An elliptic curve is a mathematical structure defined by an equation over a particular number system. In cryptography, carefully selected curves support public-key operations: users can derive public keys from private keys, while recovering the private key from the public key is computationally infeasible with known methods when parameters and implementations are sound.
Elliptic-curve cryptography is used in digital signatures and key agreement, often with smaller keys than some other public-key systems. Curve parameters are public; their documented origin is a provenance question, not a secret-key mechanism. In real attacks, weak implementations, poor randomness, or reused signature nonces can expose private keys. Defenders should use vetted libraries and approved parameters, and distinguish concerns about how a curve was generated from evidence that its cryptography is broken.