Bitcoin: Compressed vs. Uncompressed Public Keys

Introduction to Public Keys

In Bitcoin, a public key is a point on the secp256k1 elliptic curve represented as coordinates (x, y). For example, the hexadecimal public key:

04d061e9c5891f579fd548cfd22ff29f5c642714cc7e7a9215f0071ef5a5723f691757b28e31be71f09f24673eed52348e58d53bcfd26f4d96ec6bf1489eab429d

Translates to:

• x: 0xd061e9c5891f579fd548cfd22ff29f5c642714cc7e7a9215f0071ef5a5723f69
• y: 0x1757b28e31be71f09f24673eed52348e58d53bcfd26f4d96ec6bf1489eab429d

These coordinates must satisfy the elliptic curve equation:
y² = x³ + 7 (modulo the curve’s finite field).

👉 Learn more about elliptic curve cryptography

Why Do the Coordinates Satisfy the Equation?

The Bitcoin curve (secp256k1) operates within a finite field defined by prime Pcurve. Verification in Python:

Pcurve = 2**256 - 2**32 - 2**9 - 2**8 - 2**7 - 2**6 - 2**4 - 1
x = 0xd061e9c5891f579fd548cfd22ff29f5c642714cc7e7a9215f0071ef5a5723f69
y = 0x1757b28e31be71f09f24673eed52348e58d53bcfd26f4d96ec6bf1489eab429d
assert (x**3 + 7) % Pcurve == (y**2) % Pcurve

Smaller Example (Finite Field 𝔽₂₃):
For y² = x³ + 7 mod 23, the equation must hold:
((x³) + 7) % 23 == (y²) % 23.

Types of Public Keys

1. Uncompressed Public Key

   • Format: 04 (prefix) + x + y
   • Example:
     04d061e9c5891f579fd548cfd22ff29f5c642714cc7e7a9215f0071ef5a5723f691757b28e31be71f09f24673eed52348e58d53bcfd26f4d96ec6bf1489eab429d

2. Compressed Public Key

   • Format:

     • 02 + x (if y is even)
     • 03 + x (if y is odd)
   • Example (since y ends in 0xd, odd):
     03d061e9c5891f579fd548cfd22ff29f5c642714cc7e7a9215f0071ef5a5723f69

👉 Explore Bitcoin key generation

Bitcoin Address Examples

Same private key generates different addresses for compressed vs. uncompressed keys:

FAQs

Q1: Why use compressed public keys?
A1: They save space (33 bytes vs. 65 bytes) and are now the standard, reducing blockchain bloat.

Q2: Do compressed/uncompressed keys affect security?
A2: No—both derive from the same private key and offer identical security.

Q3: How can I convert between key formats?
A3: Use cryptographic libraries (e.g., OpenSSL) to recompute y from x via the curve equation.

Q4: Are uncompressed keys still used?
A4: Rarely. Most modern wallets default to compressed keys for efficiency.

Key Terms

• secp256k1: Bitcoin’s elliptic curve.
• Finite Field (Pcurve): A prime-modulo space for curve arithmetic.
• Public Key Formats: Compressed (02/03), Uncompressed (04).

For further reading:
👉 Bitcoin key mechanics

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