P256

P-256, also known as secp256r1 or prime256v1, is an elliptic curve defined over a 256-bit prime field. It is one of the NIST recommended curves and is widely used for digital signatures and key exchange. P-256 provides approximately 128-bit security and is designed to offer a favorable balance between cryptographic strength and computational efficiency.

The curve is defined by the short Weierstrass form y^2 = x^3 - 3x + b over the finite field with p = 2^256 - 2^224 + 2^192 + 2^96 - 1. The coefficient a is -3, and the constant b is 0x5AC635D8AA3A93E7B3EBBD55769886BC651D06B0CC53B0F63BCE3C3E27D2604B. The base point G has coordinates Gx = 0x6B17D1F2E12C4247F8BCE6E563A440F277037D812DEB33A0F4A13945D898C296 and Gy = 0x4FE342E2FE1A7F9B8EE7EB4A7C0F9E162BCE33576B315ECECBB6406837BF51F5. The order of the group generated by G is n = 0xFFFFFFFF00000000FFFFFFFFFFFFFFFFBCE6FAADA7179E84F3B9CAC2FC632551, and the cofactor h = 1.

Standardization and naming: P-256 is defined in FIPS 186-4 as part of the Digital Signature Standard and corresponds to the SEC2 standard as secp256r1. The curve is commonly referred to as prime256v1 in various software libraries. It is among the recommended curves in NIST documents for elliptic-curve cryptography.

Applications and usage: The curve supports ECDSA for digital signatures and ECDH for key agreement. It is widely implemented in TLS, SSH, PGP, and many cryptographic libraries and protocols, often serving as the default curve due to its established security properties and performance profile.

Security considerations: As a 256-bit curve, P-256 targets about 128-bit security. Implementations should use constant-time arithmetic, proper nonce generation in signatures, and secure parameter handling. Some deployments may opt for larger curves to meet higher future security requirements.