What Is Shor's Algorithm and Why Does It Threaten Crypto?

Shor's algorithm, published by mathematician Peter Shor in 1994, is a quantum algorithm that efficiently solves integer factorization and discrete logarithm problems. These mathematical problems form the security foundation of RSA, Diffie-Hellman, and ECDSA cryptography—including the signatures protecting virtually all current cryptocurrencies.

Classical computers require exponential time to factor large numbers or compute discrete logarithms. A 2048-bit RSA key would take classical computers longer than the age of the universe to break. Shor's algorithm reduces this to polynomial time, making these operations feasible on sufficiently powerful quantum computers.

For cryptocurrency, the threat is specific: Shor's algorithm can derive ECDSA private keys from public keys. Bitcoin, Ethereum, and most cryptocurrencies use ECDSA with the secp256k1 curve. Once public keys are exposed (which occurs when addresses are spent from), quantum computers running Shor's algorithm can compute the corresponding private keys.

Running Shor's algorithm against 256-bit ECDSA takes 1,200–1,450 logical qubits, fits inside fewer than 500,000 physical qubits, and completes in minutes — the March 2026 benchmark from Google Quantum AI, working with the Ethereum Foundation and Stanford. That is roughly half the ~2,330 logical qubits earlier estimates assumed. An independent Caltech/Oratomic analysis (Cain et al.) puts the same break at about 26,000 neutral atoms running for a few days. Factoring RSA-2048 now takes under 1 million noisy qubits and under a week (Gidney, May 2025), down from 20 million qubits in 2019 — but RSA is not the near target, because ECDSA-256 is cheaper.

Today's best public hardware is roughly 2,500 physical qubits, none of it fault-tolerant at scale. The machine that closes the gap is IBM's Blue Jay in 2033, with over 2,000 logical qubits on about 100,000 physical — preceded by Starling in 2029 at ~200 logical. Nobody knows when; published hardware roadmaps reach the needed scale between 2028 (IonQ, on paper) and 2033 (IBM Blue Jay).

Post-quantum cryptography addresses this threat by using mathematical problems that Shor's algorithm cannot solve efficiently. Lattice-based cryptography (Kyber) and hash-based signatures (SPHINCS+) have no known efficient quantum attacks.

SynX implements Kyber-768 and SPHINCS+ algorithms specifically because Shor's algorithm provides no advantage against them. It leaves no elliptic-curve or RSA key for Shor's algorithm to attack.