ECDSA

What is ECDSA?

ECDSA is a cryptographic algorithm used for digital signatures based on elliptic curve cryptography (ECC). It’s widely used in secure systems like:

• Bitcoin and Ethereum
• TLS (HTTPS)
• JWTs
• Secure messaging (e.g. Signal)

It’s a more efficient version of DSA (Digital Signature Algorithm) using elliptic curves — providing strong security with smaller keys.

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🧠 What is it used for?

ECDSA is used to:

• ✅ Sign data (e.g., a transaction, message, or file)
• ✅ Verify that the data came from a particular private key (authenticity)
• ✅ Detect if the data was changed (integrity)

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📦 Key components

1. Elliptic Curve
   • A mathematical curve over a finite field, e.g., secp256k1 (used in Bitcoin), or P-256.
   • Public/private key pairs are points on this curve.
2. Private key (d)
   • A random scalar (integer)
   • Must be kept secret
3. Public key (Q = d * G)
   • A point on the curve derived from the private key and base point G
4. Base point (G)
   • A publicly known generator point on the curve

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✍️ Signing (ECDSA Sign)

To sign a message:

1. Hash the message to get a digest z (e.g. using SHA-256)
2. Choose a random nonce k (must be unique for each signature!)
3. Compute R = k * G (a point on the curve)
4. Let r = R.x mod n (where n is the curve order)
5. Compute s = k⁻¹ * (z + r * d) mod n

The signature is the pair (r, s).

⚠️ If k is reused, the private key d can be recovered! (This happened to Sony in 2010.)

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✅ Verifying (ECDSA Verify)

Given a message msg, signature (r, s), and public key Q, you:

1. Hash the message: z = HASH(msg)
2. Compute w = s⁻¹ mod n
3. Compute u1 = z * w mod n and u2 = r * w mod n
4. Compute point R = u1 * G + u2 * Q
5. Signature is valid if R.x mod n == r

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🔁 Summary

Step	Math operation	Purpose
KeyGen	Q = d * G	Create public key
Sign	r = (k * G).x, s = k⁻¹(z + r * d) mod n	Sign message
Verify	R = u1 * G + u2 * Q, check R.x == r	Verify message