How Private Keys Generate All Addresses

Overview

In Bitcoin, a single private key can generate multiple address formats. This is possible because all address types are derived from the same underlying public key, with each format applying different encoding and hashing operations. Understanding this relationship is fundamental to Bitcoin key management and wallet implementation.

The Derivation Chain

Address generation follows a deterministic chain of cryptographic operations:

Each step in this chain is a one-way function, meaning it is computationally infeasible to reverse the process. This ensures that addresses can be freely shared without exposing the private key.

Step 1: Private Key Generation

A Bitcoin private key is a randomly generated 256-bit number (32 bytes). It must fall within the valid range for the secp256k1 elliptic curve:

Valid range: From 1 to 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEBAAEDCE6AF48A03BBFD25E8CD0364140

The private key can be represented in several formats:

• Raw Hex: 64 hexadecimal characters.
• WIF (Wallet Import Format): Base58Check encoded with version prefix.
• WIF Compressed: WIF with compression flag indicating compressed public key usage.

# Raw Hex (32 bytes)
e8f32e723decf4051aefac8e2c93c9c5b214313817cdb01a1494b917c8436b35

# WIF (Uncompressed)
5Kb8kLf9zgWQnogidDA76MzPL6TsZZY36hWXMssSzNydYXYB9KF

# WIF (Compressed)
L53fCHmQhbNp1B4JipfBtfeHZH7cAibzG9oK19XfiFzxHgAkz6JK

# Raw Hex (32 bytes)
e8f32e723decf4051aefac8e2c93c9c5b214313817cdb01a1494b917c8436b35

# WIF (Uncompressed)
5Kb8kLf9zgWQnogidDA76MzPL6TsZZY36hWXMssSzNydYXYB9KF

# WIF (Compressed)
L53fCHmQhbNp1B4JipfBtfeHZH7cAibzG9oK19XfiFzxHgAkz6JK

Step 2: Public Key Derivation

The public key is derived from the private key using elliptic curve multiplication on the secp256k1 curve:

Where G is the generator point of the secp256k1 curve. This operation is computationally easy in one direction but infeasible to reverse (the discrete logarithm problem).

Public keys can be represented in two formats:

• Uncompressed Public Key (65 bytes).
• Compressed Public Key (33 bytes).

Uncompressed Public Key

The prefix 04 indicates an uncompressed key containing both x and y coordinates.

Compressed Public Key

Compressed keys store only the x-coordinate with a prefix indicating the y-coordinate's parity:

• 02: Indicates the y-coordinate is even.
• 03: Indicates the y-coordinate is odd.

The full y-coordinate can be mathematically recovered from the x-coordinate and the curve equation.

# Uncompressed (65 bytes)
04
79be667ef9dcbbac55a06295ce870b07029bfcdb2dce28d959f2815b16f81798  # x
483ada7726a3c4655da4fbfc0e1108a8fd17b448a68554199c47d08ffb10d4b8  # y

# Uncompressed Full
0479be667ef9dcbbac55a06295ce870b07029bfcdb2dce28d959f2815b16f81798483ada7726a3c4655da4fbfc0e1108a8fd17b448a68554199c47d08ffb10d4b8

# Compressed (33 bytes)
02
79be667ef9dcbbac55a06295ce870b07029bfcdb2dce28d959f2815b16f81798

# Compressed Full
0279be667ef9dcbbac55a06295ce870b07029bfcdb2dce28d959f2815b16f81798

# Uncompressed (65 bytes)
04
79be667ef9dcbbac55a06295ce870b07029bfcdb2dce28d959f2815b16f81798  # x
483ada7726a3c4655da4fbfc0e1108a8fd17b448a68554199c47d08ffb10d4b8  # y

# Uncompressed Full
0479be667ef9dcbbac55a06295ce870b07029bfcdb2dce28d959f2815b16f81798483ada7726a3c4655da4fbfc0e1108a8fd17b448a68554199c47d08ffb10d4b8

# Compressed (33 bytes)
02
79be667ef9dcbbac55a06295ce870b07029bfcdb2dce28d959f2815b16f81798

# Compressed Full
0279be667ef9dcbbac55a06295ce870b07029bfcdb2dce28d959f2815b16f81798

Step 3: Public Key Hashing

Most Bitcoin addresses use a hash of the public key rather than the raw public key. This provides an additional layer of security and produces shorter addresses.

• HASH160: P2PKH, P2WPKH.
• SHA-256: P2TR.

Step 4: Address Encoding

The final step converts the public key or its hash into a human-readable address format. Different address types use different encoding schemes.

• Base58Check: Used by legacy addresses (P2PKH, P2SH).
• Bech32: Used by native SegWit v0 addresses (P2WPKH, P2WSH).
• Bech32m: Used by SegWit v1+ addresses (P2TR).

One Key, Multiple Addresses

A single private key produces different addresses for each format, but all addresses represent the same underlying control.

Funds sent to any of these addresses can be spent using the same private key. However, the signing process differs slightly for each address type due to different script structures and signature algorithms.

Hierarchical Deterministic (HD) Derivation

While a single private key can generate multiple address formats, modern wallets use HD derivation to generate multiple private keys from a single seed.

Each derived key follows the same process described above to generate addresses. This enables:

• Unlimited Addresses: Generate new addresses for each transaction.
• Single Backup: One mnemonic phrase recovers all derived keys.
• Account Separation: Organize keys into accounts for different purposes.
• Address Type Paths: BIP44/49/84/86 define derivation paths for each address format.

Standard Derivation Paths