# ECDSA (secp256k1) for Bitcoin Transaction

Now that you have a **high-level understanding** of BSV transactions, let’s look more closely at the **Elliptic Curve Digital Signature Algorithm (ECDSA)** with the secp256k1 curve. This is the algorithm Bitcoin SV uses for **key generation, signing, and verification**.

While the algorithm itself is standard, there are some **subtle differences in its application within BSV**.

***

### Private and Public Key Generation

The first step in digital signing is creating a **private–public key pair**.

**Equation:**

 `Public Key = Private Key × Base Point`

* The **private key** is simply a large random integer.
* The **base point** is a special point on the elliptic curve that generates the entire cyclic group when combined repeatedly.
* The **public key** is a point on the curve, represented with both **x** and **y** coordinates.

<figure><img src="/files/O1Zc5knls1nRjlkP8Zjb" alt=""><figcaption></figcaption></figure>

A private key is a **256-bit random number**. Wallets usually encode it into **WIF (Wallet Import Format)** for easier handling.

***

#### Public Key Representation

Once the private key is created, the **public key** is derived using the equation above.

* **Compressed format (33 bytes):** Stores the x-coordinate plus a prefix indicating whether y is odd or even.
* **Uncompressed format (65 bytes):** Stores both x and y coordinates explicitly.

Most wallets use the **compressed form** in practice.

***

#### Best Practices for Wallets

Wallets follow two important practices for **security, privacy, and usability**:

1. **Security & Privacy:**
   * A new key pair is generated for each transaction.
   * This prevents private key reuse (which could expose multiple transactions if compromised).
   * It also avoids easy traceability of funds across transactions.
2. **Usability:**
   * Requiring backups for every new key pair would be inconvenient.
   * To solve this, wallets use **Hierarchical Deterministic (HD) keys**:
     * All keys are derived from a single **seed key**.
     * The seed can be backed up once, instead of after every transaction.
     * The seed is often encoded into a **word list** (mnemonic phrase), easier to manage than a long hex string.

***

### Message Signing and Verification in BSV

In BSV, digital signatures are always part of a **transaction script**:

* **Inputs** contain an **unlocking script (scriptSig)** → includes the signature and public key.
* **Outputs** contain a **locking script (scriptPubKey)** → includes a public key or a **public key hash** (created by SHA-256 followed by RIPEMD-160).

\[GIF: Unlocking script combining with locking script for verification]

Transactions use a **standardized template** with these elements. Verification works by combining the unlocking and locking scripts, then evaluating the result as **true or false**.

Note: For enterprise applications, a signature may sometimes need to be checked directly against a public key before reaching the network. However, applications usually exchange **addresses** (public key hashes encoded in Base58) rather than raw public keys.

***

### Signature Format in BSV

A signature in ECDSA is a pair of integers **(r, s)**. In BSV, signatures follow the **\[DER, SIGHASH] format**:

1. **DER (Distinguished Encoding Rules):**
   * Defines how to serialize signatures in binary format.
   * Structure: `[header, length, rheader, rlength, r, sheader, slength, s]`
   * Example: `header = 0x30` (static), `rlength` = length of r, `s` = big-endian encoding of s.
2. **SIGHASH (Signature Hash):**
   * A flag that tells which parts of the transaction are signed.
   * Options include:
     * `SIGHASH_ALL (0x01)` → signs all inputs and outputs.
     * `SIGHASH_NONE (0x02)` → signs all inputs, no outputs.
     * `SIGHASH_SINGLE (0x03)` → signs inputs and only the output with the same index.
     * Variations with **ANYONECANPAY** restrict signing to a single input while leaving outputs flexible.

Most transactions use **SIGHASH\_ALL**.

***

#### Additional Flag: SIGHASH\_FORKID

In BSV, all signatures also require **SIGHASH\_FORKID**, which **adds 0x40** to the chosen SIGHASH value.

* Example: `SIGHASH_ALL (0x01)` becomes `0x41`.

***

### Key Takeaway

* **Private keys** are large random numbers; **public keys** are derived points on the elliptic curve.
* Wallets improve security and usability with **new key pairs per transaction** and **HD keys**.
* In BSV, **signatures live inside transaction scripts**, combining unlocking and locking scripts for verification.
* The **DER format** standardizes signatures, while **SIGHASH flags** control which parts of a transaction are signed.
* Most transactions use **SIGHASH\_ALL + FORKID**.


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