secp256r1

secp256r1 is an elliptic curve used in public-key cryptography. It is widely known under several names, including prime256v1 and NIST P-256, and is specified in the SECG SEC 2 standard as well as in FIPS publications. The curve is defined over a 256‑bit prime field and provides approximately 128 bits of security for elliptic-curve operations.

The curve is defined by the short Weierstrass form y^2 = x^3 + ax + b over the prime field F_p, with a = -3 and a specific constant b. The prime p is p = 2^256 − 2^224 + 2^192 + 2^96 − 1, expressed in hexadecimal as FFFFFFFF00000001000000000000000000000000FFFFFFFFFFFFFFFFFFFFFFFF. The coefficient b is 5AC635D8AA3A93E7B3EBBD55769886BC651D06B0CC53B0F63BCE3C3E27D2604B, and a is typically represented as p − 3, i.e., FFFFFFFF00000001000000000000000000000000FFFFFFFFFFFFFFFFFFFFFFFC.

The base point G (the generator) and its order n are defined in the standards. The order is n = FFFFFFFF00000000FFFFFFFFFFFFFFFFBCE6FAADA7179E84F3B9CAC2FC632551, and the cofactor h = 1. The exact coordinates of G are specified in the official specifications; together with n and h they determine the subgroup used for cryptographic operations.

secp256r1 is commonly employed for ECDSA signatures, ECDH key exchange, and other ECC-based protocols, including many TLS configurations. It enjoys broad hardware and software support and remains a standard choice for 256‑bit security requirements, often preferred for its balance of security and performance.