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use blstrs::Compress;
use crossbeam_channel::bounded;
use ff::{Field, PrimeField};
use group::{prime::PrimeCurveAffine, Curve, Group};
use log::{debug, info};
use pairing::{Engine, MultiMillerLoop};
use rayon::prelude::*;
use serde::Serialize;
use super::{
accumulator::PairingChecks,
inner_product,
prove::polynomial_evaluation_product_form_from_transcript,
structured_scalar_power,
transcript::{Challenge, Transcript},
AggregateProof, AggregateProofAndInstance, KZGOpening, VerifierSRS,
};
use crate::groth16::{
aggregate::AggregateVersion,
multiscalar::{par_multiscalar, MultiscalarPrecomp, ScalarList},
PreparedVerifyingKey,
};
use bellpepper_core::SynthesisError;
use std::default::Default;
use std::ops::{AddAssign, MulAssign, SubAssign};
use std::time::Instant;
/// Verifies the aggregated proofs thanks to the Groth16 verifying key, the
/// verifier SRS from the aggregation scheme, all the public inputs of the
/// proofs and the aggregated proof.
///
/// WARNING: transcript_include represents everything that should be included in
/// the transcript from outside the boundary of this function. This is especially
/// relevant for ALL public inputs of ALL individual proofs. In the regular case,
/// one should input ALL public inputs from ALL proofs aggregated. However, IF ALL the
/// public inputs are **fixed, and public before the aggregation time**, then there is
/// no need to hash those. The reason we specify this extra assumption is because hashing
/// the public inputs from the decoded form can take quite some time depending on the
/// number of proofs and public inputs (+100ms in our case). In the case of Filecoin, the only
/// non-fixed part of the public inputs are the challenges derived from a seed. Even though this
/// seed comes from a random beeacon, we are hashing this as a safety precaution.
pub fn verify_aggregate_proof<E, R>(
ip_verifier_srs: &VerifierSRS<E>,
pvk: &PreparedVerifyingKey<E>,
rng: R,
public_inputs: &[Vec<E::Fr>],
proof: &AggregateProof<E>,
transcript_include: &[u8],
version: AggregateVersion,
) -> Result<bool, SynthesisError>
where
E: MultiMillerLoop + std::fmt::Debug,
E::Fr: Serialize,
<E as Engine>::Gt: Compress + Serialize,
E::G1: Serialize,
E::G1Affine: Serialize,
E::G2Affine: Serialize,
R: rand_core::RngCore + Send,
{
info!("verify_aggregate_proof");
proof.parsing_check()?;
for pub_input in public_inputs {
if (pub_input.len() + 1) != pvk.ic.len() {
return Err(SynthesisError::MalformedVerifyingKey);
}
}
if public_inputs.len() != proof.tmipp.gipa.nproofs as usize {
return Err(SynthesisError::MalformedProofs(
"public inputs length does not match nproofs".to_string(),
));
}
let hcom = Transcript::<E>::new("hcom")
.write(&proof.com_ab)
.write(&proof.com_c)
.into_challenge();
// Random linear combination of proofs
let r = Transcript::<E>::new("random-r")
.write(&hcom)
.write(&transcript_include)
.into_challenge();
let pairing_checks = PairingChecks::new(rng);
let pairing_checks_copy = &pairing_checks;
// 1.Check TIPA proof ab
// 2.Check TIPA proof c
// s.spawn(move |_| {
let now = Instant::now();
verify_tipp_mipp::<E, R>(
ip_verifier_srs,
proof,
&r, // we give the extra r as it's not part of the proof itself - it is simply used on top for the groth16 aggregation
pairing_checks_copy,
&hcom,
version,
);
debug!("TIPP took {} ms", now.elapsed().as_millis(),);
// Check aggregate pairing product equation
// SUM of a geometric progression
// SUM a^i = (1 - a^n) / (1 - a) = -(1-a^n)/-(1-a)
// = (a^n - 1) / (a - 1)
info!("checking aggregate pairing");
let mut r_sum = r.pow_vartime(&[public_inputs.len() as u64]);
r_sum.sub_assign(&E::Fr::ONE);
let b = (*r - E::Fr::ONE).invert().unwrap();
r_sum.mul_assign(&b);
// The following parts 3 4 5 are independently computing the parts of the Groth16
// verification equation
// NOTE From this point on, we are only checking *one* pairing check (the Groth16
// verification equation) so we don't need to randomize as all other checks are being
// randomized already. When merging all pairing checks together, this will be the only one
// non-randomized.
//
let (r_vec_sender, r_vec_receiver) = bounded(1);
let now = Instant::now();
r_vec_sender
.send(structured_scalar_power(public_inputs.len(), &*r))
.unwrap();
let elapsed = now.elapsed().as_millis();
debug!("generation of r vector: {}ms", elapsed);
par! {
// 3. Compute left part of the final pairing equation
let left = {
let mut alpha_g1_r_sum = pvk.alpha_g1;
alpha_g1_r_sum.mul_assign(r_sum);
E::multi_miller_loop(&[(&alpha_g1_r_sum.to_affine(), &pvk.beta_g2)])
},
// 4. Compute right part of the final pairing equation
let right = {
E::multi_miller_loop(&[(
// e(c^r vector form, h^delta)
// let agg_c = inner_product::multiexponentiation::<E::G1Affine>(&c, r_vec)
&proof.agg_c.to_affine(),
&pvk.delta_g2,
)])
},
// 5. compute the middle part of the final pairing equation, the one
// with the public inputs
let middle = {
// We want to compute MUL(i:0 -> l) S_i ^ (SUM(j:0 -> n) ai,j * r^j)
// this table keeps tracks of incremental computation of each i-th
// exponent to later multiply with S_i
// The index of the table is i, which is an index of the public
// input element
// We incrementally build the r vector and the table
// NOTE: in this version it's not r^2j but simply r^j
let l = public_inputs[0].len();
let mut g_ic = pvk.ic_projective[0];
g_ic.mul_assign(r_sum);
let powers = r_vec_receiver.recv().unwrap();
let now = Instant::now();
// now we do the multi exponentiation
let getter = |i: usize| -> <E::Fr as PrimeField>::Repr {
// i denotes the column of the public input, and j denotes which public input
let mut c = public_inputs[0][i];
for j in 1..public_inputs.len() {
let mut ai = public_inputs[j][i];
ai.mul_assign(&powers[j]);
c.add_assign(&ai);
}
c.to_repr()
};
let totsi = par_multiscalar::<_, E::G1Affine>(
&ScalarList::Getter(getter, l),
&pvk.multiscalar.at_point(1),
std::mem::size_of::<<E::Fr as PrimeField>::Repr>() * 8,
);
g_ic.add_assign(&totsi);
let ml = E::multi_miller_loop(&[(&g_ic.to_affine(), &pvk.gamma_g2)]);
let elapsed = now.elapsed().as_millis();
debug!("table generation: {}ms", elapsed);
ml
}
};
pairing_checks_copy.merge_nonrandom(
vec![left, middle, right],
// final value ip_ab is what we want to compare in the groth16
// aggregated equation A * B
proof.ip_ab,
);
let res = pairing_checks.verify();
info!("aggregate verify done");
res
}
/// verification of related instances i.e. when instances are given by
/// [a1, ... , an, b1, ... , bn], [b1, ... , bn, c1, ..., cn], [c1, ..., cn, d1, ..., dn] etc
#[allow(clippy::too_many_arguments)]
pub fn verify_aggregate_proof_and_aggregate_instances<
E: Engine + std::fmt::Debug,
R: rand::RngCore + Send,
>(
ip_verifier_srs: &VerifierSRS<E>,
pvk: &PreparedVerifyingKey<E>,
rng: R,
public_inputs: &[E::Fr],
public_outputs: &[E::Fr],
aggregate_proof_and_instance: &AggregateProofAndInstance<E>,
transcript_include: &[u8],
version: AggregateVersion,
) -> Result<bool, SynthesisError>
where
E: MultiMillerLoop + std::fmt::Debug,
E::Fr: Serialize,
<E as Engine>::Gt: Compress + Serialize,
E::G1: Serialize,
E::G1Affine: Serialize,
E::G2Affine: Serialize,
R: rand_core::RngCore + Send,
{
info!("verify_aggregate_proof");
aggregate_proof_and_instance.parsing_check()?;
let proof = &aggregate_proof_and_instance.pi_agg;
if (public_inputs.len() + public_outputs.len() + 1) != pvk.ic.len() {
return Err(SynthesisError::MalformedVerifyingKey);
}
let transcript_new = Transcript::<E>::new("transcript-with-coms")
.write(&aggregate_proof_and_instance.com_f)
.write(&aggregate_proof_and_instance.com_w0)
.write(&aggregate_proof_and_instance.com_wd)
.write(&transcript_include)
.into_bytes();
let hcom = Transcript::<E>::new("hcom")
.write(&proof.com_ab)
.write(&proof.com_c)
.into_challenge();
// Random linear combination of proofs
let r = Transcript::<E>::new("random-r")
.write(&hcom)
.write(&transcript_new)
.into_challenge();
let r_f = (*r).pow_vartime(&[1u64]);
// let pairing_checks_instance: PairingChecks<E,R> = PairingChecks::new(rng2);
let pairing_checks: PairingChecks<E, R> = PairingChecks::new(rng);
let pairing_checks_copy = &pairing_checks;
for (i, public_input) in public_inputs.iter().enumerate() {
// check com_f has a_0 as zero'th coefficient: com_f - a0 * g
let d = (aggregate_proof_and_instance.com_f[i] - (ip_verifier_srs.g * public_input))
.to_affine();
pairing_checks_copy.merge_miller_inputs(
&[
(&d, &ip_verifier_srs.h.to_affine()),
(
&aggregate_proof_and_instance.com_w0[i].to_affine(),
&ip_verifier_srs.h_alpha.to_affine(),
),
],
&<E as Engine>::Gt::generator(),
);
// check com_f has bounded degree
pairing_checks_copy.merge_miller_inputs(
&[
(
&aggregate_proof_and_instance.com_f[i].to_affine(),
&ip_verifier_srs.h_alpha_d.to_affine(),
),
(
&aggregate_proof_and_instance.com_wd[i].to_affine(),
&ip_verifier_srs.h.to_affine(),
),
],
&<E as Engine>::Gt::generator(),
);
// check com_f evaluates to d2 at r d2 = F g^(-eval)
let d2 = (aggregate_proof_and_instance.com_f[i]
- (ip_verifier_srs.g * aggregate_proof_and_instance.f_eval[i]))
.to_affine();
let d2 = -d2;
let d3 = (ip_verifier_srs.h_alpha - (ip_verifier_srs.h * r_f)).to_affine();
pairing_checks_copy.merge_miller_inputs(
&[
(&d2, &ip_verifier_srs.h.to_affine()),
(
&aggregate_proof_and_instance.f_eval_proof[i].to_affine(),
&d3,
),
],
&<E as Engine>::Gt::generator(),
);
}
rayon::scope(move |_s| {
// 1.Check TIPA proof ab
// 2.Check TIPA proof c
// s.spawn(move |_| {
let now = Instant::now();
verify_tipp_mipp::<E, R>(
ip_verifier_srs,
proof,
&r, // we give the extra r as it's not part of the proof itself - it is simply used on top for the groth16 aggregation
pairing_checks_copy,
&hcom,
version,
);
debug!("TIPP took {} ms", now.elapsed().as_millis(),);
// Check aggregate pairing product equation
// SUM of a geometric progression
// SUM a^i = (1 - a^n) / (1 - a) = -(1-a^n)/-(1-a)
// = (a^n - 1) / (a - 1)
info!("checking aggregate pairing");
let mut r_sum = r.pow_vartime(&[ip_verifier_srs.n as u64]);
r_sum.sub_assign(&E::Fr::ONE);
let b = (*r - E::Fr::ONE).invert().unwrap();
r_sum.mul_assign(&b);
// The following parts 3 4 5 are independently computing the parts of the Groth16
// verification equation
// NOTE From this point on, we are only checking *one* pairing check (the Groth16
// verification equation) so we don't need to randomize as all other checks are being
// randomized already. When merging all pairing checks together, this will be the only one
// non-randomized.
//
par! {
// 3. Compute left part of the final pairing equation
let left = {
let alpha_g1_r_sum = pvk.alpha_g1 * r_sum;
E::multi_miller_loop(&[(&alpha_g1_r_sum.to_affine(), &pvk.beta_g2)])
},
let middle = {
// first public input is 1 for all circuits.
let mut g_ic = pvk.ic_projective[0] * r_sum;
for i in 0..public_inputs.len() {
// g_ic = prod_i Si^(f_i(r)) S_(i + n)^( 1/r( f_i(r) - a0) + a_n r^(n-1))
g_ic += pvk.ic[1 + i] * aggregate_proof_and_instance.f_eval[i];
// d = f(r) - a0
let mut d = aggregate_proof_and_instance.f_eval[i] - public_inputs[i];
// d = (1/r) (f(r) - a0)
d *= &r.invert().unwrap();
// d = (1/r) (f(r) - a0 ) + r^(n-1) an
let n_neg_one = (ip_verifier_srs.n - 1) as u64;
d += public_outputs[i] * r.pow_vartime(&[n_neg_one]) ;
// pk_ic_in = S_(i + m + 1)^d
let pk_ic_in = pvk.ic[1 + i + public_inputs.len()] * d;
g_ic += pk_ic_in;
}
E::multi_miller_loop(&[(&g_ic.to_affine() , &pvk.gamma_g2)])
},
// 4. Compute right part of the final pairing equation
let right = {
E::multi_miller_loop(&[(
// e(c^r vector form, h^delta)
// let agg_c = inner_product::multiexponentiation::<E::G1Affine>(&c, r_vec)
&proof.agg_c.to_affine(),
&pvk.delta_g2,
)])
}
};
pairing_checks_copy.merge_nonrandom(
vec![left, middle, right],
// final value ip_ab is what we want to compare in the groth16
// aggregated equation A * B
proof.ip_ab,
);
});
let res = pairing_checks.verify();
info!("aggregate verify done");
res
}
/// verify_tipp_mipp returns a pairing equation to check the tipp proof. $r$ is
/// the randomness used to produce a random linear combination of A and B and
/// used in the MIPP part with C
fn verify_tipp_mipp<E, R>(
v_srs: &VerifierSRS<E>,
proof: &AggregateProof<E>,
r_shift: &E::Fr,
pairing_checks: &PairingChecks<E, R>,
hcom: &Challenge<E>,
version: AggregateVersion,
) where
E: MultiMillerLoop,
E::Fr: Serialize,
<E as Engine>::Gt: Compress + Serialize,
E::G1: Serialize,
E::G1Affine: Serialize,
E::G2Affine: Serialize,
R: rand_core::RngCore + Send,
{
info!("verify with srs shift");
let now = Instant::now();
// (T,U), Z for TIPP and MIPP and all challenges
let (final_res, final_r, challenges, challenges_inv, extra_challenge) =
gipa_verify_tipp_mipp(proof, r_shift, hcom, version);
debug!(
"TIPP verify: gipa verify tipp {}ms",
now.elapsed().as_millis()
);
// Verify commitment keys wellformed
let fvkey = proof.tmipp.gipa.final_vkey;
let fwkey = proof.tmipp.gipa.final_wkey;
// we take reference so they are able to be copied in the par! macro
let final_a = &proof.tmipp.gipa.final_a;
let final_b = &proof.tmipp.gipa.final_b;
let final_c = &proof.tmipp.gipa.final_c;
let final_zab = &final_res.zab;
let final_tab = &final_res.tab;
let final_uab = &final_res.uab;
let final_tc = &final_res.tc;
let final_uc = &final_res.uc;
// KZG challenge point
let c = match version {
AggregateVersion::V1 => Transcript::<E>::new("random-z")
.write(&challenges[0])
.write(&fvkey.0)
.write(&fvkey.1)
.write(&fwkey.0)
.write(&fwkey.1)
.into_challenge(),
AggregateVersion::V2 => Transcript::<E>::new("random-z")
.write(&extra_challenge)
.write(&fvkey.0)
.write(&fvkey.1)
.write(&fwkey.0)
.write(&fwkey.1)
.write(final_a)
.write(final_b)
.write(final_c)
.into_challenge(),
};
let now = Instant::now();
par! {
// check the opening proof for v
let _vtuple = verify_kzg_v(
v_srs,
&fvkey,
&proof.tmipp.vkey_opening,
&challenges_inv,
&c,
pairing_checks,
),
// check the opening proof for w - note that w has been rescaled by $r^{-1}$
let _wtuple = verify_kzg_w(
v_srs,
&fwkey,
&proof.tmipp.wkey_opening,
&challenges,
&r_shift.invert().unwrap(),
&c,
pairing_checks,
),
//
// We create a sequence of pairing tuple that we aggregate together at
// the end to perform only once the final exponentiation.
//
// TIPP
// z = e(A,B)
let _check_z = pairing_checks.merge_miller_inputs(&[(final_a, final_b)], final_zab),
// final_aB.0 = T = e(A,v1)e(w1,B)
let _check_ab0 = pairing_checks.merge_miller_inputs(&[(final_a, &fvkey.0),(&fwkey.0, final_b)], final_tab),
// final_aB.1 = U = e(A,v2)e(w2,B)
let _check_ab1 = pairing_checks.merge_miller_inputs(&[(final_a, &fvkey.1),(&fwkey.1, final_b)], final_uab),
// MIPP
// Verify base inner product commitment
// Z == c ^ r
let final_z =
inner_product::multiexponentiation::<E::G1Affine>(&[*final_c],
&[final_r]),
// Check commiment correctness
// T = e(C,v1)
let _check_t = pairing_checks.merge_miller_inputs(&[(final_c,&fvkey.0)], final_tc),
// U = e(A,v2)
let _check_u = pairing_checks.merge_miller_inputs(&[(final_c,&fvkey.1)], final_uc)
};
match final_z {
Err(e) => pairing_checks.report_err(e),
Ok(z) => {
debug!(
"TIPP verify: parallel checks before merge: {}ms",
now.elapsed().as_millis(),
);
let b = z == final_res.zc;
// only check that doesn't require pairing so we can give a tuple
// that will render the equation wrong in case it's false
if !b {
pairing_checks.invalidate();
}
}
}
}
/// gipa_verify_tipp_mipp recurse on the proof and statement and produces the final
/// values to be checked by TIPP and MIPP verifier, namely, for TIPP for example:
/// * T,U: the final commitment values of A and B
/// * Z the final product between A and B.
/// * Challenges are returned in inverse order as well to avoid
/// repeating the operation multiple times later on.
/// * There are T,U,Z vectors as well for the MIPP relationship. Both TIPP and
/// MIPP share the same challenges however, enabling to re-use common operations
/// between them, such as the KZG proof for commitment keys.
#[allow(clippy::type_complexity)]
fn gipa_verify_tipp_mipp<E>(
proof: &AggregateProof<E>,
r_shift: &E::Fr,
hcom: &E::Fr,
version: AggregateVersion,
) -> (GipaTUZ<E>, E::Fr, Vec<E::Fr>, Vec<E::Fr>, E::Fr)
where
E: MultiMillerLoop,
E::Fr: Serialize,
<E as Engine>::Gt: Compress + Serialize,
E::G1: Serialize,
E::G1Affine: Serialize,
E::G2Affine: Serialize,
{
info!("gipa verify TIPP [version {}]", version);
let gipa = &proof.tmipp.gipa;
// COM(A,B) = PROD e(A,B) given by prover
let comms_ab = &gipa.comms_ab;
// COM(C,r) = SUM C^r given by prover
let comms_c = &gipa.comms_c;
// Z vectors coming from the GIPA proofs
let zs_ab = &gipa.z_ab;
let zs_c = &gipa.z_c;
let now = Instant::now();
let mut challenges = Vec::new();
let mut challenges_inv = Vec::new();
let mut c_inv: E::Fr = *Transcript::<E>::new("gipa-0")
.write(hcom)
.write(&proof.ip_ab)
.write(&proof.agg_c)
.write(&r_shift)
.into_challenge();
let mut c = c_inv.invert().unwrap();
// We first generate all challenges as this is the only consecutive process
// that can not be parallelized then we scale the commitments in a
// parallelized way
for (i, ((comm_ab, z_ab), (comm_c, z_c))) in comms_ab
.iter()
.zip(zs_ab.iter())
.zip(comms_c.iter().zip(zs_c.iter()))
.enumerate()
{
let (tab_l, tab_r) = comm_ab;
let (zab_l, zab_r) = z_ab;
let (tc_l, tc_r) = comm_c;
let (zc_l, zc_r) = z_c;
// Fiat-Shamir challenge
// combine both TIPP and MIPP transcript
if i == 0 {
match version {
AggregateVersion::V1 => {
// already generated c_inv and c outside of the loop
}
AggregateVersion::V2 => {
// in this version we do fiat shamir with the first inputs
c_inv = *Transcript::<E>::new("gipa-0")
.write(&c_inv)
.write(&zab_l)
.write(&zab_r)
.write(&zc_l)
.write(&zc_r)
.write(&tab_l.0)
.write(&tab_l.1)
.write(&tab_r.0)
.write(&tab_r.1)
.write(&tc_l.0)
.write(&tc_l.1)
.write(&tc_r.0)
.write(&tc_r.1)
.into_challenge();
c = c_inv.invert().unwrap();
}
}
} else {
c_inv = *Transcript::<E>::new(&format!("gipa-{}", i))
.write(&c_inv)
.write(&zab_l)
.write(&zab_r)
.write(&zc_l)
.write(&zc_r)
.write(&tab_l.0)
.write(&tab_l.1)
.write(&tab_r.0)
.write(&tab_r.1)
.write(&tc_l.0)
.write(&tc_l.1)
.write(&tc_r.0)
.write(&tc_r.1)
.into_challenge();
c = c_inv.invert().unwrap();
}
challenges.push(c);
challenges_inv.push(c_inv);
info!("verify: challenge {} -> {:?}", i, c);
}
debug!(
"TIPP verify: gipa challenge gen took {}ms",
now.elapsed().as_millis()
);
let now = Instant::now();
// output of the pair commitment T and U in TIPP -> COM((v,w),A,B)
let (t_ab, u_ab) = proof.com_ab;
let z_ab = proof.ip_ab; // in the end must be equal to Z = A^r * B
let (final_zab_l, final_zab_r) = proof.tmipp.gipa.z_ab.last().unwrap();
let (final_zc_l, final_zc_r) = proof.tmipp.gipa.z_c.last().unwrap();
let (final_tab_l, final_tab_r) = proof.tmipp.gipa.comms_ab.last().unwrap();
let (final_tuc_l, final_tuc_r) = proof.tmipp.gipa.comms_c.last().unwrap();
// COM(v,C)
let (t_c, u_c) = proof.com_c;
let z_c = proof.agg_c; // in the end must be equal to Z = C^r
let mut final_res = GipaTUZ {
tab: t_ab,
uab: u_ab,
zab: z_ab,
tc: t_c,
uc: u_c,
zc: z_c,
};
// This extra challenge is simply done to make the bridge between the
// MIPP/TIPP proofs and the KZG proofs, but is not used in TIPP/MIPP.
let extra_challenge = *Transcript::<E>::new("gipa-extra-link")
.write(&challenges.last().unwrap())
.write(&proof.tmipp.gipa.final_a)
.write(&proof.tmipp.gipa.final_b)
.write(&proof.tmipp.gipa.final_c)
.write(&final_zab_l)
.write(&final_zab_r)
.write(&final_zc_l)
.write(&final_zc_r)
.write(&final_tab_l.0)
.write(&final_tab_l.1)
.write(&final_tab_r.0)
.write(&final_tab_r.1)
.write(&final_tuc_l.0)
.write(&final_tuc_l.1)
.write(&final_tuc_r.0)
.write(&final_tuc_r.1)
.into_challenge();
debug!("verify: extra challenge {:?}", extra_challenge);
// we first multiply each entry of the Z U and L vectors by the respective
// challenges independently
// Since at the end we want to multiple all "t" values together, we do
// multiply all of them in parrallel and then merge then back at the end.
// same for u and z.
#[allow(clippy::upper_case_acronyms)]
enum Op<'a, E>
where
E: MultiMillerLoop,
{
TAB(&'a <E as Engine>::Gt, &'a E::Fr),
UAB(&'a <E as Engine>::Gt, &'a E::Fr),
ZAB(&'a <E as Engine>::Gt, &'a E::Fr),
TC(&'a <E as Engine>::Gt, &'a E::Fr),
UC(&'a <E as Engine>::Gt, &'a E::Fr),
ZC(&'a E::G1, &'a E::Fr),
}
let res = comms_ab
.par_iter()
.zip(zs_ab.par_iter())
.zip(comms_c.par_iter().zip(zs_c.par_iter()))
.zip(challenges.par_iter().zip(challenges_inv.par_iter()))
.flat_map(|(((comm_ab, z_ab), (comm_c, z_c)), (c, c_inv))| {
// T and U values for right and left for AB part
let ((tab_l, uab_l), (tab_r, uab_r)) = comm_ab;
let (zab_l, zab_r) = z_ab;
// T and U values for right and left for C part
let ((tc_l, uc_l), (tc_r, uc_r)) = comm_c;
let (zc_l, zc_r) = z_c;
// we multiple left side by x and right side by x^-1
vec![
Op::TAB::<E>(tab_l, c),
Op::TAB(tab_r, c_inv),
Op::UAB(uab_l, c),
Op::UAB(uab_r, c_inv),
Op::ZAB(zab_l, c),
Op::ZAB(zab_r, c_inv),
Op::TC::<E>(tc_l, c),
Op::TC(tc_r, c_inv),
Op::UC(uc_l, c),
Op::UC(uc_r, c_inv),
Op::ZC(zc_l, c),
Op::ZC(zc_r, c_inv),
]
})
.fold(GipaTUZ::<E>::default, |mut res, op: Op<E>| {
match op {
Op::TAB(tx, c) => {
let tx = *tx * c;
res.tab += tx;
}
Op::UAB(ux, c) => {
let ux = *ux * c;
res.uab += ux;
}
Op::ZAB(zx, c) => {
let zx = *zx * c;
res.zab += zx;
}
Op::TC(tx, c) => {
let tx = *tx * c;
res.tc += tx;
}
Op::UC(ux, c) => {
let ux = *ux * c;
res.uc += ux;
}
Op::ZC(zx, c) => {
let zx = *zx * c;
res.zc += zx;
}
}
res
})
.reduce(GipaTUZ::default, |mut acc_res, res| {
acc_res.merge(&res);
acc_res
});
// we reverse the order because the polynomial evaluation routine expects
// the challenges in reverse order.Doing it here allows us to compute the final_r
// in log time. Challenges are used as well in the KZG verification checks.
challenges.reverse();
challenges_inv.reverse();
let ref_final_res = &mut final_res;
let ref_challenges_inv = &challenges_inv;
ref_final_res.merge(&res);
let final_r = polynomial_evaluation_product_form_from_transcript(
ref_challenges_inv,
r_shift,
&E::Fr::ONE,
);
debug!(
"TIPP verify: gipa prep and accumulate took {}ms",
now.elapsed().as_millis()
);
(
final_res,
final_r,
challenges,
challenges_inv,
extra_challenge,
)
}
/// verify_kzg_opening_g2 takes a KZG opening, the final commitment key, SRS and
/// any shift (in TIPP we shift the v commitment by r^-1) and returns a pairing
/// tuple to check if the opening is correct or not.
pub fn verify_kzg_v<E, R>(
v_srs: &VerifierSRS<E>,
final_vkey: &(E::G2Affine, E::G2Affine),
vkey_opening: &KZGOpening<E::G2Affine>,
challenges: &[E::Fr],
kzg_challenge: &E::Fr,
pairing_checks: &PairingChecks<E, R>,
) where
E: MultiMillerLoop,
R: rand_core::RngCore + Send,
{
// f_v(z)
let vpoly_eval_z =
polynomial_evaluation_product_form_from_transcript(challenges, kzg_challenge, &E::Fr::ONE);
// -g such that when we test a pairing equation we only need to check if
// it's equal 1 at the end:
// e(a,b) = e(c,d) <=> e(a,b)e(-c,d) = 1
// e(A,B) = e(C,D) <=> e(A,B)e(-C,D) == 1 <=> e(A,B)e(C,D)^-1 == 1
let ng = (-v_srs.g).to_affine();
par! {
// e(g, C_f * h^{-y}) == e(v1 * g^{-x}, \pi) = 1
let _check1 = kzg_check_v::<E, R>(
v_srs,
ng,
*kzg_challenge,
vpoly_eval_z,
final_vkey.0.to_curve(),
v_srs.g_alpha,
vkey_opening.0,
pairing_checks,
),
// e(g, C_f * h^{-y}) == e(v2 * g^{-x}, \pi) = 1
let _check2 = kzg_check_v::<E, R>(
v_srs,
ng,
*kzg_challenge,
vpoly_eval_z,
final_vkey.1.to_curve(),
v_srs.g_beta,
vkey_opening.1,
pairing_checks,
)
};
}
#[allow(clippy::too_many_arguments)]
fn kzg_check_v<E, R>(
v_srs: &VerifierSRS<E>,
ng: E::G1Affine,
x: E::Fr,
y: E::Fr,
cf: E::G2,
vk: E::G1,
pi: E::G2Affine,
pairing_checks: &PairingChecks<E, R>,
) where
E: MultiMillerLoop,
R: rand_core::RngCore + Send,
{
// KZG Check: e(g, C_f * h^{-y}) = e(vk * g^{-x}, \pi)
// Transformed, such that
// e(-g, C_f * h^{-y}) * e(vk * g^{-x}, \pi) = 1
// C_f - (y * h)
let b = (cf - (v_srs.h * y)).to_affine();
// vk - (g * x)
let c = (vk - (v_srs.g * x)).to_affine();
pairing_checks.merge_miller_inputs(&[(&ng, &b), (&c, &pi)], &<E as Engine>::Gt::generator());
}
/// Similar to verify_kzg_opening_g2 but for g1.
pub fn verify_kzg_w<E, R>(
v_srs: &VerifierSRS<E>,
final_wkey: &(E::G1Affine, E::G1Affine),
wkey_opening: &KZGOpening<E::G1Affine>,
challenges: &[E::Fr],
r_shift: &E::Fr,
kzg_challenge: &E::Fr,
pairing_checks: &PairingChecks<E, R>,
) where
E: MultiMillerLoop,
R: rand_core::RngCore + Send,
{
// compute in parallel f(z) and z^n and then combines into f_w(z) = z^n * f(z)
par! {
let fz = polynomial_evaluation_product_form_from_transcript(challenges, kzg_challenge, r_shift),
let zn = kzg_challenge.pow_vartime(&[v_srs.n as u64])
};
let mut fwz = fz;
fwz.mul_assign(&zn);
let nh = -v_srs.h;
let nh = nh.to_affine();
par! {
// e(C_f * g^{-y}, h) = e(\pi, w1 * h^{-x})
let _check1 = kzg_check_w::<E, R>(
v_srs,
nh,
*kzg_challenge,
fwz,
final_wkey.0.to_curve(),
v_srs.h_alpha,
wkey_opening.0,
pairing_checks,
),
// e(C_f * g^{-y}, h) = e(\pi, w2 * h^{-x})
let _check2 = kzg_check_w::<E, R>(
v_srs,
nh,
*kzg_challenge,
fwz,
final_wkey.1.to_curve(),
v_srs.h_beta,
wkey_opening.1,
pairing_checks,
)
};
}
#[allow(clippy::too_many_arguments)]
fn kzg_check_w<E, R>(
v_srs: &VerifierSRS<E>,
nh: E::G2Affine,
x: E::Fr,
y: E::Fr,
cf: E::G1,
wk: E::G2,
pi: E::G1Affine,
pairing_checks: &PairingChecks<E, R>,
) where
E: MultiMillerLoop,
R: rand_core::RngCore + Send,
{
// KZG Check: e(C_f * g^{-y}, h) = e(\pi, wk * h^{-x})
// Transformed, such that
// e(C_f * g^{-y}, -h) * e(\pi, wk * h^{-x}) = 1
// C_f - (y * g)
let a = (cf - (v_srs.g * y)).to_affine();
// wk - (x * h)
let d = (wk - (v_srs.h * x)).to_affine();
pairing_checks.merge_miller_inputs(&[(&a, &nh), (&pi, &d)], &<E as Engine>::Gt::generator());
}
/// Keeps track of the variables that have been sent by the prover and must
/// be multiplied together by the verifier. Both MIPP and TIPP are merged
/// together.
#[allow(clippy::upper_case_acronyms)]
struct GipaTUZ<E>
where
E: MultiMillerLoop,
{
pub tab: <E as Engine>::Gt,
pub uab: <E as Engine>::Gt,
pub zab: <E as Engine>::Gt,
pub tc: <E as Engine>::Gt,
pub uc: <E as Engine>::Gt,
pub zc: E::G1,
}
impl<E> Default for GipaTUZ<E>
where
E: MultiMillerLoop,
{
fn default() -> Self {
Self {
tab: <E as Engine>::Gt::identity(),
uab: <E as Engine>::Gt::identity(),
zab: <E as Engine>::Gt::identity(),
tc: <E as Engine>::Gt::identity(),
uc: <E as Engine>::Gt::identity(),
zc: E::G1::identity(),
}
}
}
impl<E> GipaTUZ<E>
where
E: MultiMillerLoop,
{
fn merge(&mut self, other: &Self) {
self.tab += &other.tab;
self.uab += &other.uab;
self.zab += &other.zab;
self.tc += &other.tc;
self.uc += &other.uc;
self.zc += &other.zc;
}
}