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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.

2025-12-02 19:14:39 +00:00

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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

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