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jubjub-elgamal

Build Status Repository Documentation

This crate provides a Rust implementation of the ElGamal encryption scheme implemented for elements of the JubJub elliptic curve to be used natively and as part of a Zero-Knowledge circuit using plonk. This implementation is designed by the Dusk team.

About

The ElGamal encryption system is an asymmetric key encryption algorithm for public-key cryptography based on the Diffie-Hellman key exchange. Its security relies on the difficulty of computing discrete logarithms over finite fields. The implementation has been created using the field elements of the jubjub elliptic curve.

Algorithm

Notation

In the following:

  • Multiplication of a point $P$ by a scalar $s$ stands for adding $P$ $s$-times to itself.
  • $\mathbb{F}_q$ is the prime finite field of order $q$
  • for a prime $q$: $\mathbb{F}_q^× = \mathbb{F}_q \setminus 0$ contains all nonzero elements of $\mathbb{F}_q$.

Setup

Since we implement our ElGamal encryption scheme on the jubjub elliptic curve we have:

  • a finite field $\mathbb{F}_q$ over prime $q$, which corresponds to the scalar field of the elliptic curve BLS12-381
  • an elliptic curve $E / \mathbb{F}_q$, in our case this is the jubjub elliptic curve
  • a subgroup $\mathbb{G} \in E(\mathbb{F}_q)$ of curve points, with prime order $p$
  • a fixed generator point $G \in \mathbb{G}$

Key generation

  • Choose a private signing key, $sk \in \mathbb{F}_p^×$.
  • Compute the matching public key, $PK = skG \in \mathbb{G}$.

Encrypting

Suppose Alice wants to send Bob an encrypted message $m \in \mathbb{F}_q^×$. To encrypt the message Alice will use Bob's public-key $PK_B$:

  • Choose a random blinder nonce $r \in \mathbb{F}_p^×$.
  • Compute first part of the ciphertext $c_1 = R = rG$.
  • Compute second part of the ciphertext $c_2 = m + PK_B * r$.
  • Send the ciphertext $(c_1, c_2)$ to Bob.

Decrypting

To decrypt the ciphertext $(c_1, c_2)$ Bob will use his secret-key $sk_B$:

  • Compute $c_2 - c_1 * sk_B = m$

This is true because:

$$c_2 - c_1 * sk_B = m + PK_B * r - (r * G * sk_B) = m + PK_B * r - PK_B * r = m$$

Example

A basic example demonstrating how to encrypt and decrypt a message using ElGamal:

use dusk_jubjub::{JubJubScalar, GENERATOR_EXTENDED};
use ff::Field;
use jubjub_elgamal::{decrypt, encrypt};
use rand::rngs::StdRng;
use rand::SeedableRng;

let mut rng = StdRng::seed_from_u64(0xc0b);

let sk = JubJubScalar::random(&mut rng);
let pk = GENERATOR_EXTENDED * &sk;

let message = GENERATOR_EXTENDED * JubJubScalar::from(1234u64);

// Encrypt using a fresh random value 'blinder'
let r = JubJubScalar::random(&mut rng);
let (c1, c2) = encrypt(&pk, &message, &r);

// Assert decryption
let dec_message = decrypt(&sk, &(c1, c2));
assert_eq!(message, dec_message);

Licensing

This Source Code Form is subject to the terms of the Mozilla Public License, v. 2.0. If a copy of the MPL was not distributed with this file, You can obtain one at http://mozilla.org/MPL/2.0/.

Copyright (c) DUSK NETWORK. All rights reserved.