1use r1cs_core::SynthesisError;
  2
  3use super::PairingVar as PG;
  4
  5use crate::{
  6    fields::{fp::FpVar, fp2::Fp2Var, fp4::Fp4Var, FieldVar},
  7    groups::mnt4::{
  8        AteAdditionCoefficientsVar, AteDoubleCoefficientsVar, G1PreparedVar, G1Var, G2PreparedVar,
  9        G2ProjectiveExtendedVar, G2Var,
 10    },
 11};
 12use algebra::{
 13    curves::mnt4::{MNT4Parameters, MNT4},
 14    fields::BitIteratorBE,
 15};
 16use core::marker::PhantomData;
 17
 18/// Specifies the constraints for computing a pairing in a MNT4 bilinear group.
 19pub struct PairingVar<P: MNT4Parameters>(PhantomData<P>);
 20
 21type Fp2G<P> = Fp2Var<<P as MNT4Parameters>::Fp2Params>;
 22type Fp4G<P> = Fp4Var<<P as MNT4Parameters>::Fp4Params>;
 23/// A variable corresponding to `algebra_core::mnt4::GT`.
 24pub type GTVar<P> = Fp4G<P>;
 25
 26impl<P: MNT4Parameters> PairingVar<P> {
 27    #[tracing::instrument(target = "r1cs", skip(r))]
 28    pub(crate) fn doubling_step_for_flipped_miller_loop(
 29        r: &G2ProjectiveExtendedVar<P>,
 30    ) -> Result<(G2ProjectiveExtendedVar<P>, AteDoubleCoefficientsVar<P>), SynthesisError> {
 31        let a = r.t.square()?;
 32        let b = r.x.square()?;
 33        let c = r.y.square()?;
 34        let d = c.square()?;
 35        let e = (&r.x + &c).square()? - &b - &d;
 36        let f = (b.double()? + &b) + &a * P::TWIST_COEFF_A;
 37        let g = f.square()?;
 38
 39        let d_eight = d.double()?.double()?.double()?;
 40
 41        let e2 = e.double()?;
 42        let x = &g - &e2.double()?;
 43
 44        let y = &f * (&e2 - &x) - &d_eight;
 45        let z = (&r.y + &r.z).square()? - &c - &r.z.square()?;
 46        let t = z.square()?;
 47
 48        let r2 = G2ProjectiveExtendedVar { x, y, z, t };
 49        let c_h = (&r2.z + &r.t).square()? - &r2.t - &a;
 50        let c_4c = c.double()?.double()?;
 51        let c_j = (&f + &r.t).square()? - &g - &a;
 52        let c_l = (&f + &r.x).square()? - &g - &b;
 53        let coeff = AteDoubleCoefficientsVar {
 54            c_h,
 55            c_4c,
 56            c_j,
 57            c_l,
 58        };
 59
 60        Ok((r2, coeff))
 61    }
 62
 63    #[tracing::instrument(target = "r1cs", skip(r))]
 64    pub(crate) fn mixed_addition_step_for_flipped_miller_loop(
 65        x: &Fp2G<P>,
 66        y: &Fp2G<P>,
 67        r: &G2ProjectiveExtendedVar<P>,
 68    ) -> Result<(G2ProjectiveExtendedVar<P>, AteAdditionCoefficientsVar<P>), SynthesisError> {
 69        let a = y.square()?;
 70        let b = &r.t * x;
 71        let d = ((&r.z + y).square()? - &a - &r.t) * &r.t;
 72        let h = &b - &r.x;
 73        let i = h.square()?;
 74        let e = i.double()?.double()?;
 75        let j = &h * &e;
 76        let v = &r.x * &e;
 77        let ry2 = r.y.double()?;
 78        let l1 = &d - &ry2;
 79
 80        let x = l1.square()? - &j - &v.double()?;
 81        let y = &l1 * &(&v - &x) - j * &ry2;
 82        let z = (&r.z + &h).square()? - &r.t - &i;
 83        let t = z.square()?;
 84
 85        let r2 = G2ProjectiveExtendedVar {
 86            x,
 87            y,
 88            z: z.clone(),
 89            t,
 90        };
 91        let coeff = AteAdditionCoefficientsVar { c_l1: l1, c_rz: z };
 92
 93        Ok((r2, coeff))
 94    }
 95
 96    #[tracing::instrument(target = "r1cs", skip(p, q))]
 97    pub(crate) fn ate_miller_loop(
 98        p: &G1PreparedVar<P>,
 99        q: &G2PreparedVar<P>,
100    ) -> Result<Fp4G<P>, SynthesisError> {
101        let l1_coeff = Fp2G::<P>::new(p.x.clone(), FpVar::<P::Fp>::zero()) - &q.x_over_twist;
102
103        let mut f = Fp4G::<P>::one();
104
105        let mut dbl_idx: usize = 0;
106        let mut add_idx: usize = 0;
107
108        // code below gets executed for all bits (EXCEPT the MSB itself) of
109        // mnt6_param_p (skipping leading zeros) in MSB to LSB order
110        for bit in BitIteratorBE::without_leading_zeros(P::ATE_LOOP_COUNT).skip(1) {
111            let dc = &q.double_coefficients[dbl_idx];
112            dbl_idx += 1;
113
114            let g_rr_at_p = Fp4G::<P>::new(
115                &dc.c_l - &dc.c_4c - &dc.c_j * &p.x_twist,
116                &dc.c_h * &p.y_twist,
117            );
118
119            f = f.square()? * &g_rr_at_p;
120
121            if bit {
122                let ac = &q.addition_coefficients[add_idx];
123                add_idx += 1;
124
125                let g_rq_at_p = Fp4G::<P>::new(
126                    &ac.c_rz * &p.y_twist,
127                    (&q.y_over_twist * &ac.c_rz + &l1_coeff * &ac.c_l1).negate()?,
128                );
129                f *= &g_rq_at_p;
130            }
131        }
132
133        if P::ATE_IS_LOOP_COUNT_NEG {
134            let ac = &q.addition_coefficients[add_idx];
135
136            let g_rnegr_at_p = Fp4G::<P>::new(
137                &ac.c_rz * &p.y_twist,
138                (&q.y_over_twist * &ac.c_rz + &l1_coeff * &ac.c_l1).negate()?,
139            );
140            f = (&f * &g_rnegr_at_p).inverse()?;
141        }
142
143        Ok(f)
144    }
145
146    #[tracing::instrument(target = "r1cs", skip(value))]
147    pub(crate) fn final_exponentiation(value: &Fp4G<P>) -> Result<GTVar<P>, SynthesisError> {
148        let value_inv = value.inverse()?;
149        let value_to_first_chunk = Self::final_exponentiation_first_chunk(value, &value_inv)?;
150        let value_inv_to_first_chunk = Self::final_exponentiation_first_chunk(&value_inv, value)?;
151        Self::final_exponentiation_last_chunk(&value_to_first_chunk, &value_inv_to_first_chunk)
152    }
153
154    #[tracing::instrument(target = "r1cs", skip(elt, elt_inv))]
155    fn final_exponentiation_first_chunk(
156        elt: &Fp4G<P>,
157        elt_inv: &Fp4G<P>,
158    ) -> Result<Fp4G<P>, SynthesisError> {
159        // (q^2-1)
160
161        // elt_q2 = elt^(q^2)
162        let elt_q2 = elt.unitary_inverse()?;
163        // elt_q2_over_elt = elt^(q^2-1)
164        Ok(elt_q2 * elt_inv)
165    }
166
167    #[tracing::instrument(target = "r1cs", skip(elt, elt_inv))]
168    fn final_exponentiation_last_chunk(
169        elt: &Fp4G<P>,
170        elt_inv: &Fp4G<P>,
171    ) -> Result<Fp4G<P>, SynthesisError> {
172        let elt_clone = elt.clone();
173        let elt_inv_clone = elt_inv.clone();
174
175        let mut elt_q = elt.clone();
176        elt_q.frobenius_map_in_place(1)?;
177
178        let w1_part = elt_q.cyclotomic_exp(&P::FINAL_EXPONENT_LAST_CHUNK_1)?;
179        let w0_part = if P::FINAL_EXPONENT_LAST_CHUNK_W0_IS_NEG {
180            elt_inv_clone.cyclotomic_exp(&P::FINAL_EXPONENT_LAST_CHUNK_ABS_OF_W0)?
181        } else {
182            elt_clone.cyclotomic_exp(&P::FINAL_EXPONENT_LAST_CHUNK_ABS_OF_W0)?
183        };
184
185        Ok(w1_part * &w0_part)
186    }
187}
188
189impl<P: MNT4Parameters> PG<MNT4<P>, P::Fp> for PairingVar<P> {
190    type G1Var = G1Var<P>;
191    type G2Var = G2Var<P>;
192    type G1PreparedVar = G1PreparedVar<P>;
193    type G2PreparedVar = G2PreparedVar<P>;
194    type GTVar = GTVar<P>;
195
196    #[tracing::instrument(target = "r1cs")]
197    fn miller_loop(
198        ps: &[Self::G1PreparedVar],
199        qs: &[Self::G2PreparedVar],
200    ) -> Result<Self::GTVar, SynthesisError> {
201        let mut result = Fp4G::<P>::one();
202        for (p, q) in ps.iter().zip(qs) {
203            result *= Self::ate_miller_loop(p, q)?;
204        }
205
206        Ok(result)
207    }
208
209    #[tracing::instrument(target = "r1cs")]
210    fn final_exponentiation(r: &Self::GTVar) -> Result<Self::GTVar, SynthesisError> {
211        Self::final_exponentiation(r)
212    }
213
214    #[tracing::instrument(target = "r1cs")]
215    fn prepare_g1(p: &Self::G1Var) -> Result<Self::G1PreparedVar, SynthesisError> {
216        Self::G1PreparedVar::from_group_var(p)
217    }
218
219    #[tracing::instrument(target = "r1cs")]
220    fn prepare_g2(q: &Self::G2Var) -> Result<Self::G2PreparedVar, SynthesisError> {
221        Self::G2PreparedVar::from_group_var(q)
222    }
223}