1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164

use std::iter;

use serde::{Deserialize, Serialize};

use curv::arithmetic::traits::{Modulo, Samplable};
use curv::BigInt;
use paillier::traits::{Add, Mul};
use paillier::EncryptWithChosenRandomness;
use paillier::Paillier;
use paillier::{EncryptionKey, Randomness, RawCiphertext, RawPlaintext};

use super::errors::IncorrectProof;

/// This proof shows that a paillier ciphertext was constructed correctly
///
/// The proof is taken from https://www.brics.dk/RS/00/14/BRICS-RS-00-14.pdf 9.1.3
/// Given a ciphertext c and a prover encryption key , a prover wants to prove that it knows (x,r) such that c = Enc(x,r)
/// 1) P picks x',r' at random, and computes c' = Enc(x', r')
/// 2) P computes z1 = x' + ex , z2 = r' *r^e  (e is a varifier challenge)
/// 3) P sends, c' , z1,z2
/// 4) V accepts if 1) Enc(z1,z2 ) = c' * c^e
#[derive(Clone, PartialEq, Debug, Serialize, Deserialize)]
pub struct CiphertextProof {
    pub z1: BigInt,
    pub z2: BigInt,
    pub c_prime: BigInt,
}

#[derive(Clone, PartialEq, Debug, Serialize, Deserialize)]
pub struct CiphertextWitness {
    pub x: BigInt,
    pub r: BigInt,
}

#[derive(Clone, PartialEq, Debug, Serialize, Deserialize)]
pub struct CiphertextStatement {
    pub ek: EncryptionKey,
    pub c: BigInt,
}

impl CiphertextProof {
    pub fn prove(witness: &CiphertextWitness, statement: &CiphertextStatement) -> Self {
        let x_prime = BigInt::sample_below(&statement.ek.n);
        let r_prime = BigInt::sample_below(&statement.ek.n);
        let c_prime = Paillier::encrypt_with_chosen_randomness(
            &statement.ek,
            RawPlaintext::from(x_prime.clone()),
            &Randomness(r_prime.clone()),
        )
        .0
        .into_owned();

        let e = super::compute_digest(
            iter::once(&statement.ek.n)
                .chain(iter::once(&statement.c))
                .chain(iter::once(&c_prime)),
        );

        let z1 = &x_prime + &witness.x * &e;
        let r_e = BigInt::mod_pow(&witness.r, &e, &statement.ek.nn);
        let z2 = BigInt::mod_mul(&r_prime, &r_e, &statement.ek.nn);

        CiphertextProof { z1, z2, c_prime }
    }

    pub fn verify(&self, statement: &CiphertextStatement) -> Result<(), IncorrectProof> {
        let e = super::compute_digest(
            iter::once(&statement.ek.n)
                .chain(iter::once(&statement.c))
                .chain(iter::once(&self.c_prime)),
        );

        let c_z = Paillier::encrypt_with_chosen_randomness(
            &statement.ek,
            RawPlaintext::from(self.z1.clone()),
            &Randomness(self.z2.clone()),
        )
        .0
        .into_owned();

        let c_e = Paillier::mul(
            &statement.ek,
            RawPlaintext::from(e),
            RawCiphertext::from(statement.c.clone()),
        );
        let c_z_test = Paillier::add(
            &statement.ek,
            c_e,
            RawCiphertext::from(self.c_prime.clone()),
        )
        .0
        .into_owned();

        match c_z == c_z_test {
            true => Ok(()),
            false => Err(IncorrectProof),
        }
    }
}

#[cfg(test)]
mod tests {
    use curv::arithmetic::traits::*;
    use curv::BigInt;
    use paillier::core::Randomness;
    use paillier::traits::EncryptWithChosenRandomness;
    use paillier::traits::KeyGeneration;
    use paillier::Paillier;
    use paillier::RawPlaintext;

    use crate::zkproofs::correct_ciphertext::CiphertextProof;
    use crate::zkproofs::correct_ciphertext::CiphertextStatement;
    use crate::zkproofs::correct_ciphertext::CiphertextWitness;

    #[test]
    fn test_ciphertext_proof() {
        let (ek, _) = Paillier::keypair().keys();
        let x = BigInt::sample_below(&ek.n);
        let r = BigInt::sample_below(&ek.n);

        let c = Paillier::encrypt_with_chosen_randomness(
            &ek,
            RawPlaintext::from(x.clone()),
            &Randomness(r.clone()),
        )
        .0
        .into_owned();

        let witness = CiphertextWitness { x, r };

        let statement = CiphertextStatement { ek, c };

        let proof = CiphertextProof::prove(&witness, &statement);
        let verify = proof.verify(&statement);
        assert!(verify.is_ok());
    }

    #[test]
    #[should_panic]
    fn test_bad_ciphertext_proof() {
        let (ek, _) = Paillier::keypair().keys();
        let x = BigInt::sample_below(&ek.n);
        let r = BigInt::sample_below(&ek.n);

        let c = Paillier::encrypt_with_chosen_randomness(
            &ek,
            RawPlaintext::from(x.clone()),
            &Randomness(r.clone()),
        )
        .0
        .into_owned();

        let witness = CiphertextWitness {
            x,
            r: r + BigInt::one(),
        };

        let statement = CiphertextStatement { ek, c };

        let proof = CiphertextProof::prove(&witness, &statement);
        let verify = proof.verify(&statement);
        assert!(verify.is_ok());
    }
}