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laicrypto - LAI Crypto

pqcrypto

Post-Quantum Lemniscate-AGM Isogeny (LAI) Encryption

A Python package providing a reference implementation of the Lemniscate-AGM Isogeny (LAI) encryption scheme. LAI is a promising post-quantum cryptosystem based on isogenies of elliptic curves over lemniscate lattices, offering resistance against quantum-capable adversaries.

Project Overview

This library implements the core mathematical primitives and high-level API of the LAI scheme, including:

  • Key Generation: Derivation of a private scalar and corresponding public point via binary exponentiation of the LAI transformation.
  • Encryption: Secure encryption of integer messages modulo a prime.
  • Decryption: Accurate recovery of plaintext via inverse transform.

The code is annotated with direct correspondence to the mathematical definitions and pseudocode, making it suitable for research, educational use, and further development.

Mathematical Formulation

1. Hash-Based Seed Function

Define:

$$ H(x, y, s) \;=\; \mathrm{SHA256}\bigl(x \,|\, y \,|\, s\bigr) \bmod p $$

where \$x,y,s \in \mathbb{Z}_p\$ and \$|\$ denotes byte-string concatenation.

2. Modular Square Root (Tonelli–Shanks)

Compute \$z = \sqrt{a} \bmod p\$ for prime \$p\$:

  • If \$p \equiv 3 \pmod{4}\$: $z \;=\; a^{\frac{p+1}{4}} \bmod p$
  • Otherwise, use the full Tonelli–Shanks algorithm for general primes.

3. LAI Transformation \$T\$

Given a point \$(x,y) \in \mathbb{F}_p^2\$, parameter \$a\$, and seed index \$s\$, define:

$$ \begin{aligned} h &= H(x,y,s), [4pt] x' &= \frac{x + a + h}{2} \bmod p, [4pt] y' &= \sqrt{x \, y + h} \bmod p. \end{aligned} $$

Thus,

$T\bigl((x,y), s; a, p\bigr) = (\,x', y').$

4. Binary Exponentiation of \$T\$

To compute \$T^k(P_0)\$ efficiently, use exponentiation by squaring:

text function pow_T(P, k): result ← P base ← P s ← 1 while k > 0: if (k mod 2) == 1: result ← T(result, s) base ← T(base, s) k ← k >> 1 s ← s + 1 return result

5. API Algorithms

Key Generation

text function keygen(p, a, P0): k ← random integer in [1, p−1] Q ← pow_T(P0, k) return (k, Q)

Encryption

text function encrypt(m, Q, p, a, P0): r ← random integer in [1, p−1] C1 ← pow_T(P0, r) Sr ← pow_T(Q, r) M ← (m mod p, 0) C2 ← ( (M.x + Sr.x) mod p, (M.y + Sr.y) mod p ) return (C1, C2)

Decryption

text function decrypt(C1, C2, k, a, p): S ← pow_T(C1, k) M.x ← (C2.x − S.x) mod p return M.x

Features

  1. Pure Python implementation: no external dependencies for core routines (uses hashlib & secrets).
  2. Mathematically Annotated: formulas and pseudocode directly reference the original scheme.
  3. Modular Design: separation of primitives (H, sqrt_mod, T) and high-level API (keygen, encrypt, decrypt).
  4. General & Optimized: Tonelli–Shanks for any prime, plus branch for \$p\equiv3\pmod4\$.
  5. Automated Testing: pytest suite for end-to-end verification.
  6. CI/CD Ready: PyPI publication via GitHub Actions.

Installation

From PyPI

bash pip install pqcrypto

From Source

bash git clone https://github.com/username/pqcrypto.git cd pqcrypto pip install .

Usage Example

```python from pqcrypto import keygen, encrypt, decrypt

Parameters

a = 5 p = 10007 P0 = (1, 0)

Key generation

privatek, publicQ = keygen(p, a, P0)

Encryption

text = 1234 C1, C2 = encrypt(text, public_Q, p, a, P0)

Decryption

mout = decrypt(C1, C2, privatek, a, p) assert mout == text print("Recovered message:", mout) ```

API Reference

| Function | Description | | ------------------------------------ | --------------------------------------- | | H(x, y, s, p) -> int | Hash-based seed modulo \$p\$. | | sqrt_mod(a, p) -> int | Modular square root via Tonelli–Shanks. | | T(point, s, a, p) -> (int, int) | One LAI transform step. | | keygen(p, a, P0) -> (k, Q) | Generate private key and public point. | | encrypt(m, Q, p, a, P0) -> (C1,C2) | Encrypt integer message. | | decrypt(C1, C2, k, a, p) -> int | Decrypt ciphertext to integer. |

Testing

bash pytest --disable-warnings -q

Contributing & Development

  1. Fork the repo
  2. Create branch: git checkout -b feature/xyz
  3. Implement changes with corresponding tests
  4. Run tests: pytest
  5. Submit Pull Request

Please follow PEP 8 and include unit tests for new functionality.

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Published by 4211421036 9 months ago