🖋️ECDSA

Elliptic Curve Digital Signature Algorithm

Signature algorithm

• Use the secret scalar $e$ to compute the public point$P$, by doing a scalar multiplication with $G$: $eG = P$.
• Pick a random (secret) scalar $k$, and perform a scalar multiplication with $G$ to get a random point$R$. $kG = R$
• Use the above variables in the general equation of ECDSA is: $uG + vP = R$, where, $u = z/s$, $v=r/s$ $z$ is the 256-bit message being signed
• Simplify the resulting equation to get the $s$ component of the signature:

  $s = (z + re) / k$

from random import randint

@dataclass
class PrivateKey:
    secret: int
    
    def sign(self, z: int) -> "Signature":
        e = self.secret
        k = randint(0, N)
        R = k * G
        r = R.x.value
        k_inv = pow(k, -1, N)  # Python 3.8+
        s = ((z + r*e) * k_inv) % N
        
        return Signature(r, s)

Apart from the fact that e is a secret number, the security of ECDSA also relies on the condition that k is also very random and secret.

We'll learn about the consequences of not having a random kin the next section.

Verification algorithm

• Given: (r, s) is the signature, z is the 256 bit message being signed, and P is the public key of the signer.

• Calculate: $u = z/s$, $v=r/s$.
• Calculate $uG + vP = R$.
• Signature is valid is $Rx$ is equal to $r$.

@dataclass
class Signature:
    r: int
    s: int
    
    def verify(self, z: int, pub_key: Point) -> bool:
        s_inv = pow(self.s, -1, N)  # Python 3.8+
        u = (z * s_inv) % N
        v = (self.r * s_inv) % N
        
        return (u*G + v*pub_key).x.value == self.r

Testing our ECDSA implementation

pub = Point(
    x=0x887387E452B8EACC4ACFDE10D9AAF7F6D9A0F975AABB10D006E4DA568744D06C,
    y=0x61DE6D95231CD89026E286DF3B6AE4A894A3378E393E93A0F45B666329A0AE34,
    curve=secp256k1
)

# Test case 1: verify authenticity
z = 0xEC208BAA0FC1C19F708A9CA96FDEFF3AC3F230BB4A7BA4AEDE4942AD003C0F60
r = 0xAC8D1C87E51D0D441BE8B3DD5B05C8795B48875DFFE00B7FFCFAC23010D3A395
s = 0x68342CEFF8935EDEDD102DD876FFD6BA72D6A427A3EDB13D26EB0781CB423C4

assert Signature(r, s).verify(z, pub)

# Test case 2: verify authenticity for different signature w/ same P
z = 0x7C076FF316692A3D7EB3C3BB0F8B1488CF72E1AFCD929E29307032997A838A3D
r = 0xEFF69EF2B1BD93A66ED5219ADD4FB51E11A840F404876325A1E8FFE0529A2C
s = 0xC7207FEE197D27C618AEA621406F6BF5EF6FCA38681D82B2F06FDDBDCE6FEAB6
assert Signature(r, s).verify(z, pub)

# Test case 3: sign and verify
e = PrivateKey(randint(0, N))  # generate a private key
pub = e.secret * G  # public point corresponding to e
z = randint(0, 2 ** 256)  # generate a random message for testing
signature: Signature = e.sign(z)
assert signature.verify(z, pub)

Resources

• https://www.instructables.com/id/Understanding-how-ECDSA-protects-your-data