Shor's Algorithm

Definition

Shor's algorithm is a quantum algorithm discovered by Peter Shor in 1994 that efficiently factors large integers and computes discrete logarithms. When run on a sufficiently powerful quantum computer, it breaks RSA, ECDSA, and all elliptic curve cryptography used by current cryptocurrencies.

Technical Explanation

Shor's algorithm exploits quantum superposition and interference to find periodicities in modular exponentiation. For factoring an n-bit integer, classical algorithms require exponential time; Shor's algorithm requires only O(n³) operations—polynomial rather than exponential.

For ECDSA (used by Bitcoin, Ethereum, and most cryptocurrencies), Shor's algorithm solves the discrete logarithm problem on elliptic curves. Given a public key, the private key can be computed in polynomial time. A quantum computer with approximately 2,500-4,000 logical qubits could break 256-bit ECDSA.

SynX Relevance

SynX was designed specifically to resist Shor's algorithm. Kyber-768 key encapsulation and SPHINCS+ signatures use mathematical problems (lattice and hash-based) that Shor's algorithm cannot solve efficiently. SynX transactions remain secure regardless of quantum computing advances.

Frequently Asked Questions

When will Shor's algorithm break Bitcoin?: Estimates range from 2030-2040 for cryptographically relevant quantum computers; timeline is uncertain.
Does Shor's algorithm break all cryptography?: No—only cryptography based on factoring or discrete logarithms. Lattice and hash-based schemes are immune.
Can Shor's algorithm be improved?: Optimizations exist, but the fundamental capability requires large-scale quantum computers.

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