🔐 1) Bitcoin Uses Elliptic Curve Cryptography (ECC)

Bitcoin employs the secp256k1 elliptic curve. A private key k is simply a 256-bit random number. The public key K is obtained via:

    \[ K = k \cdot G \]

where:

  • G is the base point on the elliptic curve,
  • \cdot denotes elliptic curve point multiplication.

The security of this system relies on the Elliptic Curve Discrete Logarithm Problem (ECDLP): given K and G, find k. This is believed to be mathematically hard for properly chosen curves.

🧼 2) Brute-Forcing the Private Key Space

The keyspace has size:

    \[ 2^{256} \approx 1.16 \times 10^{77} \]

For comparison:

  • The estimated number of atoms in the observable universe is \sim 10^{80}.
  • Even with a supercomputer performing 10^{18} guesses per second, it would take on the order of 10^{51} years to exhaust the keyspace.

⚡ 3) No Known Shortcut Algorithms

For secp256k1:

  • There are no known sub-exponential algorithms for solving the ECDLP.
  • Baby-step giant-step or Pollard’s rho algorithm could reduce the problem from O(2^{256}) to about O(2^{128}), but:
    • 2^{128} is still astronomically large.
    • Storing 2^{128} points or computing that many iterations is beyond conceivable technology.

🧠 4) Quantum Computers (Shor’s Algorithm)

In theory, a sufficiently large, error-corrected quantum computer could run Shor’s algorithm to solve the discrete log in polynomial time. But to break secp256k1, estimates suggest:

  • ~1,500–3,000 logical qubits,
  • tens of millions of physical qubits due to error correction overhead,
  • plus long, stable coherence times.

Such a machine does not exist today and is not expected in the immediate future. Motivation for post-quantum crypto

📝 5) Address Type Caveat

If Satoshi’s coins were sent to P2PK (pay-to-public-key) addresses (as some early coins were), the public key is on-chain, so it’s theoretically visible.

For P2PKH (pay-to-public-key-hash) addresses, the public key is not revealed until the first spend. So if those coins remain unspent, only the hash of the public key is public — making quantum attacks even harder.

đŸš« In Summary

To figure out Satoshi’s private key from the public key, you’d have to:

  • Solve the elliptic curve discrete log problem on secp256k1, or
  • Use a large fault-tolerant quantum computer (which does not yet exist), or
  • Exploit a flaw in key generation (e.g., bad RNG or reused nonces), which is unlikely in Satoshi’s case.

Without one of these, it’s effectively impossible. ✅

 

Anuj holds professional certifications in Google Cloud, AWS as well as certifications in Docker and App Performance Tools such as New Relic. He specializes in Cloud Security, Data Encryption and Container Technologies.

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Anuj Varma – who has written posts on Anuj Varma, Hands-On Technology Architect, Clean Air Activist.

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