Project_Overview

Project Overview

This library implements all core mathematical primitives and high-level APIs for LAI (Lemniscate-AGM Isogeny):

• Hash-Based Seed Function

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

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

• LAI Transformation

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

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

• 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.

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High-Level Components

1. Low-Level Primitives

   • H(x, y, s): hash-based seed
   • sqrt_mod(a, p): modular square root (Tonelli–Shanks)
   • T(point, s, a, p): one LAI transform step

2. Binary Exponentiation
   Implements exponentiation by squaring for repeated application of $$(T)$$.

3. High-Level API

   • keygen(p, a, P0) → (k, Q)
   • encrypt(m, Q, k, p, a, P0) → (C1, C2, r)
   • decrypt(C1, C2, k, r, a, p) → m
   • decryptAll(jsonPayload) → byte[]

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