = (0x79be6bcf55d3da4f2ff8eaf682ae23ee7, 0x1)
To compress it using the P-128 scheme, simply call the `compress()` method on your compression object:
# To compress a point using the P-128 scheme, we need to call the `compress()` method on our compression object.
# First, we need to create a compression object using the `compression` module.
compression_object = compression()
# Next, we need to pass the point we want to compress to the `compress()` method.
compressed_point = compression_object.compress(point)
# Finally, we can print the compressed point, which will be a 32-byte compressed point in binary format.
print(compressed_point)
And that’s it! You now have a compressed version of your original elliptic curve point, which can be used for various purposes such as storing data on disk or transmitting it over the network.
But what about decompressing these points? Well, NumMaster makes that easy too! To do so, simply create a new instance of the `EllipticCurvePointDecompression` class and pass it your desired compression scheme:
# Import the necessary module
import NumMaster as nm
# Create an instance of the EllipticCurvePointDecompression class
decompression = nm.EllipticCurvePointDecompression()
# Specify the desired compression scheme
compression_scheme = nm.P128
# Pass the compression scheme as an argument to the EllipticCurvePointDecompression instance
decompression.set_compression_scheme(compression_scheme)
# Now the decompression instance is ready to be used for decompressing points
# For example, we can decompress a point using the decompress_point() method
decompressed_point = decompression.decompress_point(compressed_point)
# The decompressed point can then be used for various purposes such as storing data on disk or transmitting it over the network.
Now, let’s say you have a compressed point that looks like this:
# Define a variable "compressed_point" and assign it a 32-byte compressed point in binary format
compressed_point = b'\x03\x79\xbe\x6b\xcf\x55\xd3\xda\x4f\x2ff\x8e\xaf\x68\x2a\xe2\x3e\x7e'
# The "b" prefix indicates that the string is in binary format
# Print the compressed point
print(compressed_point)
# The "print" function outputs the value of the variable "compressed_point"
# Convert the compressed point from binary format to hexadecimal format
hex_compressed_point = compressed_point.hex()
# The "hex" method converts the binary string to a hexadecimal string
# Print the hexadecimal compressed point
print(hex_compressed_point)
# The "print" function outputs the value of the variable "hex_compressed_point"
# Convert the hexadecimal compressed point to a list of bytes
byte_list = [int(hex_compressed_point[i:i+2], 16) for i in range(0, len(hex_compressed_point), 2)]
# The "int" function converts the hexadecimal string to an integer
# The "range" function creates a range of numbers from 0 to the length of the hexadecimal string
# The "for" loop iterates through the range and converts each pair of characters to an integer using base 16 (hexadecimal)
# The "byte_list" variable is assigned a list of the converted integers
# Print the list of bytes
print(byte_list)
# The "print" function outputs the value of the variable "byte_list"
To decompress it using the P-128 scheme, simply call the `decompress()` method on your decompression object:
# Decompress the compressed point using the P-128 scheme
decompressed_point = decompression.decompress(compressed_point)
# Print the decompressed point in big-endian format (x and y coordinates)
print(decompressed_point)
And that’s it! You now have the original elliptic curve point, which can be used for various purposes such as verifying signatures or performing cryptographic operations.