Add foundational resources for elliptic curve operations and distributed systems

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.

.claude/skills/elliptic-curves/references/secp256k1-parameters.md

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+# 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 |