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

// Copyright (C) 2019-2023 Aleo Systems Inc.
// This file is part of the snarkVM library.

// Licensed under the Apache License, Version 2.0 (the "License");
// you may not use this file except in compliance with the License.
// You may obtain a copy of the License at:
// http://www.apache.org/licenses/LICENSE-2.0

// Unless required by applicable law or agreed to in writing, software
// distributed under the License is distributed on an "AS IS" BASIS,
// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
// See the License for the specific language governing permissions and
// limitations under the License.

use super::*;

impl<E: Environment> Elligator2<E> {
    /// Returns the decoded field element, given the encoded affine group element and sign.
    pub fn decode(group: &Group<E>, sign_high: bool) -> Result<Field<E>> {
        ensure!(
            Group::<E>::EDWARDS_D.legendre().is_qnr(),
            "D on the twisted Edwards curve must be a quadratic nonresidue"
        );
        ensure!(!group.is_zero(), "Inputs to Elligator2 must be nonzero (inverses will fail)");
        ensure!((**group).to_affine().is_on_curve(), "Inputs to Elligator2 must be on the twisted Edwards curve");

        // Compute the coefficients for the Weierstrass form: v^2 == u^3 + A * u^2 + B * u.
        let (montgomery_b_inverse, a, b) = match Group::<E>::MONTGOMERY_B.inverse() {
            Ok(b_inverse) => (b_inverse, Group::MONTGOMERY_A * b_inverse, b_inverse.square()),
            Err(_) => bail!("Montgomery B must be invertible in order to use Elligator2"),
        };

        let (x, y) = group.to_xy_coordinates();

        // Ensure that x != -A.
        ensure!(x != -a, "Elligator2 failed: x == -A");

        // Ensure that if y is 0, then x is 0.
        if y.is_zero() {
            ensure!(x.is_zero(), "Elligator2 failed: y == 0 but x != 0");
        }

        // Convert the twisted Edwards element (x, y) to the Weierstrass element (u, v)
        let (u, v) = {
            let one = Field::<E>::one();

            let numerator = one + y;
            let denominator = one - y;

            // Compute u = (1 + y) / (1 - y).
            let u = numerator * denominator.inverse().map_err(|_| anyhow!("Elligator2 failed: (1 - y) == 0"))?;

            // Compute v = (1 + y) / ((1 - y) * x).
            let v =
                numerator * (denominator * x).inverse().map_err(|_| anyhow!("Elligator2 failed: x * (1 - y) == 0"))?;

            // Ensure (u, v) is a valid Montgomery element on: B * v^2 == u^3 + A * u^2 + u
            let u2 = u.square();
            ensure!(
                Group::MONTGOMERY_B * v.square() == (u2 * u) + (Group::MONTGOMERY_A * u2) + u,
                "Elligator2 failed: B * v^2 != u^3 + A * u^2 + u"
            );

            let u = u * montgomery_b_inverse;
            let v = v * montgomery_b_inverse;

            // Ensure (u, v) is a valid Weierstrass element on: v^2 == u^3 + A * u^2 + B * u
            let u2 = u.square();
            ensure!(v.square() == (u2 * u) + (a * u2) + (b * u), "Elligator2 failed: v^2 != u^3 + A * u^2 + B * u");

            (u, v)
        };

        // Ensure -D * u * (u + A) is a residue.
        let du = Group::EDWARDS_D * u;
        let u_plus_a = u + a;
        ensure!((-du * u_plus_a).legendre().is_qr(), "Elligator2 failed: -D * u * (u + A) is not a quadratic residue");

        let v_reconstructed =
            v.square().square_root().map_err(|_| anyhow!("Elligator2 failed: cannot square root v^2"))?;
        let exists_in_sqrt_fq2 = v_reconstructed == v;

        let element = match exists_in_sqrt_fq2 {
            // Let element = sqrt(-u / ((u + A) * D)).
            true => {
                -u * (u_plus_a * Group::EDWARDS_D)
                    .inverse()
                    .map_err(|_| anyhow!("Elligator2 failed: (u+A) * D == 0"))?
            }
            // Let element = sqrt(-(u + A) / Du)).
            false => -u_plus_a * du.inverse().map_err(|_| anyhow!("Elligator2 failed: D * u == 0"))?,
        }
        .square_root()
        .map_err(|_| anyhow!("Elligator2 failed: cannot compute the square root for the element"))?;

        match sign_high {
            true => Ok(cmp::max(element, -element)),
            false => Ok(cmp::min(element, -element)),
        }
    }
}

#[cfg(test)]
mod tests {
    use super::*;
    use snarkvm_console_types::environment::Console;

    type CurrentEnvironment = Console;

    pub(crate) const ITERATIONS: usize = 10000;

    #[test]
    fn test_encode_and_decode() -> Result<()> {
        let rng = &mut TestRng::default();

        let mut high_ctr = 0usize;
        let mut low_ctr = 0usize;

        for _ in 0..ITERATIONS {
            let expected = Uniform::rand(rng);

            let (encoded, sign_high) = Elligator2::<CurrentEnvironment>::encode_without_cofactor_clear(&expected)?;
            let decoded = Elligator2::<CurrentEnvironment>::decode(&encoded, sign_high)?;
            assert_eq!(expected, decoded);

            match sign_high {
                true => high_ctr += 1,
                false => low_ctr += 1,
            }
        }

        println!("Sign high: {high_ctr}, sign low: {low_ctr}");
        Ok(())
    }

    #[test]
    fn test_zero_fails() {
        let encode = Elligator2::<CurrentEnvironment>::encode(&Zero::zero());
        assert!(encode.is_err());

        let decode = Elligator2::<CurrentEnvironment>::decode(&Zero::zero(), true);
        assert!(decode.is_err());
        let decode = Elligator2::<CurrentEnvironment>::decode(&Zero::zero(), false);
        assert!(decode.is_err());
    }
}