# Elliptic Curve Diffie-Hellman (ECDH) and the Elliptic Curve Digital Signature Algorithm (ECDSA)

#### Why Move to Elliptic Curves?

Both **ECDH** (Elliptic Curve Diffie-Hellman) and **ECDSA** (Elliptic Curve Digital Signature Algorithm) follow the **same principles** as standard Diffie-Hellman and DSA.

The main difference:

* **Standard DH/DSA:** Use **exponentiation** (gᵃ mod n).
* **Elliptic Curve versions:** Use **multiplication on elliptic curve points**.

👉 The benefit: Elliptic curve math offers **equivalent security with much smaller key sizes**.

* 3,072-bit keys in DH/DSA ≈ 256-bit keys in ECDH/ECDSA.

***

#### Key Exchange: DH vs. ECDH

**Standard Diffie-Hellman:**

`sv = (gᵃ)ᵇ mod 2³⁰⁷² = (gᵇ)ᵃ mod 2³⁰⁷²`

 

**Elliptic Curve Diffie-Hellman (ECDH):**

`sv = g·b·a mod 2²⁵⁶ = g·a·b mod 2²⁵⁶`

 

👉 **Result:** Both methods allow Alice and Bob to compute the same **shared secret (sv)**, but elliptic curves achieve this with much **smaller numbers**.

***

#### Digital Signatures: DSA vs. ECDSA

**Standard DSA (on message hash m):**

1. Private key: `a` (random integer < 2³⁰⁷²).
2. Public key: `A = gᵃ`.
3. Random one-time key: `k`.
4. Signature values:
   * `r = (gᵏ mod p) mod q`
   * `s = (k⁻¹(m + a·r)) mod q`

&#x20;

**ECDSA (on message hash m):**

1. Private key: `a` (random integer < 2²⁵⁶).
2. Public key: `A = g·a`.
3. Random one-time key: `k`.
4. Signature values:
   * `R = k·g`
   * `r = Rₓ mod q` (x-coordinate of R)
   * `s = (m + a·r)·k⁻¹ mod q`

👉 **Key difference:** Elliptic curve math replaces exponentiation with **point multiplication**.

***

#### Signature Verification

**Standard DSA verification:**

```
z = s⁻¹ mod q
w₁ = m·z mod q
w₂ = r·z mod q
v  = (gʷ¹ · Aʷ² mod p) mod q
Valid if v == r
```

&#x20;

**ECDSA verification:**

```
z = s⁻¹ mod q
w₁ = m·z mod q
w₂ = r·z mod q
v  = (g·w₁) + (A·w₂)
Valid if vₓ == r mod q
```

***

#### Why It Matters for BSV

Although **BSV’s UTXO model** is not about encrypting all communications, it is still built on two fundamental cryptographic ideas:

1. **Shared secrets** (ECDH provides a secure foundation).
2. **Digital signatures** (ECDSA secures ownership and transaction authenticity).

✅ **Summary:** By using elliptic curve cryptography, BSV achieves **stronger security with smaller keys**, making transactions more efficient while preserving the essential principles of **trust and integrity**.


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