1use ark_relations::r1cs::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 ark_ec::mnt4::{MNT4Config, MNT4};
 13
 14use core::marker::PhantomData;
 15
 16/// Specifies the constraints for computing a pairing in a MNT4 bilinear group.
 17pub struct PairingVar<P: MNT4Config>(PhantomData<P>);
 18
 19type Fp2G<P> = Fp2Var<<P as MNT4Config>::Fp2Config>;
 20type Fp4G<P> = Fp4Var<<P as MNT4Config>::Fp4Config>;
 21/// A variable corresponding to `ark_ec::mnt4::GT`.
 22pub type GTVar<P> = Fp4G<P>;
 23
 24impl<P: MNT4Config> PairingVar<P> {
 25    #[tracing::instrument(target = "r1cs", skip(r))]
 26    pub(crate) fn doubling_step_for_flipped_miller_loop(
 27        r: &G2ProjectiveExtendedVar<P>,
 28    ) -> Result<(G2ProjectiveExtendedVar<P>, AteDoubleCoefficientsVar<P>), SynthesisError> {
 29        let a = r.t.square()?;
 30        let b = r.x.square()?;
 31        let c = r.y.square()?;
 32        let d = c.square()?;
 33        let e = (&r.x + &c).square()? - &b - &d;
 34        let f = (b.double()? + &b) + &a * P::TWIST_COEFF_A;
 35        let g = f.square()?;
 36
 37        let d_eight = d.double()?.double()?.double()?;
 38
 39        let e2 = e.double()?;
 40        let x = &g - &e2.double()?;
 41
 42        let y = &f * (&e2 - &x) - &d_eight;
 43        let z = (&r.y + &r.z).square()? - &c - &r.z.square()?;
 44        let t = z.square()?;
 45
 46        let r2 = G2ProjectiveExtendedVar { x, y, z, t };
 47        let c_h = (&r2.z + &r.t).square()? - &r2.t - &a;
 48        let c_4c = c.double()?.double()?;
 49        let c_j = (&f + &r.t).square()? - &g - &a;
 50        let c_l = (&f + &r.x).square()? - &g - &b;
 51        let coeff = AteDoubleCoefficientsVar {
 52            c_h,
 53            c_4c,
 54            c_j,
 55            c_l,
 56        };
 57
 58        Ok((r2, coeff))
 59    }
 60
 61    #[tracing::instrument(target = "r1cs", skip(r))]
 62    pub(crate) fn mixed_addition_step_for_flipped_miller_loop(
 63        x: &Fp2G<P>,
 64        y: &Fp2G<P>,
 65        r: &G2ProjectiveExtendedVar<P>,
 66    ) -> Result<(G2ProjectiveExtendedVar<P>, AteAdditionCoefficientsVar<P>), SynthesisError> {
 67        let a = y.square()?;
 68        let b = &r.t * x;
 69        let d = ((&r.z + y).square()? - &a - &r.t) * &r.t;
 70        let h = &b - &r.x;
 71        let i = h.square()?;
 72        let e = i.double()?.double()?;
 73        let j = &h * &e;
 74        let v = &r.x * &e;
 75        let ry2 = r.y.double()?;
 76        let l1 = &d - &ry2;
 77
 78        let x = l1.square()? - &j - &v.double()?;
 79        let y = &l1 * &(&v - &x) - j * &ry2;
 80        let z = (&r.z + &h).square()? - &r.t - &i;
 81        let t = z.square()?;
 82
 83        let r2 = G2ProjectiveExtendedVar {
 84            x,
 85            y,
 86            z: z.clone(),
 87            t,
 88        };
 89        let coeff = AteAdditionCoefficientsVar { c_l1: l1, c_rz: z };
 90
 91        Ok((r2, coeff))
 92    }
 93
 94    #[tracing::instrument(target = "r1cs", skip(p, q))]
 95    pub(crate) fn ate_miller_loop(
 96        p: &G1PreparedVar<P>,
 97        q: &G2PreparedVar<P>,
 98    ) -> Result<Fp4G<P>, SynthesisError> {
 99        let l1_coeff = Fp2G::<P>::new(p.x.clone(), FpVar::<P::Fp>::zero()) - &q.x_over_twist;
100
101        let mut f = Fp4G::<P>::one();
102
103        let mut add_idx: usize = 0;
104
105        // code below gets executed for all bits (EXCEPT the MSB itself) of
106        // mnt6_param_p (skipping leading zeros) in MSB to LSB order
107        let y_over_twist_neg = &q.y_over_twist.negate()?;
108        for (dbl_idx, bit) in P::ATE_LOOP_COUNT.iter().skip(1).enumerate() {
109            let dc = &q.double_coefficients[dbl_idx];
110
111            let g_rr_at_p = Fp4G::<P>::new(
112                &dc.c_l - &dc.c_4c - &dc.c_j * &p.x_twist,
113                &dc.c_h * &p.y_twist,
114            );
115
116            f = f.square()? * &g_rr_at_p;
117
118            let g_rq_at_p;
119            // Compute l_{R,Q}(P) if bit == 1, and l_{R,-Q}(P) if bit == -1
120            if *bit == 1 {
121                let ac = &q.addition_coefficients[add_idx];
122                add_idx += 1;
123
124                g_rq_at_p = Fp4G::<P>::new(
125                    &ac.c_rz * &p.y_twist,
126                    (&q.y_over_twist * &ac.c_rz + &l1_coeff * &ac.c_l1).negate()?,
127                );
128            } else if *bit == -1 {
129                let ac = &q.addition_coefficients[add_idx];
130                add_idx += 1;
131
132                g_rq_at_p = Fp4G::<P>::new(
133                    &ac.c_rz * &p.y_twist,
134                    (y_over_twist_neg * &ac.c_rz + &l1_coeff * &ac.c_l1).negate()?,
135                );
136            } else {
137                continue;
138            }
139
140            f *= &g_rq_at_p;
141        }
142
143        if P::ATE_IS_LOOP_COUNT_NEG {
144            let ac = &q.addition_coefficients[add_idx];
145
146            let g_rnegr_at_p = Fp4G::<P>::new(
147                &ac.c_rz * &p.y_twist,
148                (&q.y_over_twist * &ac.c_rz + &l1_coeff * &ac.c_l1).negate()?,
149            );
150            f = (&f * &g_rnegr_at_p).inverse()?;
151        }
152
153        Ok(f)
154    }
155
156    #[tracing::instrument(target = "r1cs", skip(value))]
157    pub(crate) fn final_exponentiation(value: &Fp4G<P>) -> Result<GTVar<P>, SynthesisError> {
158        let value_inv = value.inverse()?;
159        let value_to_first_chunk = Self::final_exponentiation_first_chunk(value, &value_inv)?;
160        let value_inv_to_first_chunk = Self::final_exponentiation_first_chunk(&value_inv, value)?;
161        Self::final_exponentiation_last_chunk(&value_to_first_chunk, &value_inv_to_first_chunk)
162    }
163
164    #[tracing::instrument(target = "r1cs", skip(elt, elt_inv))]
165    fn final_exponentiation_first_chunk(
166        elt: &Fp4G<P>,
167        elt_inv: &Fp4G<P>,
168    ) -> Result<Fp4G<P>, SynthesisError> {
169        // (q^2-1)
170
171        // elt_q2 = elt^(q^2)
172        let elt_q2 = elt.unitary_inverse()?;
173        // elt_q2_over_elt = elt^(q^2-1)
174        Ok(elt_q2 * elt_inv)
175    }
176
177    #[tracing::instrument(target = "r1cs", skip(elt, elt_inv))]
178    fn final_exponentiation_last_chunk(
179        elt: &Fp4G<P>,
180        elt_inv: &Fp4G<P>,
181    ) -> Result<Fp4G<P>, SynthesisError> {
182        let elt_clone = elt.clone();
183        let elt_inv_clone = elt_inv.clone();
184
185        let mut elt_q = elt.clone();
186        elt_q.frobenius_map_in_place(1)?;
187
188        let w1_part = elt_q.cyclotomic_exp(&P::FINAL_EXPONENT_LAST_CHUNK_1)?;
189        let w0_part = if P::FINAL_EXPONENT_LAST_CHUNK_W0_IS_NEG {
190            elt_inv_clone.cyclotomic_exp(&P::FINAL_EXPONENT_LAST_CHUNK_ABS_OF_W0)?
191        } else {
192            elt_clone.cyclotomic_exp(&P::FINAL_EXPONENT_LAST_CHUNK_ABS_OF_W0)?
193        };
194
195        Ok(w1_part * &w0_part)
196    }
197}
198
199impl<P: MNT4Config> PG<MNT4<P>> for PairingVar<P> {
200    type G1Var = G1Var<P>;
201    type G2Var = G2Var<P>;
202    type G1PreparedVar = G1PreparedVar<P>;
203    type G2PreparedVar = G2PreparedVar<P>;
204    type GTVar = GTVar<P>;
205
206    #[tracing::instrument(target = "r1cs")]
207    fn miller_loop(
208        ps: &[Self::G1PreparedVar],
209        qs: &[Self::G2PreparedVar],
210    ) -> Result<Self::GTVar, SynthesisError> {
211        let mut result = Fp4G::<P>::one();
212        for (p, q) in ps.iter().zip(qs) {
213            result *= Self::ate_miller_loop(p, q)?;
214        }
215
216        Ok(result)
217    }
218
219    #[tracing::instrument(target = "r1cs")]
220    fn final_exponentiation(r: &Self::GTVar) -> Result<Self::GTVar, SynthesisError> {
221        Self::final_exponentiation(r)
222    }
223
224    #[tracing::instrument(target = "r1cs")]
225    fn prepare_g1(p: &Self::G1Var) -> Result<Self::G1PreparedVar, SynthesisError> {
226        Self::G1PreparedVar::from_group_var(p)
227    }
228
229    #[tracing::instrument(target = "r1cs")]
230    fn prepare_g2(q: &Self::G2Var) -> Result<Self::G2PreparedVar, SynthesisError> {
231        Self::G2PreparedVar::from_group_var(q)
232    }
233}