Shor’s algorithm is the mathematical proof that quantum computers will break Bitcoin. Published by Peter Shor in 1994, it demonstrates that a quantum computer can efficiently solve the integer factorisation and discrete logarithm problems that underpin RSA and ECDSA. Since Bitcoin’s wallet security relies entirely on ECDSA secp256k1 — a discrete logarithm problem — Shor’s algorithm breaks it completely. This is not theoretical risk. It is proven mathematics applied to known cryptographic foundations.
Shor’s algorithm is a quantum algorithm that solves integer factorisation and discrete logarithm problems in polynomial time. On a classical computer, factoring a large integer or computing a discrete logarithm takes exponential time — so long it is computationally infeasible. On a quantum computer with sufficient qubits and error correction, Shor’s algorithm solves the same problems in hours. The security of RSA relies on factoring difficulty. The security of ECDSA relies on discrete logarithm difficulty. Both are broken by Shor’s algorithm.
Bitcoin’s ECDSA secp256k1 works on the elliptic curve discrete logarithm problem (ECDLP): given a point Q = k*G on the curve, where G is the generator point, find k (the private key) given only Q (the public key). On a classical computer this takes roughly 2^128 operations — effectively impossible. Shor’s algorithm applied to ECDLP reduces this to polynomial time on a quantum computer with sufficient logical qubits. The private key is derived from the public key in hours. Every Bitcoin address that has ever broadcast a transaction — exposing its public key — becomes directly stealable.
| Target | Physical Qubits Required (est.) | Current Best (2026) | Gap |
|---|---|---|---|
| 256-bit ECDSA (Bitcoin) | ~4,000 logical / ~4M physical | IBM ~1,000 physical | ~4,000x |
| 2048-bit RSA | ~4,000 logical / ~20M physical | IBM ~1,000 physical | ~20,000x |
| 128-bit AES | Grover only — much higher bar | N/A | Much larger |
Grover’s algorithm provides a quadratic speedup for searching unsorted databases — reducing the effective security of symmetric cryptography and hash functions by half. For Bitcoin’s SHA-256 mining, this means effective security drops from 256-bit to 128-bit. This is weakening but not catastrophic — SHA-256 can be upgraded by doubling key length. ECDSA cannot be saved this way: Shor’s provides an exponential speedup that no key length increase can defeat.
Current quantum computers have ~1,000 physical qubits with high error rates. Breaking 256-bit ECDSA requires approximately 4,000 error-corrected logical qubits — each logical qubit requiring roughly 1,000 physical qubits for error correction at current rates, giving ~4 million physical qubits needed. IBM projects doubling qubit counts annually. Conservative estimates: 2031-2036 for a CRQC capable of running Shor’s algorithm against Bitcoin. But harvest-now-decrypt-later attacks collect on-chain data today for that future decryption.
CRYSTALS-Kyber and CRYSTALS-Dilithium — BMIC’s cryptographic foundation — are based on the Module Learning With Errors (MLWE) lattice problem. No variant of Shor’s algorithm applies to MLWE. No other known quantum algorithm efficiently solves MLWE. NIST confirmed this after seven years of evaluation by the world’s top cryptographers. BMIC also uses ERC-4337 to hide public keys entirely — so even if Shor’s algorithm were run, there would be no public keys to process against BMIC wallets.
What is Shor’s algorithm?
A quantum algorithm that solves integer factorisation and discrete logarithm problems in polynomial time — breaking RSA and ECDSA. Published by Peter Shor in 1994. Proven to work mathematically; requires only sufficient quantum hardware.
How does Shor’s algorithm break Bitcoin?
It solves the elliptic curve discrete logarithm problem (ECDLP) efficiently — deriving a Bitcoin private key from an exposed public key in hours. Every address that has ever made a transaction is vulnerable.
How many qubits are needed to break Bitcoin?
Approximately 4,000 error-corrected logical qubits — requiring roughly 4 million physical qubits at current error rates. IBM currently has ~1,000 physical qubits. Timeline: expert consensus 2031-2036.
Does Shor’s algorithm break BMIC?
No. BMIC uses CRYSTALS-Kyber and Dilithium — based on MLWE lattice problems. No variant of Shor’s algorithm applies. NIST confirmed this after seven years of global evaluation.
How do I protect my Bitcoin from Shor’s algorithm?
Buy BMIC in the presale at $0.049999. CRYSTALS-Kyber and Dilithium plus ERC-4337 hidden keys. The only presale wallet token immune to Shor’s algorithm.
Shor’s Algorithm Cannot Break BMIC — The Only Presale With Proof
CRYSTALS-Kyber + Dilithium + ERC-4337. Presale $0.049999.
Buy BMIC Now