Montgomery’s Binary Algorithm for Scalar Multiplication in E(Fp)

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It’s called Montgomery’s Binary Algorithm for Scalar Multiplication in E(Fp), and it’s pretty cool.

So, let’s say you have an elliptic curve over a finite field Fp with equation y^2 = x^3 + ax + b (where p is prime and all that jazz). And let’s say you want to compute the point P * k for some integer k. This can be done using the traditional method, which involves adding up a bunch of points on the curve until you get to your final destination. But if k has lots of digits (which it often does in cryptography), this can take forever.

Enter Montgomery’s Binary Algorithm! Instead of doing all that point-adding, we’re going to use some fancy math tricks to make things faster and easier. Here’s how it works:

1. First, let’s convert k into binary form (i.e., write it as a string of 0s and 1s). For example, if k = 27, we can write that as 11001 in binary.

2. Next, we’re going to compute the “Montgomery ladder” for each bit in our binary representation. This involves doing some fancy math stuff with points on the curve and doubling them up until we get to a certain point (which is called the “basepoint”). Here’s an example:

Let k = 10110, so we want to compute P * 2^3 * 5. First, let’s convert that into binary form: 10000. Then, for each bit in our binary representation (starting from the right), we’re going to do a “Montgomery ladder” using the following steps:

– If the current bit is 1, compute P * R and add it to our running total (which starts at the basepoint). Otherwise (i.e., if the current bit is 0), just double up our running total until we get back to the basepoint. Here’s what that looks like in code:

# Import necessary libraries
from cryptography.hazmat.primitives import serialization, ec
import random

# Generate a private key and corresponding public key for an elliptic curve over Fp (where p is prime)
# Choose a random integer between 1 and p-1 as our private key
private_key = random.randint(1, p-1) 

# Compute the basepoint using the curve's parameters (i.e., a and b)
basepoint = ec.curve.CurvePoint(x=0, y=0).multiply(ec.curve.FieldElement(a), ec.curve.FieldElement(b)) 

# Calculate our public key by multiplying the basepoint with our private key modulo p
public_key = basepoint * private_key % p 

# The following code implements the "Montgomery ladder" algorithm for scalar multiplication on an elliptic curve
# This algorithm is used to efficiently compute the scalar multiplication of a point on the curve
# It involves doubling and adding the point based on the binary representation of the scalar
# If the current bit is 1, compute P * R and add it to our running total (which starts at the basepoint)
# Otherwise (i.e., if the current bit is 0), just double up our running total until we get back to the basepoint
# Here's what that looks like in code:

# Initialize a running total starting at the basepoint
running_total = basepoint 

# Loop through each bit in the binary representation of the private key
for bit in bin(private_key)[2:]: # [2:] removes the "0b" prefix from the binary representation
    if bit == '1': # If the current bit is 1, compute P * R and add it to our running total
        running_total = running_total + basepoint 
    else: # Otherwise (i.e., if the current bit is 0), just double up our running total until we get back to the basepoint
        running_total = running_total.double() 

# The final value of the running total is our public key
public_key = running_total % p

And that’s it! With Montgomery’s Binary Algorithm, we can compute scalar multiplication on elliptic curves over finite fields much faster than traditional methods. So next time you need to do some fancy math stuff for your cryptography project, give this algorithm a try and see how it works out for you!

Of course, there are plenty of other algorithms out there that are even more efficient (like the “Miller-Rabin” or “Baby-Step Giant-Step” methods), but Montgomery’s Binary Algorithm is still pretty cool in its own right. And who knows? Maybe someday it’ll become the go-to algorithm for all your elliptic curve needs!

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