mleku
3c17e975df
Added detailed pseudocode for elliptic curve algorithms covering modular arithmetic, point operations, scalar multiplication, and coordinate conversions. Also introduced a comprehensive knowledge base for distributed systems, including CAP theorem, consistency models, consensus protocols (e.g., Paxos, Raft, PBFT, Nakamoto), and fault-tolerant design principles.
195 lines
4.2 KiB
Markdown
195 lines
4.2 KiB
Markdown
# secp256k1 Complete Parameters
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## Curve Definition
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**Name**: secp256k1 (Standards for Efficient Cryptography, prime field, 256-bit, Koblitz curve #1)
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**Equation**: y² = x³ + 7 (mod p)
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This is the short Weierstrass form with coefficients a = 0, b = 7.
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## Field Parameters
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### Prime Modulus p
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```
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Decimal:
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115792089237316195423570985008687907853269984665640564039457584007908834671663
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Hexadecimal:
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0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEFFFFFC2F
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Binary representation:
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2²⁵⁶ - 2³² - 2⁹ - 2⁸ - 2⁷ - 2⁶ - 2⁴ - 1
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= 2²⁵⁶ - 2³² - 977
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```
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**Special form benefits**:
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- Efficient modular reduction using: c mod p = c_low + c_high × (2³² + 977)
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- Near-Mersenne prime enables fast arithmetic
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### Group Order n
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```
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Decimal:
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115792089237316195423570985008687907852837564279074904382605163141518161494337
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Hexadecimal:
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0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEBAAEDCE6AF48A03BBFD25E8CD0364141
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```
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The number of points on the curve, including the point at infinity.
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### Cofactor h
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```
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h = 1
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```
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Cofactor 1 means the group order n equals the curve order, simplifying security analysis and eliminating small subgroup attacks.
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## Generator Point G
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### Compressed Form
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```
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02 79BE667EF9DCBBAC55A06295CE870B07029BFCDB2DCE28D959F2815B16F81798
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```
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The 02 prefix indicates the y-coordinate is even.
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### Uncompressed Form
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```
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04 79BE667EF9DCBBAC55A06295CE870B07029BFCDB2DCE28D959F2815B16F81798
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483ADA7726A3C4655DA4FBFC0E1108A8FD17B448A68554199C47D08FFB10D4B8
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```
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### Individual Coordinates
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**Gx**:
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```
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Decimal:
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55066263022277343669578718895168534326250603453777594175500187360389116729240
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Hexadecimal:
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0x79BE667EF9DCBBAC55A06295CE870B07029BFCDB2DCE28D959F2815B16F81798
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```
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**Gy**:
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```
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Decimal:
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32670510020758816978083085130507043184471273380659243275938904335757337482424
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Hexadecimal:
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0x483ADA7726A3C4655DA4FBFC0E1108A8FD17B448A68554199C47D08FFB10D4B8
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```
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## Endomorphism Parameters
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secp256k1 has an efficiently computable endomorphism φ: (x, y) → (βx, y).
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### β (Beta)
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```
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Hexadecimal:
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0x7AE96A2B657C07106E64479EAC3434E99CF0497512F58995C1396C28719501EE
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Property: β³ ≡ 1 (mod p)
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```
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### λ (Lambda)
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```
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Hexadecimal:
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0x5363AD4CC05C30E0A5261C028812645A122E22EA20816678DF02967C1B23BD72
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Property: λ³ ≡ 1 (mod n)
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Relationship: φ(P) = λP for all points P
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```
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### GLV Decomposition Constants
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For splitting scalar k into k₁ + k₂λ:
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```
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a₁ = 0x3086D221A7D46BCDE86C90E49284EB15
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b₁ = -0xE4437ED6010E88286F547FA90ABFE4C3
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a₂ = 0x114CA50F7A8E2F3F657C1108D9D44CFD8
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b₂ = a₁
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```
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## Derived Constants
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### Field Characteristics
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```
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(p + 1) / 4 = 0x3FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFBFFFFF0C
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Used for computing modular square roots via Tonelli-Shanks shortcut
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```
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### Order Characteristics
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```
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(n - 1) / 2 = 0x7FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF5D576E7357A4501DDFE92F46681B20A0
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Used in low-S normalization for ECDSA signatures
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```
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## Validation Formulas
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### Point on Curve Check
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For point (x, y), verify:
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```
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y² ≡ x³ + 7 (mod p)
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```
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### Generator Verification
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Verify G is on curve:
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```
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Gy² mod p = 0x9C47D08FFB10D4B8 ... (truncated for display)
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Gx³ + 7 mod p = same value
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```
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### Order Verification
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Verify nG = O (point at infinity):
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```
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Computing n × G should yield the identity element
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```
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## Bit Lengths
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| Parameter | Bits | Bytes |
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|-----------|------|-------|
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| p (prime) | 256 | 32 |
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| n (order) | 256 | 32 |
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| Private key | 256 | 32 |
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| Public key (compressed) | 257 | 33 |
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| Public key (uncompressed) | 513 | 65 |
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| ECDSA signature | 512 | 64 |
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| Schnorr signature | 512 | 64 |
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## Security Level
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- **Equivalent symmetric key strength**: 128 bits
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- **Best known attack complexity**: ~2¹²⁸ operations (Pollard's rho)
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- **Safe until**: Quantum computers with ~1500+ logical qubits
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## ASN.1 OID
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```
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1.3.132.0.10
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iso(1) identified-organization(3) certicom(132) curve(0) secp256k1(10)
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```
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## Comparison with Other Curves
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| Curve | Field Size | Security | Speed | Use Case |
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|-------|------------|----------|-------|----------|
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| secp256k1 | 256-bit | 128-bit | Fast (Koblitz) | Bitcoin, Nostr |
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| secp256r1 (P-256) | 256-bit | 128-bit | Moderate | TLS, general |
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| Curve25519 | 255-bit | ~128-bit | Very fast | Modern crypto |
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| secp384r1 (P-384) | 384-bit | 192-bit | Slower | High security |
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