r1cs 0.4.7

A library for building R1CS gadgets
Documentation 
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
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221

//! This module extends GadgetBuilder with native field arithmetic methods.

use crate::expression::{BooleanExpression, Expression};
use crate::field::{Element, Field};
use crate::gadget_builder::GadgetBuilder;
use crate::wire_values::WireValues;
use crate::util::concat;

impl<F: Field> GadgetBuilder<F> {
    /// The product of two `Expression`s `x` and `y`, i.e. `x * y`.
    pub fn product(&mut self, x: &Expression<F>, y: &Expression<F>) -> Expression<F> {
        if let Some(c) = x.as_constant() {
            return y * c;
        }
        if let Some(c) = y.as_constant() {
            return x * c;
        }

        let product = self.wire();
        let product_exp = Expression::from(product);
        self.assert_product(x, y, &product_exp);

        {
            let x = x.clone();
            let y = y.clone();
            self.generator(
                concat(&[x.dependencies(), y.dependencies()]),
                move |values: &mut WireValues<F>| {
                    let product_value = x.evaluate(values) * y.evaluate(values);
                    values.set(product, product_value);
                },
            );
        }

        product_exp
    }

    /// `x^p` for a constant `p`.
    pub fn exponentiation(&mut self, x: &Expression<F>, p: &Element<F>) -> Expression<F> {
        // This is exponentiation by squaring. For each 1 bit of p, multiply by the associated
        // square power.
        let mut product_exp = Expression::one();
        let mut last_square = Expression::zero();

        for i in 0..p.bits() {
            let square = if i == 0 {
                x.clone()
            } else {
                self.product(&last_square, &last_square)
            };

            if p.bit(i) {
                product_exp = self.product(&product_exp, &square);
            }

            last_square = square;
        }
        product_exp
    }

    /// Returns `1 / x`, assuming `x` is non-zero. If `x` is zero, the gadget will not be
    /// satisfiable.
    pub fn inverse(&mut self, x: &Expression<F>) -> Expression<F> {
        let x_inv = self.wire();
        self.assert_product(x, &Expression::from(x_inv), &Expression::one());

        let x = x.clone();
        self.generator(
            x.dependencies(),
            move |values: &mut WireValues<F>| {
                let x_value = x.evaluate(values);
                let inverse_value = x_value.multiplicative_inverse();
                values.set(x_inv, inverse_value);
            },
        );

        x_inv.into()
    }

    /// Like `inverse`, except that zero is mapped to itself rather than being prohibited.
    pub fn inverse_or_zero(&mut self, x: &Expression<F>) -> Expression<F> {
        let x_inv_or_zero = self.wire();
        let nonzero = self.nonzero(x);
        self.assert_product(x, &Expression::from(x_inv_or_zero), nonzero.expression());

        let x = x.clone();
        self.generator(
            x.dependencies(),
            move |values: &mut WireValues<F>| {
                let x_value = x.evaluate(values);
                values.set(x_inv_or_zero, x_value.multiplicative_inverse_or_zero());
            },
        );

        x_inv_or_zero.into()
    }

    /// Returns `x / y`, assuming `y` is non-zero. If `y` is zero, the gadget will not be
    /// satisfiable.
    pub fn quotient(&mut self, x: &Expression<F>, y: &Expression<F>) -> Expression<F> {
        let y_inv = self.inverse(y);
        self.product(x, &y_inv)
    }

    /// Returns `x / y`, assuming `y` is non-zero. This is equivalent to `quotient` except that it
    /// allows `0 / 0`. This method will panic if it encounters `0 / 0`, but a malicious prover
    /// would be able to supply an arbitrary quotient.
    ///
    /// This method uses a single constraint, whereas `quotient` uses two, so this method may be
    /// preferable in cases where `0 / 0` cannot possibly arise.
    pub fn quotient_unsafe(&mut self, x: &Expression<F>, y: &Expression<F>) -> Expression<F> {
        let q = self.wire();
        let x = x.clone();
        let y = y.clone();
        self.generator(
            [x.dependencies(), y.dependencies()].concat(),
            move |values: &mut WireValues<F>| {
                let x_value = x.evaluate(values);
                let y_value = y.evaluate(values);
                assert!(y_value.is_nonzero(), "Division by zero");
                let q_value = x_value / y_value;
                values.set(q, q_value)
            }
        );
        Expression::from(q)
    }

    /// Returns `x mod y`, assuming `y` is non-zero. If `y` is zero, the gadget will not be
    /// satisfiable.
    pub fn modulus(&mut self, x: &Expression<F>, y: &Expression<F>) -> Expression<F> {
        // We will non-deterministically compute a quotient q and remainder r such that:
        //     y * q = x - r
        //     r < y

        let q = self.wire();
        let r = self.wire();
        self.assert_product(y, &Expression::from(q), &(x - Expression::from(r)));
        self.assert_lt(&Expression::from(r), y);

        {
            let x = x.clone();
            let y = y.clone();
            self.generator(
                concat(&[x.dependencies(), y.dependencies()]),
                move |values: &mut WireValues<F>| {
                    let x_value = x.evaluate(values);
                    let y_value = y.evaluate(values);
                    values.set(q, x_value.integer_division(&y_value));
                    values.set(r, x_value.integer_modulus(&y_value));
                },
            );
        }

        r.into()
    }

    /// Returns whether `x` divides `y`, i.e. `x | y`.
    pub fn divides(&mut self, x: &Expression<F>, y: &Expression<F>) -> BooleanExpression<F> {
        let m = self.modulus(y, x);
        self.zero(&m)
    }
}

#[cfg(test)]
mod tests {
    use crate::expression::Expression;
    use crate::field::Element;
    use crate::gadget_builder::GadgetBuilder;
    use crate::test_util::{assert_eq_false, assert_eq_true, F257};

    #[test]
    fn exp() {
        let mut builder = GadgetBuilder::<F257>::new();
        let x = builder.wire();
        let x_exp_0 = builder.exponentiation(&Expression::from(x), &Element::from(0u8));
        let x_exp_1 = builder.exponentiation(&Expression::from(x), &Element::from(1u8));
        let x_exp_2 = builder.exponentiation(&Expression::from(x), &Element::from(2u8));
        let x_exp_3 = builder.exponentiation(&Expression::from(x), &Element::from(3u8));
        let gadget = builder.build();

        let mut values = values!(x => 3u8.into());
        assert!(gadget.execute(&mut values));
        assert_eq!(Element::from(1u8), x_exp_0.evaluate(&values));
        assert_eq!(Element::from(3u8), x_exp_1.evaluate(&values));
        assert_eq!(Element::from(9u8), x_exp_2.evaluate(&values));
        assert_eq!(Element::from(27u8), x_exp_3.evaluate(&values));
    }

    #[test]
    #[should_panic]
    fn invert_zero() {
        let mut builder = GadgetBuilder::<F257>::new();
        let x = builder.wire();
        builder.inverse(&Expression::from(x));
        let gadget = builder.build();

        let mut values = values!(x => 0u8.into());
        gadget.execute(&mut values);
    }

    #[test]
    fn divides() {
        let mut builder = GadgetBuilder::<F257>::new();
        let x = builder.wire();
        let y = builder.wire();
        let divides = builder.divides(&Expression::from(x), &Expression::from(y));
        let gadget = builder.build();

        let mut values_1_1 = values!(x => 1u8.into(), y => 1u8.into());
        assert!(gadget.execute(&mut values_1_1));
        assert_eq_true(&divides, &values_1_1);

        let mut values_3_6 = values!(x => 3u8.into(), y => 6u8.into());
        assert!(gadget.execute(&mut values_3_6));
        assert_eq_true(&divides, &values_3_6);

        let mut values_3_7 = values!(x => 3u8.into(), y => 7u8.into());
        assert!(gadget.execute(&mut values_3_7));
        assert_eq_false(&divides, &values_3_7);
    }
}