A truly minimal Python secp256k1 ECDSA? I wrote one.

Step 1: The Finite Field and The Curve Equation

First, we define the parameters of the secp256k1 curve. The equation is y² ≡ x³ + 7 (mod p). We need to define the massive prime `p`, the generator point `G`, and the order of the group `n`.

# secp256k1 curve parameters
P = 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEFFFFFC2F
A = 0
B = 7
# Generator point G
Gx = 0x79BE667EF9DCBBAC55A06295CE870B07029BFCDB2DCE28D959F2815B16F81798
Gy = 0x483ADA7726A3C4655DA4FBFC0E1108A8FD17B448A68554199C47D08FFB10D4B8
G = (Gx, Gy)
# Order of the group
N = 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEBAAEDCE6AF48A03BBFD25E8CD0364141

Step 2: Point Addition and Scalar Multiplication

This is the core of ECC. We need a function for point addition (`P1 + P2`) and a function for scalar multiplication (`k * P`). Point addition has a few cases: adding a point to the point at infinity (our identity element, `None`), adding a point to itself (point doubling), and adding two different points.

The math involves calculating slopes and performing modular arithmetic. A key helper function is the modular inverse, which we can calculate efficiently using Python 3.8+'s `pow(x, -1, p)`.

def modular_inverse(a, n):
    return pow(a, -1, n)

def point_add(P, Q):
    # ... (implementation of geometric point addition)

def scalar_multiply(k, P):
    result = None # Point at infinity
    current = P
    while k > 0:
        if k & 1: # If bit is 1
            result = point_add(result, current)
        current = point_add(current, current) # Double the point
        k >>= 1 # Move to the next bit
    return result

The `scalar_multiply` function uses the efficient double-and-add algorithm. This is our one-way function: easy to compute, hard to reverse.

Step 3: Generating Keys (Private and Public)

This step is surprisingly simple once you have scalar multiplication:

1. Private Key (d): A cryptographically secure random integer between 1 and `N-1`.
2. Public Key (Q): The result of scalar multiplication: `Q = d * G`.

import secrets

def generate_keypair():
    private_key = secrets.randbelow(N - 1) + 1
    public_key = scalar_multiply(private_key, G)
    return private_key, public_key

That's it! `private_key` is just a big number, and `public_key` is a point `(x, y)` on the curve.

Step 4: The ECDSA Signing Algorithm

To sign a message, you need the message hash (`z`) and your private key (`d`). The process is as follows:

1. Generate a secret, single-use random number `k` (the nonce). This must be truly random and never reused with the same private key!
2. Calculate the point `R = k * G`. The x-coordinate of this point is `r`.
3. Calculate `s = (z + r * d) * k⁻¹ (mod N)`.
4. The signature is the pair `(r, s)`.

import hashlib

def sign(private_key, message):
    z = int.from_bytes(hashlib.sha256(message).digest(), 'big')
    k = secrets.randbelow(N - 1) + 1
    
    R = scalar_multiply(k, G)
    r = R[0] % N
    
    k_inv = modular_inverse(k, N)
    s = (k_inv * (z + r * private_key)) % N
    
    return (r, s)

Step 5: The ECDSA Verification Algorithm

To verify a signature `(r, s)`, you need the message hash `z` and the public key `Q`. The magic of the math allows you to verify the signature without ever knowing the private key.

1. Calculate `w = s⁻¹ (mod N)`.
2. Calculate `u1 = z * w (mod N)`.
3. Calculate `u2 = r * w (mod N)`.
4. Calculate the point `P = u1 * G + u2 * Q`.
5. The signature is valid if the x-coordinate of `P` is equal to `r`.

Why does this work? With a bit of algebraic substitution, you can see that if the signature is valid, `P` will always equal the original `R` point (`k * G`) calculated during signing.

def verify(public_key, message, signature):
    z = int.from_bytes(hashlib.sha256(message).digest(), 'big')
    r, s = signature
    
    w = modular_inverse(s, N)
    u1 = (z * w) % N
    u2 = (r * w) % N
    
    P1 = scalar_multiply(u1, G)
    P2 = scalar_multiply(u2, public_key)
    P = point_add(P1, P2)
    
    return P[0] == r

Putting It All Together: A Complete Example

Let's combine everything into a simple script that generates a keypair, signs a message, and verifies it.

if __name__ == '__main__':
    # 1. Generate keys
    my_private_key, my_public_key = generate_keypair()
    print(f"My Private Key: {hex(my_private_key)}")
    print(f"My Public Key (x, y): ({hex(my_public_key[0])}, {hex(my_public_key[1])})")

    # 2. Sign a message
    message = b"A truly minimal Python secp256k1 ECDSA? I wrote one."
    signature = sign(my_private_key, message)
    print(f"\nSignature (r, s): ({hex(signature[0])}, {hex(signature[1])})")

    # 3. Verify the signature
    is_valid = verify(my_public_key, message, signature)
    print(f"\nSignature valid? {is_valid}")

    # 4. Tamper with the message and verify again
    tampered_message = b"Someone else wrote this blog post."
    is_tampered_valid = verify(my_public_key, tampered_message, signature)
    print(f"Tampered message signature valid? {is_tampered_valid}")

Running this code, you'll see the signature successfully verifies for the original message but fails for the tampered one, proving our implementation works!

Why You Absolutely Shouldn't Use This in Production

I cannot stress this enough: this code is for educational purposes only. It is completely insecure for real-world applications. Production-grade libraries are incredibly complex for good reason. Here's a comparison:

Feature	Our Minimal Implementation	Production Library (e.g., `coincurve`)
Goal	Education & Understanding	Security & Performance
Dependencies	Pure Python	Optimized C library (libsecp256k1)
Timing Attacks	Vulnerable	Hardened with constant-time operations
Nonce Generation (k)	Simple (`secrets` module)	RFC 6979 deterministic nonces to prevent reuse
Code Size	~100 lines	Thousands of lines of C and Python
Use Case	Learning, Tinkering	Production Applications

The most famous vulnerability related to a poor implementation is the Sony PS3 hack, which resulted from using a static value for the nonce `k`. This allowed hackers to calculate the console's private key directly from a single signature. Our code is better than that, but it's not hardened against the myriad of other side-channel attacks.

Conclusion: From Black Box to Clear Glass

Building secp256k1 ECDSA from the ground up was an incredibly rewarding experience. What once seemed like impenetrable cryptographic magic is now a logical, step-by-step process rooted in elegant mathematics. We've seen how a private key is just a number, a public key is a point on a curve, and how the properties of elliptic curve arithmetic allow us to create and verify digital signatures securely.

While we should always rely on battle-tested, peer-reviewed libraries for any real application, understanding the fundamentals is empowering. It transforms you from a mere user of technology to someone who grasps its core principles. I encourage you to play with the code, trace the functions, and truly see how the numbers flow. The black box is now open.