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4.2 KiB
4.2 KiB
secp256k1 Complete Parameters
Curve Definition
Name: secp256k1 (Standards for Efficient Cryptography, prime field, 256-bit, Koblitz curve #1)
Equation: y² = x³ + 7 (mod p)
This is the short Weierstrass form with coefficients a = 0, b = 7.
Field Parameters
Prime Modulus p
Decimal:
115792089237316195423570985008687907853269984665640564039457584007908834671663
Hexadecimal:
0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEFFFFFC2F
Binary representation:
2²⁵⁶ - 2³² - 2⁹ - 2⁸ - 2⁷ - 2⁶ - 2⁴ - 1
= 2²⁵⁶ - 2³² - 977
Special form benefits:
- Efficient modular reduction using: c mod p = c_low + c_high × (2³² + 977)
- Near-Mersenne prime enables fast arithmetic
Group Order n
Decimal:
115792089237316195423570985008687907852837564279074904382605163141518161494337
Hexadecimal:
0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEBAAEDCE6AF48A03BBFD25E8CD0364141
The number of points on the curve, including the point at infinity.
Cofactor h
h = 1
Cofactor 1 means the group order n equals the curve order, simplifying security analysis and eliminating small subgroup attacks.
Generator Point G
Compressed Form
02 79BE667EF9DCBBAC55A06295CE870B07029BFCDB2DCE28D959F2815B16F81798
The 02 prefix indicates the y-coordinate is even.
Uncompressed Form
04 79BE667EF9DCBBAC55A06295CE870B07029BFCDB2DCE28D959F2815B16F81798
483ADA7726A3C4655DA4FBFC0E1108A8FD17B448A68554199C47D08FFB10D4B8
Individual Coordinates
Gx:
Decimal:
55066263022277343669578718895168534326250603453777594175500187360389116729240
Hexadecimal:
0x79BE667EF9DCBBAC55A06295CE870B07029BFCDB2DCE28D959F2815B16F81798
Gy:
Decimal:
32670510020758816978083085130507043184471273380659243275938904335757337482424
Hexadecimal:
0x483ADA7726A3C4655DA4FBFC0E1108A8FD17B448A68554199C47D08FFB10D4B8
Endomorphism Parameters
secp256k1 has an efficiently computable endomorphism φ: (x, y) → (βx, y).
β (Beta)
Hexadecimal:
0x7AE96A2B657C07106E64479EAC3434E99CF0497512F58995C1396C28719501EE
Property: β³ ≡ 1 (mod p)
λ (Lambda)
Hexadecimal:
0x5363AD4CC05C30E0A5261C028812645A122E22EA20816678DF02967C1B23BD72
Property: λ³ ≡ 1 (mod n)
Relationship: φ(P) = λP for all points P
GLV Decomposition Constants
For splitting scalar k into k₁ + k₂λ:
a₁ = 0x3086D221A7D46BCDE86C90E49284EB15
b₁ = -0xE4437ED6010E88286F547FA90ABFE4C3
a₂ = 0x114CA50F7A8E2F3F657C1108D9D44CFD8
b₂ = a₁
Derived Constants
Field Characteristics
(p + 1) / 4 = 0x3FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFBFFFFF0C
Used for computing modular square roots via Tonelli-Shanks shortcut
Order Characteristics
(n - 1) / 2 = 0x7FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF5D576E7357A4501DDFE92F46681B20A0
Used in low-S normalization for ECDSA signatures
Validation Formulas
Point on Curve Check
For point (x, y), verify:
y² ≡ x³ + 7 (mod p)
Generator Verification
Verify G is on curve:
Gy² mod p = 0x9C47D08FFB10D4B8 ... (truncated for display)
Gx³ + 7 mod p = same value
Order Verification
Verify nG = O (point at infinity):
Computing n × G should yield the identity element
Bit Lengths
| Parameter | Bits | Bytes |
|---|---|---|
| p (prime) | 256 | 32 |
| n (order) | 256 | 32 |
| Private key | 256 | 32 |
| Public key (compressed) | 257 | 33 |
| Public key (uncompressed) | 513 | 65 |
| ECDSA signature | 512 | 64 |
| Schnorr signature | 512 | 64 |
Security Level
- Equivalent symmetric key strength: 128 bits
- Best known attack complexity: ~2¹²⁸ operations (Pollard's rho)
- Safe until: Quantum computers with ~1500+ logical qubits
ASN.1 OID
1.3.132.0.10
iso(1) identified-organization(3) certicom(132) curve(0) secp256k1(10)
Comparison with Other Curves
| Curve | Field Size | Security | Speed | Use Case |
|---|---|---|---|---|
| secp256k1 | 256-bit | 128-bit | Fast (Koblitz) | Bitcoin, Nostr |
| secp256r1 (P-256) | 256-bit | 128-bit | Moderate | TLS, general |
| Curve25519 | 255-bit | ~128-bit | Very fast | Modern crypto |
| secp384r1 (P-384) | 384-bit | 192-bit | Slower | High security |