Efficient Scalar Multiplication in ECC Signing

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Alright, one of the most important aspects of ECC signing: efficient scalar multiplication. This is where things get really juicy (or nerdy, depending on your perspective). Chill out, don’t worry, because we’re going to break it down in a way that even your grandma could understand!

First off, let’s start with the basics. When you sign something using ECC, you need to multiply a secret number (your private key) by a point on the curve (the public key). This is called scalar multiplication because we’re essentially scaling up or down the point based on our secret number.

Now, here’s where things get interesting. If your secret number is really long and complicated (like it should be for security reasons), multiplying it by a point can take forever. But there are some clever tricks that allow us to do this much more efficiently.

One of the most popular methods is called windowing, which involves breaking up our secret number into smaller chunks and then multiplying each chunk separately. This reduces the number of operations we need to perform and makes things a lot faster.

Here’s an example in Python using the `tinyec` library:

# Import the necessary libraries
from tinyec import secp256k1, Point, Gf2x

# Define our private key (a secret number)
priv_key = 0b101100110001011001010001101100100101011000001000000000000
# Convert our private key to a list of chunks (each chunk is 4 bits)
chunks = [int(digit) for digit in bin(priv_key)[2:].zfill(512)] # Convert the private key to binary and fill it with leading zeros to make it 512 bits long, then convert each chunk to an integer and store them in a list

# Define the base point on the curve (our public key)
base_point = Point(secp256k1.G, secp256k1.Fp) # Initialize the base point using the secp256k1 curve and the finite field Fp

# Initialize our result to be zero
result = Point(0, 0) # Initialize the result point to be the point at infinity (0,0)

# Loop through each chunk and multiply it by the base point using windowing
for i in range(len(chunks)):
    # Calculate the x-coordinate of the current chunk (as a polynomial over GF(2^8))
    x_poly = Gf2x([0] * 16) # Initialize a polynomial with 16 coefficients, all set to 0
    for j, bit in enumerate(reversed(bin(chunks[i])[2:])): # Loop through each bit in the binary representation of the current chunk, starting from the least significant bit
        if int(bit): # If the bit is 1
            x_poly[j] = secp256k1.Fp(1) # Set the corresponding coefficient in the polynomial to 1, using the finite field Fp
    
    # Calculate the y-coordinate of the current chunk (as a polynomial over GF(2^8))
    y_poly = Gf2x([0] * 16) # Initialize a polynomial with 16 coefficients, all set to 0
    for j, bit in enumerate(reversed(bin(chunks[i])[2:])): # Loop through each bit in the binary representation of the current chunk, starting from the least significant bit
        if int(bit): # If the bit is 1
            y_poly[j] = secp256k1.Fp(1) # Set the corresponding coefficient in the polynomial to 1, using the finite field Fp
    
    # Calculate the point on the curve corresponding to this chunk (using a precomputation table for efficiency)
    chunk_point = base_point * Gf2x([0] * 16, y_poly) # Multiply the base point by the polynomial representing the y-coordinate of the current chunk, using a precomputation table for efficiency
    
    # Add the result of each chunk together using another clever trick called projective coordinates
    result += chunk_point.affine() # Convert the chunk point to affine coordinates and add it to the result point

# Convert our final result back to a point on the curve (in affine coordinates)
result = Point(int(result[0][1]), int(result[0][0])) # Convert the result point to affine coordinates and store it as the final result

Woaaw!, that was quite an adventure! But now you know how to perform efficient scalar multiplication in ECC signing. And who knows? Maybe one day your grandma will be able to do it too (although we can’t guarantee her success).

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