pqcrypto

Post-Quantum Lemniscate-AGM Isogeny (LAI) Encryption

A multi-language 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 conjectured resistance against quantum-capable adversaries.

Table of Contents

1. Project Overview
2. Mathematical Formulation
3. Features
4. Releases & Package Managers
   4.1. Python (PyPI)
   4.2. JavaScript (npm)
   4.3. Ruby (RubyGems)
   4.4. .NET (NuGet)
   4.5. Java (Maven)
   4.6. Rust (crates.io)
5. Usage Examples
   5.1. Python
   5.2. JavaScript
   5.3. Ruby
   5.4. .NET (C#)
   5.5. Java
   5.6. Rust
6. API Reference
7. Testing
8. Contributing & Development
9. License

Project Overview

This library implements all core mathematical primitives and high-level APIs for LAI:

• Hash-Based Seed Function
  $$( H(x, y, s) = \mathrm{SHA256}\bigl(x,|,y,|,s\bigr) \bmod p )$$

• Modular Square Root via Tonelli–Shanks (with fast branch if $$(p \equiv 3 \pmod 4)$$).

• LAI Transformation

  $$[ T\bigl((x,y),,s;,a,,p\bigr) ;=; \Bigl(, x' ;=; \tfrac{x + a + h}{2} \bmod p,;; y' ;=; \sqrt{x,y + h}\bmod p \Bigr) ] $$

  where $$(h = H(x,y,s))$$.

• Binary Exponentiation of $$(T)$$ to compute $$(T^k(P_0))$$ in $$(O(\log k)$$) time.

• Key Generation, Encryption, and Decryption routines for integer messages $$(0 \le m < p)$$.

• Bulk JSON Decryption: decrypt an entire JSON payload into raw bytes (e.g., to reconstruct a file or UTF-8 text).

All language‐specific wrappers expose identical API semantics under the hood. This makes pqcrypto ideal for cross-platform experiments, research, and educational purposes.

Mathematical Formulation

1. Hash-Based Seed Function

For $$(x, y, s \in \mathbb{Z}_p)$$, define:

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

where $$“(|)”$$ denotes concatenation of the big-endian byte representations.

2. Modular Square Root (Tonelli–Shanks)

Solve $$(z^2 \equiv a \pmod p) for prime (p)$$:

• If $$(p \equiv 3 \pmod 4)$$:

$$ [ z = a^{\frac{p+1}{4}} \bmod p. ] $$

• Otherwise: apply the general Tonelli–Shanks algorithm in $$(O(\log^2 p))$$ time.

3. LAI Transformation $$(T)$$

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

$$ \begin{cases} h = H(x,,y,,s),[6pt] x' = \dfrac{x + a + h}{2}\bmod p,[6pt] y' = \sqrt{x,y + h};\bmod p. \end{cases} $$

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

4. Binary Exponentiation of $$(T)$$

To compute $$(T^k(P_0))$$ efficiently:

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. Algorithmic API

Key Generation

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

Encryption

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

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

Bulk Decryption (JSON)

function decryptAll(jsonPayload):
   parse p, a, P0, k, blocks[]
   for each block in blocks:
      (x1,y1) = block.C1
      (x2,y2) = block.C2
      r       = block.r
      M_int   = decrypt((x1,y1),(x2,y2),k,r,a,p)
      convert M_int into fixed-length big-endian B-byte chunk
      append to output byte buffer
   return outputBuffer

Features

1. Pure Implementations (no native code)

   • Python: only uses hashlib, secrets (stdlib).
   • JavaScript: pure JS/BigInt.
   • Ruby: pure Ruby + openssl.
   • C#: uses System.Numerics.BigInteger (no external C/C++).
   • Java: uses java.math.BigInteger + Jackson for JSON.

2. Mathematically Annotated
   Every function corresponds exactly to the paper’s formulas.

3. Modular Design
   Separation of low‐level primitives (H, sqrt_mod, T) from high‐level API (keygen, encrypt, decrypt).

4. General & Optimized

   • Fast branch for $$(p\equiv3\pmod4)$$.
   • Full Tonelli–Shanks fallback for any odd prime.

5. Bulk JSON Decryption
   Produce or consume large ciphertext payloads (e.g., encrypted files, JavaScript code, JSON blobs).

6. CI/CD Ready

   • Python: auto‐publish to PyPI via GitHub Actions.
   • JS: auto‐publish to npm.
   • Ruby: auto‐publish to RubyGems.
   • C#: auto‐publish to NuGet & GitHub Packages.
   • Java: auto‐publish to GitHub Packages (Maven).

Releases & Package Managers

Python (PyPI)

pip install laicrypto

JavaScript (npm)

npm install @galihru/pqlaicrypto

Ruby (RubyGems)

gem install laicrypto

.NET (NuGet)

<PackageReference Include="PQCrypto.Lai" Version="0.1.0" />

Rust

Add to your Cargo.toml:

[dependencies]
laicrypto = "0.1.4"

Or via Cargo CLI:

bash

cargo add laicrypto

Build:

cargo build

• Crates.io: https://crates.io/crates/laicrypto

• Documentation: https://docs.rs/laicrypto

Java (Maven Central / GitHub Packages)

<dependency>
  <groupId>com.pelajaran.pqcrypto</groupId>
  <artifactId>laicrypto</artifactId>
  <version>0.1.0</version>
</dependency>

Usage Examples

Below are minimal “hello, world”-style code snippets for each language wrapper.

Python

import math
from pqcrypto import keygen, encrypt, decrypt

# 1. Setup parameters
p = 10007
a = 5
P0 = (1, 0)

# 2. Generate keypair
private_k, public_Q = keygen(p, a, P0)
print("Private k:", private_k)
print("Public  Q:", public_Q)

# 3. Encrypt integer m
message = 2024
C1, C2 = encrypt(message, public_Q, p, a, P0)
print("C1:", C1, " C2:", C2)

# 4. Decrypt using private_k
recovered = decrypt(C1, C2, private_k, a, p)
print("Recovered:", recovered)
assert recovered == message

If you need to encrypt an entire text/file, convert it to integer blocks via int.from_bytes(...), then call encrypt(...) on each block. See the Python demo in this README for details.

JavaScripts

// Install: npm install pqlaicrypto

const { keygen, encrypt, decrypt } = require("pqlaicrypto");

const p = 10007n;
const a = 5n;
const P0 = [1n, 0n];

// 1. Generate keypair
const { k, Q } = keygen(p, a, P0);
console.log("Private k:", k.toString());
console.log("Public  Q:", Q);

// 2. Encrypt a small integer
const m = 2024n;
const { C1, C2, r } = encrypt(m, Q, k, p, a, P0);
console.log("C1:", C1, "C2:", C2, "r:", r.toString());

// 3. Decrypt
const recovered = decrypt(C1, C2, k, r, a, p);
console.log("Recovered:", recovered.toString());

Use BigInt-aware file/block conversions to encrypt larger messages or files.

Ruby

# Install: gem install laicrypto
require "laicrypto"

p  = 10007
a  = 5
P0 = [1, 0]

# 1. Generate keypair
k, Q = LAI.keygen(p, a, P0)
puts "Private k: #{k}"
puts "Public  Q: #{Q.inspect}"

# 2. Encrypt integer
message = 2024
C1, C2, r = LAI.encrypt(message, Q, k, p, a, P0)
puts "C1: #{C1.inspect}  C2: #{C2.inspect}  r: #{r}"

# 3. Decrypt
recovered = LAI.decrypt(C1, C2, k, r, a, p)
puts "Recovered: #{recovered}"

Similar to Python, convert larger text to integer blocks using String#bytes and Integer().

.NET (C#)

// Install via NuGet: 
//   <PackageReference Include="PQCrypto.Lai" Version="0.1.0" />

using System;
using System.Numerics;
using PQCrypto; // namespace containing LaiCrypto

class Demo {
    static void Main(string[] args) {
        // 1. Setup parameters
        BigInteger p = 10007;
        BigInteger a = 5;
        LaiCrypto.Point P0 = new LaiCrypto.Point(1, 0);

        // 2. Generate keypair
        var kp = LaiCrypto.KeyGen(p, a, P0);
        Console.WriteLine($"Private k: {kp.k}");
        Console.WriteLine($"Public  Q: ({kp.Q.x}, {kp.Q.y})");

        // 3. Encrypt integer
        BigInteger message = 2024;
        var ct = LaiCrypto.Encrypt(message, kp.Q, p, a, P0);
        Console.WriteLine($"C1: ({ct.C1.x}, {ct.C1.y})  C2: ({ct.C2.x}, {ct.C2.y})  r: {ct.r}");

        // 4. Decrypt
        BigInteger recovered = LaiCrypto.Decrypt(ct.C1, ct.C2, kp.k, ct.r, a, p);
        Console.WriteLine($"Recovered: {recovered}");
        if (recovered != message) throw new Exception("Decryption mismatch!");
    }
}

To decrypt a JSON payload:

using System.IO;
using Newtonsoft.Json.Linq; // or System.Text.Json

var json = File.ReadAllText("ciphertext.json");
var jNode = JObject.Parse(json);
byte[] plaintextBytes = LaiCrypto.DecryptAll(jNode);
string plaintext = System.Text.Encoding.UTF8.GetString(plaintextBytes);

Java

<!-- In your pom.xml -->
<dependency>
  <groupId>com.pelajaran.pqcrypto</groupId>
  <artifactId>laicrypto</artifactId>
  <version>0.1.0</version>
</dependency>

import com.pelajaran.pqcrypto.LaiCrypto;
import com.pelajaran.pqcrypto.LaiCrypto.Point;
import com.pelajaran.pqcrypto.LaiCrypto.KeyPair;
import com.pelajaran.pqcrypto.LaiCrypto.Ciphertext;

import java.math.BigInteger;

public class LAIDemo {
    public static void main(String[] args) throws Exception {
        // 1. Setup
        BigInteger p = BigInteger.valueOf(10007);
        BigInteger a = BigInteger.valueOf(5);
        Point P0 = new Point(BigInteger.ONE, BigInteger.ZERO);

        // 2. Generate key pair
        KeyPair kp = LaiCrypto.keyGen(p, a, P0);
        System.out.println("Private k: " + kp.k);
        System.out.println("Public  Q: (" + kp.Q.x + ", " + kp.Q.y + ")");

        // 3. Encrypt integer
        BigInteger message = BigInteger.valueOf(2024);
        Ciphertext ct = LaiCrypto.encrypt(message, kp.Q, p, a, P0);
        System.out.println("C1: (" + ct.C1.x + ", " + ct.C1.y + ")");
        System.out.println("C2: (" + ct.C2.x + ", " + ct.C2.y + ")");
        System.out.println("r:  " + ct.r);

        // 4. Decrypt
        BigInteger recovered = LaiCrypto.decrypt(ct.C1, ct.C2, kp.k, ct.r, a, p);
        System.out.println("Recovered: " + recovered);
    }
}

To decrypt a JSON payload in Java:

import com.fasterxml.jackson.databind.ObjectMapper;
import com.fasterxml.jackson.databind.JsonNode;

// ...
ObjectMapper mapper = new ObjectMapper();
JsonNode root = mapper.readTree(new File("ciphertext.json"));
byte[] plaintextBytes = LaiCrypto.decryptAll(root);
String plaintext = new String(plaintextBytes, StandardCharsets.UTF_8);

Rust

// Initialize engine with prime modulus
let prime = 340_282_366_920_938_463_463_374_607_431_768_211_297; // 2^128 - 159
let mut engine = LaiCryptoEngine::new(prime, 10, (5, 10))?;

// Generate keys
let (priv_key, pub_key) = engine.keygen()?;

// Encrypt message
let message = 12345;
let (c1, c2, r) = engine.encrypt(message, pub_key, priv_key)?;

// Decrypt message
let decrypted = engine.decrypt(c1, c2, priv_key)?;

// Generate performance graphs
let timeline_graph = engine.generate_perf_graph(GraphStyle::Line);
println!("{}", timeline_graph.render_ascii(80, 24)?);

let complexity_graph = engine.generate_complexity_graph();
println!("{}", complexity_graph.render_ascii(60, 20)?);

// Print detailed trace
engine.print_trace();

API Reference

Testing

Each language wrapper includes its own test suite:

• Python:

  pytest --disable-warnings -q
  

• JavaScript:

  npm test
  

• Ruby:

  bundle exec rspec
  

• .NET (C#):

  dotnet test
  

• Java (Maven):

  mvn test
  

Make sure all tests pass locally before opening a pull request.

Contributing & Development

1. Fork the repository

2. Create a feature branch

   git checkout -b feature/your_feature
   

3. Implement changes

   • Add or fix primitives/pseudo-code as needed.
   • Add unit tests for any new functionality.

4. Run tests in all supported languages.

5. Commit & push, then open a pull request.

Please follow PEP 8 style in Python, StandardJS in JavaScript, Ruby Style Guide, C# coding conventions, and Java conventions. Include thorough documentation for any new API.

License

This project is licensed under the MIT License.