Add Elliptic Curve Cryptography in Celestial Mechanics

[Copilot]

Overview

This PR implements a comprehensive Elliptic Curve Cryptography (ECC) system with a celestial mechanics theme, demonstrating the beautiful mathematical connection between cryptographic elliptic curves and Kepler's orbital mechanics.

Implementation Details

New Module: elliptic_celestial_crypto.py

The implementation includes two main classes:

1. EllipticCurve Class

Complete implementation of elliptic curve arithmetic over finite fields (y² = x³ + ax + b mod p):

• Point Addition: Combines two points on the curve using geometric line intersection, analogous to gravitational interactions between celestial bodies
• Scalar Multiplication: Efficient double-and-add algorithm for computing k·P, similar to calculating orbital positions after k periods
• Curve Validation: Ensures curves are non-singular and points lie on the curve
• Edge Case Handling: Properly handles point at infinity, zero denominators, and inverse points

2. OrbitalCryptoSystem Class

Cryptographic operations using the industry-standard secp256k1 curve (same as Bitcoin):

• Key Pair Generation: Creates public/private key pairs where the private key acts as secret orbital parameters and the public key represents observable positions
• ECDH (Elliptic Curve Diffie-Hellman): Enables two parties to compute a shared secret, analogous to finding orbital resonance
• Digital Signatures: ECDSA-style signatures with retry logic to handle rare edge cases where r or s equals zero
• Educational Bridge: Demonstrates the conceptual connection between cryptography and celestial mechanics

Mathematical Bridge

The implementation shows how cryptographic concepts mirror celestial mechanics:

Cryptography	Celestial Mechanics
Elliptic Curve (y² = x³ + ax + b)	Elliptical Orbit (Kepler's laws)
Point Addition	Gravitational Interaction
Scalar Multiplication (k·P)	Orbital Period Calculation
Generator Point G	The Sun (Central Body)
Private Key	Orbital Parameters (Secret)
Public Key	Observable Position

Testing

Comprehensive Test Suite: test_elliptic_celestial_crypto.py

17 test cases covering:

• ✅ Curve creation and validation (including singular curve rejection)
• ✅ Point-on-curve verification
• ✅ Point addition identity and inverse properties
• ✅ Point doubling and scalar multiplication
• ✅ Mathematical properties (distributivity, commutativity)
• ✅ Key pair generation and ECDH shared secrets
• ✅ Digital signature generation and verification
• ✅ Signature rejection with wrong keys or tampered messages
• ✅ Edge cases (zero denominators, invalid signatures)

All tests pass successfully with 100% coverage of implemented functionality.

Security

• ✅ CodeQL Security Scan: 0 vulnerabilities found
• Uses cryptographically secure random number generation (secrets module)
• Implements proper edge case handling to prevent crashes
• Based on secp256k1, a well-studied and battle-tested elliptic curve
• Signature generation includes retry logic for rare invalid cases

Documentation

Updated README.md with:

• Complete usage instructions for the new module
• Feature descriptions and examples
• Visual table showing the mathematical bridge between cryptography and physics
• Code examples demonstrating key operations

Example Usage

import elliptic_celestial_crypto as ecc

# Initialize the orbital cryptographic system
crypto = ecc.OrbitalCryptoSystem()

# Generate key pairs (like orbital parameters)
alice_private, alice_public = crypto.generate_keypair()
bob_private, bob_public = crypto.generate_keypair()

# Compute shared secret via ECDH (orbital resonance)
alice_shared = crypto.compute_shared_secret(alice_private, bob_public)
bob_shared = crypto.compute_shared_secret(bob_private, alice_public)
assert alice_shared == bob_shared  # Both parties arrive at the same secret!

# Create and verify digital signatures
message = "The celestial dance reveals the hidden truth"
signature = crypto.orbital_signature(message, alice_private)
is_valid = crypto.verify_signature(message, signature, alice_public)
assert is_valid == True

Code Quality

• 437 lines of implementation code
• 329 lines of comprehensive tests
• Clean, well-documented code with docstrings
• Follows Python conventions and best practices
• No breaking changes to existing functionality

Backward Compatibility

This PR is purely additive and does not modify any existing code. All original Da Vinci Code tests continue to pass.

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This implementation brings together mathematics, cryptography, and physics in an elegant demonstration of how the same mathematical structures appear across different domains of science.

Original prompt

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This section details on the original issue you should resolve

<issue_title>Elliptic (Algebraic) Curve Cryptography in Celestial Mechanics</issue_title>
<issue_description></issue_description>

Comments on the Issue (you are @copilot in this section)

Fixes #8

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