Welcome to BitDevsUyo's Thirteenth Socratic Seminar!

Thank you to our sponsor Btrust

📍Details

  • Date: July 25, 2026
  • Time: 14:00- 16:00
  • Location: CafeOne UYO, 162 Oron Road, Uyo (Map)

Agenda

1. Digital Signature Fundamentals & Functional Flow

  • Digital Signatures (ECDSA)
  • High-Level Signature Function:
    • Signature = Sign(Hash(Message), Private_Key)
  • Verification Function:
    • Verify(Message, Signature, Public_Key) -> True / False

2. Practical Math Breakdown (Step-by-Step Whiteboard Example)

  • Alice's Setup:

    • Private Key (x) = 3
    • Generator (G) = 5
    • Public Key (Xg) = x * G = 3 * 5 = 15
    • Private Nonce (k) = 7
    • Public Nonce (kG) = k * G = 7 * 5 = 35
    • Message Challenge (e) = 11

  • Signing Process:

    • Formula: S = k + (e * x)
    • Calculation: S = 7 + (11 * 3) = 7 + 33 = 40
    • Output Signature: S = 40

  • Verification Process:

    • Left-Hand Side: S * G = 40 * 5 = 200
    • Term 2: e * Xg = 11 * 15 = 165
    • Verify Equation: S * G == kG + (e * Xg)
    • Calculation Check: 200 == 35 + 165
    • Result: 200 == 200 (Signature Verified!)

3. ECDSA Mathematics: Signing Step-by-Step

  • Generating a Signature (r, s)
  • Step 1: Message Digest: Compute z = SHA256(SHA256(message))
  • Step 2: Ephemeral Nonce: Select random secret scalar k
  • Step 3: Calculate r: Compute point R = k * G, set r = R.x mod n
  • Step 4: Calculate s: Compute s = k^-1 * (z + r * d) mod n
  • Signature Output Pair: (r, s)

4. ECDSA Mathematics: Verification Step-by-Step

  • Signature Verification
  • Step 1: Compute Inverse: w = s^-1 mod n
  • Step 2: Compute Scalar Multipliers:
    • u1 = z * w mod n
    • u2 = r * w mod n
  • Step 3: Reconstruct Verification Point: P = (u1 * G) + (u2 * Q)
  • Verification Rule: Signature is valid if P.x mod n == r

5. Nonce Security & Practical Vulnerabilities

  • Nonce Reuse and Why It's Dangerous
  • Private Key Extraction from Nonce Reuse:
    • k = (z1 - z2) / (s1 - s2) mod n
    • d = ((s1 * k) - z1) / r mod n
  • Deterministic Nonces (RFC 6979): k = HMAC-SHA256(private_key, message_hash)
  • Low-s Rule (BIP 62): Require s <= n / 2 to prevent signature malleability

🏛 House Rules