Elliptic curve cryptography (ECC)

ECC relies on a mathematical problem that is easy in one direction and practically impossible to reverse: multiplying a starting point on a curve by a large secret number is fast, but recovering that number from the result is not. The secret number is the private key; the resulting point is the public key. ECC offers the same security as older systems like RSA with much shorter keys, which saves space and computation.

Bitcoin and Ethereum use the secp256k1 curve, with ECDSA signatures and, on Bitcoin since Taproot, Schnorr signatures. Solana and many other chains use Ed25519, a different curve with its own signature scheme. Ethereum validators use BLS signatures over another curve. Wallets, hardware signers and key derivation standards all build on these primitives.

The math is considered secure, but implementations fail. Reusing or predicting the random nonce in an ECDSA signature lets attackers compute the private key, which has led to real thefts, including a 2013 Android random number bug affecting Bitcoin wallets. ECC would also be broken by a large quantum computer, which is the main reason for post-quantum migration plans.