Add Elliptic Curve Cryptography with Divine Proportions

[Copilot]

Overview

This PR implements a complete Elliptic Curve Cryptography (ECC) system inspired by the Golden Ratio (Phi) and Fibonacci sequences, extending the "Da Vinci Code" mathematical theme with modern cryptographic concepts.

The implementation addresses the issue "Elliptical curve cryopathy in the divine realm" by creating a bridge between ancient mathematical beauty (the divine proportion) and modern cryptographic security.

What's New

Core Implementation (elliptic_curve.py)

A fully-featured elliptic curve cryptography library (527 lines) that includes:

• Elliptic Curve Operations: Complete implementation of point addition, doubling, and scalar multiplication over finite fields
• The Divine Curve: An elliptic curve with parameters derived from the Golden Ratio (Phi ≈ 1.618)
• Cryptographic Key Generation: Public/private key pairs generated using Fibonacci sequences and the Golden Ratio
• Encryption/Decryption: ElGamal-style encryption scheme on elliptic curves
• Advanced Algorithms: Tonelli-Shanks algorithm for computing modular square roots

Example Usage

from elliptic_curve import divine_curve, generate_divine_keypair, divine_encrypt, divine_decrypt

# Create the divine curve (parameters derived from Phi)
curve = divine_curve()
# Curve: y² = x³ + 1x + 16 (mod 163)

# Generate keys using Fibonacci and Golden Ratio
private_key, public_key = generate_divine_keypair(curve)

# Encrypt a message
message = 42
C1, C2 = divine_encrypt(message, public_key, generator)

# Decrypt
decrypted = divine_decrypt(C1, C2, private_key)

Comprehensive Testing (test_elliptic_curve.py)

14 test cases covering:

• Curve creation and validation
• Point operations (addition, doubling, scalar multiplication)
• Cryptographic properties (associativity, commutativity)
• Key generation and encryption/decryption
• Edge cases and error handling

All tests pass ✅

Integration with Existing Code

The implementation seamlessly integrates with the existing da_vinci_code.py module:

import da_vinci_code

# Divine Curve uses the Golden Ratio from da_vinci_code
phi = da_vinci_code.golden_ratio(100)

# Key generation incorporates Fibonacci sequences
fib = da_vinci_code.fibonacci(10)
private_key = int(sum(fib) * phi) % curve.p

Mathematical Background

The implementation connects classical and modern mathematics:

• Fibonacci Sequence: 1, 1, 2, 3, 5, 8, 13, 21, ... (from The Da Vinci Code)
• Golden Ratio (Phi): 1.618033988749895 (the "divine proportion")
• Elliptic Curves: y² = x³ + ax + b (modern cryptography)

The "Divine Curve" uses a = floor(Phi) = 1 and b = floor(Phi * 10) = 16, creating a cryptographic system rooted in divine proportions.

Security Notes

This implementation is designed for educational and demonstration purposes to showcase the mathematical beauty connecting ancient proportions with modern cryptography. Key security considerations:

• ⚠️ Uses deterministic key generation (Fibonacci + Phi) for thematic consistency
• ⚠️ Includes demonstration code that displays private keys for educational purposes
• ⚠️ Should not be used for production cryptographic systems without modifications

For production use, the code includes clear documentation recommending:

• Cryptographically secure random number generators (CSPRNG)
• Never logging or displaying private keys
• Following established cryptographic standards

Documentation

Updated README.md with:

• New section on Elliptic Curve Cryptography
• Usage examples for both classical and modern implementations
• Feature comparison between Da Vinci Code mathematics and ECC

Testing Results

• ✅ All 17 tests pass (3 original + 14 new)
• ✅ Original functionality preserved
• ✅ Code review feedback addressed
• ✅ Security warnings added

Files Changed

• Added: elliptic_curve.py (527 lines)
• Added: test_elliptic_curve.py (300 lines)
• Modified: README.md (documentation update)

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"The geometry of the elliptic curve reflects divine harmony..."

Original prompt

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This section details on the original issue you should resolve

<issue_title>Elliptical curve cryopathy in the divine realm</issue_title>
<issue_description></issue_description>

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

Fixes #6

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