ECDSA (Elliptic Curve Digital Signature Algorithm)

The Elliptic Curve Digital Signature Algorithm (ECDSA) is a digital signature scheme that adapts the older Digital Signature Algorithm (DSA) to elliptic curve cryptography. A digital signature proves that a message came from the holder of a particular private key and was not altered in transit. ECDSA achieves this using the algebra of points on an elliptic curve over a finite field; its security rests on the elliptic curve discrete logarithm problem, the fact that multiplying a known base point by a secret integer is easy, but recovering that secret integer from the result is believed infeasible. Conceptually, signing hashes the message and chooses a single-use random number called the nonce (written k). The nonce is multiplied by the curve's base point to produce a point whose x-coordinate becomes the first signature value (r); the message hash, private key, nonce, and r are then combined into the second value (s). Verification reverses this: using only the public key, the message hash, and the pair (r, s), anyone can recompute the point and confirm it matches r, without ever learning the private key. ECDSA's main advantage over RSA is key size. Because attacking elliptic curves is harder per bit, it reaches a given security level with far smaller keys: a 256-bit key offers protection comparable to a 3072-bit RSA key. Smaller keys mean smaller signatures and faster computation, which matters for embedded devices, TLS handshakes, and blockchains. The scheme has one notorious failure mode: the nonce must be unpredictable and never reused. If the same nonce signs two messages, or if it is predictable, the private key can be solved for directly with simple algebra. In 2010 the group fail0verflow showed that Sony reused a static nonce when signing PlayStation 3 software, letting them extract the master signing key and run unsigned code. In 2013 faulty randomness in some Android Bitcoin wallets produced repeated nonces, and attackers drained the affected addresses. These incidents show why ECDSA depends on CSPRNGs: Why Cryptographically Secure Random Numbers Are Essential for Security; a predictable generator, as explored in Reversing Pseudo-Random Number Generators: Why PRNGs Are Predictable and Hackable, breaks the whole scheme. The modern mitigation, RFC 6979, derives the nonce deterministically from the message and private key so it can never accidentally repeat. ECDSA is among the most widely deployed signature schemes today. It authenticates TLS certificates, signs Bitcoin and Ethereum transactions over the secp256k1 curve, and underpins trading APIs such as those in Building a Polymarket CLOB Client in Elixir: Architecture and Cryptography.