enabling batch opening and mock tests (#80)

- add mock circuits
- add vanilla and jellyfish plonk gates
- performance tuning

pcs/Cargo.toml

@@ -0,0 +1,51 @@
+[package]
+name = "pcs"
+version = "0.1.0"
+edition = "2021"
+
+# See more keys and their definitions at https://doc.rust-lang.org/cargo/reference/manifest.html
+
+[dependencies]
+
+ark-std = { version = "^0.3.0", default-features = false }
+ark-serialize = { version = "^0.3.0", default-features = false, features = [ "derive" ] }
+ark-ff = { version = "^0.3.0", default-features = false }
+ark-ec = { version = "^0.3.0", default-features = false }
+ark-poly = {version = "^0.3.0", default-features = false }
+ark-sponge = {version = "^0.3.0", default-features = false}
+ark-bls12-381 = { version = "0.3.0", default-features = false, features = [ "curve" ] }
+
+displaydoc = { version = "0.2.3", default-features = false }
+derivative = { version = "2", features = ["use_core"] }
+
+arithmetic = { path = "../arithmetic" }
+transcript = { path = "../transcript" }
+util = { path = "../util" }
+
+rayon = { version = "1.5.2", default-features = false, optional = true }
+itertools = { version = "0.10.4", optional = true }
+
+# Benchmarks
+[[bench]]
+name = "pcs-benches"
+path = "benches/bench.rs"
+harness = false
+
+[features]
+# default = [ "parallel", "print-trace" ]
+default = [ "parallel",]
+extensive_sanity_checks = [ ]
+parallel = [ 
+    "rayon",
+    "itertools",
+    "ark-std/parallel", 
+    "ark-ff/parallel",  
+    "ark-poly/parallel", 
+    "ark-ec/parallel",
+    "util/parallel",
+    "arithmetic/parallel", 
+    ]
+print-trace = [ 
+    "ark-std/print-trace",
+    "arithmetic/print-trace",
+    ] 

pcs/benches/bench.rs

@@ -0,0 +1,88 @@
+use ark_bls12_381::{Bls12_381, Fr};
+use ark_ff::UniformRand;
+use ark_poly::{DenseMultilinearExtension, MultilinearExtension};
+use ark_std::{rc::Rc, test_rng};
+use pcs::{
+    prelude::{MultilinearKzgPCS, PCSError, PolynomialCommitmentScheme},
+    StructuredReferenceString,
+};
+use std::time::Instant;
+
+fn main() -> Result<(), PCSError> {
+    bench_pcs()
+}
+
+fn bench_pcs() -> Result<(), PCSError> {
+    let mut rng = test_rng();
+
+    // normal polynomials
+    let uni_params = MultilinearKzgPCS::<Bls12_381>::gen_srs_for_testing(&mut rng, 18)?;
+
+    for nv in 4..19 {
+        let repetition = if nv < 10 {
+            100
+        } else if nv < 20 {
+            50
+        } else {
+            10
+        };
+
+        let poly = Rc::new(DenseMultilinearExtension::rand(nv, &mut rng));
+        let (ml_ck, ml_vk) = uni_params.0.trim(nv)?;
+        let (uni_ck, uni_vk) = uni_params.1.trim(nv)?;
+        let ck = (ml_ck, uni_ck);
+        let vk = (ml_vk, uni_vk);
+
+        let point: Vec<_> = (0..nv).map(|_| Fr::rand(&mut rng)).collect();
+
+        // commit
+        let com = {
+            let start = Instant::now();
+            for _ in 0..repetition {
+                let _commit = MultilinearKzgPCS::commit(&ck, &poly)?;
+            }
+
+            println!(
+                "KZG commit for {} variables: {} ns",
+                nv,
+                start.elapsed().as_nanos() / repetition as u128
+            );
+
+            MultilinearKzgPCS::commit(&ck, &poly)?
+        };
+
+        // open
+        let (proof, value) = {
+            let start = Instant::now();
+            for _ in 0..repetition {
+                let _open = MultilinearKzgPCS::open(&ck, &poly, &point)?;
+            }
+
+            println!(
+                "KZG open for {} variables: {} ns",
+                nv,
+                start.elapsed().as_nanos() / repetition as u128
+            );
+            MultilinearKzgPCS::open(&ck, &poly, &point)?
+        };
+
+        // verify
+        {
+            let start = Instant::now();
+            for _ in 0..repetition {
+                assert!(MultilinearKzgPCS::verify(
+                    &vk, &com, &point, &value, &proof
+                )?);
+            }
+            println!(
+                "KZG verify for {} variables: {} ns",
+                nv,
+                start.elapsed().as_nanos() / repetition as u128
+            );
+        }
+
+        println!("====================================");
+    }
+
+    Ok(())
+}

pcs/readme.md

@@ -0,0 +1,7 @@
+KZG based multilinear polynomial commitment
+-----
+
+# Compiling features:
+- `parallel`: use multi-threading when possible.
+- `print-trace`: print out user friendly information about the running time for each micro component.
+- `extensive_sanity_checks`: runs additional sanity checks that is not essential and will slow down the scheme.

pcs/src/errors.rs

@@ -0,0 +1,50 @@
+// Copyright (c) 2022 Espresso Systems (espressosys.com)
+// This file is part of the Jellyfish library.
+
+// You should have received a copy of the MIT License
+// along with the Jellyfish library. If not, see <https://mit-license.org/>.
+
+//! Error module.
+
+use arithmetic::ArithErrors;
+use ark_serialize::SerializationError;
+use ark_std::string::String;
+use displaydoc::Display;
+use transcript::TranscriptError;
+
+/// A `enum` specifying the possible failure modes of the PCS.
+#[derive(Display, Debug)]
+pub enum PCSError {
+    /// Invalid Prover: {0}
+    InvalidProver(String),
+    /// Invalid Verifier: {0}
+    InvalidVerifier(String),
+    /// Invalid Proof: {0}
+    InvalidProof(String),
+    /// Invalid parameters: {0}
+    InvalidParameters(String),
+    /// An error during (de)serialization: {0}
+    SerializationError(SerializationError),
+    /// Transcript error {0}
+    TranscriptError(TranscriptError),
+    /// ArithErrors error {0}
+    ArithErrors(ArithErrors),
+}
+
+impl From<SerializationError> for PCSError {
+    fn from(e: ark_serialize::SerializationError) -> Self {
+        Self::SerializationError(e)
+    }
+}
+
+impl From<TranscriptError> for PCSError {
+    fn from(e: TranscriptError) -> Self {
+        Self::TranscriptError(e)
+    }
+}
+
+impl From<ArithErrors> for PCSError {
+    fn from(e: ArithErrors) -> Self {
+        Self::ArithErrors(e)
+    }
+}

pcs/src/lib.rs

@@ -0,0 +1,190 @@
+mod errors;
+mod multilinear_kzg;
+mod structs;
+mod univariate_kzg;
+
+pub mod prelude;
+
+use ark_ec::PairingEngine;
+use ark_ff::Field;
+use ark_serialize::{CanonicalDeserialize, CanonicalSerialize};
+use ark_std::rand::{CryptoRng, RngCore};
+use errors::PCSError;
+use std::{borrow::Borrow, fmt::Debug, hash::Hash};
+
+/// This trait defines APIs for polynomial commitment schemes.
+/// Note that for our usage of PCS, we do not require the hiding property.
+pub trait PolynomialCommitmentScheme<E: PairingEngine> {
+    /// Prover parameters
+    type ProverParam: Clone;
+    /// Verifier parameters
+    type VerifierParam: Clone + CanonicalSerialize + CanonicalDeserialize;
+    /// Structured reference string
+    type SRS: Clone + Debug;
+    /// Polynomial and its associated types
+    type Polynomial: Clone
+        + Debug
+        + Hash
+        + PartialEq
+        + Eq
+        + CanonicalSerialize
+        + CanonicalDeserialize;
+    /// Polynomial input domain
+    type Point: Clone + Ord + Debug + Sync + Hash + PartialEq + Eq;
+    /// Polynomial Evaluation
+    type Evaluation: Field;
+    /// Commitments
+    type Commitment: Clone + CanonicalSerialize + CanonicalDeserialize + Debug + PartialEq + Eq;
+    /// Batch commitments
+    type BatchCommitment: Clone + CanonicalSerialize + CanonicalDeserialize + Debug + PartialEq + Eq;
+    /// Proofs
+    type Proof: Clone + CanonicalSerialize + CanonicalDeserialize + Debug + PartialEq + Eq;
+    /// Batch proofs
+    type BatchProof: Clone + CanonicalSerialize + CanonicalDeserialize + Debug + PartialEq + Eq;
+
+    /// Build SRS for testing.
+    ///
+    /// - For univariate polynomials, `supported_size` is the maximum degree.
+    /// - For multilinear polynomials, `supported_size` is the number of
+    ///   variables.
+    ///
+    /// WARNING: THIS FUNCTION IS FOR TESTING PURPOSE ONLY.
+    /// THE OUTPUT SRS SHOULD NOT BE USED IN PRODUCTION.
+    fn gen_srs_for_testing<R: RngCore + CryptoRng>(
+        rng: &mut R,
+        supported_size: usize,
+    ) -> Result<Self::SRS, PCSError>;
+
+    /// Trim the universal parameters to specialize the public parameters.
+    /// Input both `supported_degree` for univariate and
+    /// `supported_num_vars` for multilinear.
+    /// ## Note on function signature
+    /// Usually, data structure like SRS and ProverParam are huge and users
+    /// might wish to keep them in heap using different kinds of smart pointers
+    /// (instead of only in stack) therefore our `impl Borrow<_>` interface
+    /// allows for passing in any pointer type, e.g.: `trim(srs: &Self::SRS,
+    /// ..)` or `trim(srs: Box<Self::SRS>, ..)` or `trim(srs: Arc<Self::SRS>,
+    /// ..)` etc.
+    fn trim(
+        srs: impl Borrow<Self::SRS>,
+        supported_degree: usize,
+        supported_num_vars: Option<usize>,
+    ) -> Result<(Self::ProverParam, Self::VerifierParam), PCSError>;
+
+    /// Generate a commitment for a polynomial
+    /// ## Note on function signature
+    /// Usually, data structure like SRS and ProverParam are huge and users
+    /// might wish to keep them in heap using different kinds of smart pointers
+    /// (instead of only in stack) therefore our `impl Borrow<_>` interface
+    /// allows for passing in any pointer type, e.g.: `commit(prover_param:
+    /// &Self::ProverParam, ..)` or `commit(prover_param:
+    /// Box<Self::ProverParam>, ..)` or `commit(prover_param:
+    /// Arc<Self::ProverParam>, ..)` etc.
+    fn commit(
+        prover_param: impl Borrow<Self::ProverParam>,
+        poly: &Self::Polynomial,
+    ) -> Result<Self::Commitment, PCSError>;
+
+    /// Generate a commitment for a list of polynomials
+    fn multi_commit(
+        prover_param: impl Borrow<Self::ProverParam>,
+        polys: &[Self::Polynomial],
+    ) -> Result<Self::BatchCommitment, PCSError>;
+
+    /// On input a polynomial `p` and a point `point`, outputs a proof for the
+    /// same.
+    fn open(
+        prover_param: impl Borrow<Self::ProverParam>,
+        polynomial: &Self::Polynomial,
+        point: &Self::Point,
+    ) -> Result<(Self::Proof, Self::Evaluation), PCSError>;
+
+    /// Input a list of multilinear extensions, and a same number of points, and
+    /// a transcript, compute a multi-opening for all the polynomials.
+    fn multi_open(
+        prover_param: impl Borrow<Self::ProverParam>,
+        multi_commitment: &Self::BatchCommitment,
+        polynomials: &[Self::Polynomial],
+        points: &[Self::Point],
+    ) -> Result<(Self::BatchProof, Vec<Self::Evaluation>), PCSError>;
+
+    /// Input a multilinear extension, and a number of points, and
+    /// a transcript, compute a multi-opening for all the polynomials.
+    fn multi_open_single_poly(
+        prover_param: impl Borrow<Self::ProverParam>,
+        commitment: &Self::Commitment,
+        polynomials: &Self::Polynomial,
+        points: &[Self::Point],
+    ) -> Result<(Self::BatchProof, Vec<Self::Evaluation>), PCSError>;
+
+    /// Verifies that `value` is the evaluation at `x` of the polynomial
+    /// committed inside `comm`.
+    fn verify(
+        verifier_param: &Self::VerifierParam,
+        commitment: &Self::Commitment,
+        point: &Self::Point,
+        value: &E::Fr,
+        proof: &Self::Proof,
+    ) -> Result<bool, PCSError>;
+
+    /// Verifies that `value_i` is the evaluation at `x_i` of the polynomial
+    /// `poly_i` committed inside `comm`.
+    fn batch_verify<R: RngCore + CryptoRng>(
+        verifier_param: &Self::VerifierParam,
+        multi_commitment: &Self::BatchCommitment,
+        points: &[Self::Point],
+        values: &[E::Fr],
+        batch_proof: &Self::BatchProof,
+        rng: &mut R,
+    ) -> Result<bool, PCSError>;
+
+    /// Verifies that `value_i` is the evaluation at `x_i` of the polynomial
+    /// `poly` committed inside `comm`.
+    fn batch_verify_single_poly(
+        verifier_param: &Self::VerifierParam,
+        commitment: &Self::Commitment,
+        points: &[Self::Point],
+        values: &[E::Fr],
+        batch_proof: &Self::BatchProof,
+    ) -> Result<bool, PCSError>;
+}
+
+/// API definitions for structured reference string
+pub trait StructuredReferenceString<E: PairingEngine>: Sized {
+    /// Prover parameters
+    type ProverParam;
+    /// Verifier parameters
+    type VerifierParam;
+
+    /// Extract the prover parameters from the public parameters.
+    fn extract_prover_param(&self, supported_size: usize) -> Self::ProverParam;
+    /// Extract the verifier parameters from the public parameters.
+    fn extract_verifier_param(&self, supported_size: usize) -> Self::VerifierParam;
+
+    /// Trim the universal parameters to specialize the public parameters
+    /// for polynomials to the given `supported_size`, and
+    /// returns committer key and verifier key.
+    ///
+    /// - For univariate polynomials, `supported_size` is the maximum degree.
+    /// - For multilinear polynomials, `supported_size` is 2 to the number of
+    ///   variables.
+    ///
+    /// `supported_log_size` should be in range `1..=params.log_size`
+    fn trim(
+        &self,
+        supported_size: usize,
+    ) -> Result<(Self::ProverParam, Self::VerifierParam), PCSError>;
+
+    /// Build SRS for testing.
+    ///
+    /// - For univariate polynomials, `supported_size` is the maximum degree.
+    /// - For multilinear polynomials, `supported_size` is the number of
+    ///   variables.
+    ///
+    /// WARNING: THIS FUNCTION IS FOR TESTING PURPOSE ONLY.
+    /// THE OUTPUT SRS SHOULD NOT BE USED IN PRODUCTION.
+    fn gen_srs_for_testing<R: RngCore + CryptoRng>(
+        rng: &mut R,
+        supported_size: usize,
+    ) -> Result<Self, PCSError>;
+}

pcs/src/multilinear_kzg/batching/mod.rs

@@ -0,0 +1,11 @@
+// Copyright (c) 2022 Espresso Systems (espressosys.com)
+// This file is part of the Jellyfish library.
+
+// You should have received a copy of the MIT License
+// along with the Jellyfish library. If not, see <https://mit-license.org/>.
+
+mod multi_poly;
+mod single_poly;
+
+pub(crate) use multi_poly::*;
+pub(crate) use single_poly::*;

pcs/src/multilinear_kzg/batching/multi_poly.rs

@@ -0,0 +1,426 @@
+use crate::{
+    multilinear_kzg::{
+        open_internal,
+        srs::{MultilinearProverParam, MultilinearVerifierParam},
+        util::compute_w_circ_l,
+        verify_internal, MultilinearKzgBatchProof,
+    },
+    prelude::{Commitment, UnivariateProverParam, UnivariateVerifierParam},
+    univariate_kzg::UnivariateKzgPCS,
+    PCSError, PolynomialCommitmentScheme,
+};
+use arithmetic::{build_l, get_uni_domain, merge_polynomials};
+use ark_ec::PairingEngine;
+use ark_poly::{DenseMultilinearExtension, EvaluationDomain, MultilinearExtension, Polynomial};
+use ark_std::{end_timer, format, rc::Rc, start_timer, string::ToString, vec, vec::Vec};
+use transcript::IOPTranscript;
+
+/// Input
+/// - the prover parameters for univariate KZG,
+/// - the prover parameters for multilinear KZG,
+/// - a list of MLEs,
+/// - a commitment to all MLEs
+/// - and a same number of points,
+/// compute a multi-opening for all the polynomials.
+///
+/// For simplicity, this API requires each MLE to have only one point. If
+/// the caller wish to use more than one points per MLE, it should be
+/// handled at the caller layer, and utilize 'multi_open_same_poly_internal'
+/// API.
+///
+/// Returns an error if the lengths do not match.
+///
+/// Returns the proof, consists of
+/// - the multilinear KZG opening
+/// - the univariate KZG commitment to q(x)
+/// - the openings and evaluations of q(x) at omega^i and r
+///
+/// Steps:
+/// 1. build `l(points)` which is a list of univariate polynomials that goes
+/// through the points
+/// 2. build MLE `w` which is the merge of all MLEs.
+/// 3. build `q(x)` which is a univariate polynomial `W circ l`
+/// 4. commit to q(x) and sample r from transcript
+/// transcript contains: w commitment, points, q(x)'s commitment
+/// 5. build q(omega^i) and their openings
+/// 6. build q(r) and its opening
+/// 7. get a point `p := l(r)`
+/// 8. output an opening of `w` over point `p`
+/// 9. output `w(p)`
+pub(crate) fn multi_open_internal<E: PairingEngine>(
+    uni_prover_param: &UnivariateProverParam<E::G1Affine>,
+    ml_prover_param: &MultilinearProverParam<E>,
+    polynomials: &[Rc<DenseMultilinearExtension<E::Fr>>],
+    multi_commitment: &Commitment<E>,
+    points: &[Vec<E::Fr>],
+) -> Result<(MultilinearKzgBatchProof<E>, Vec<E::Fr>), PCSError> {
+    let open_timer = start_timer!(|| "multi open");
+
+    // ===================================
+    // Sanity checks on inputs
+    // ===================================
+    let points_len = points.len();
+    if points_len == 0 {
+        return Err(PCSError::InvalidParameters("points is empty".to_string()));
+    }
+
+    if points_len != polynomials.len() {
+        return Err(PCSError::InvalidParameters(
+            "polynomial length does not match point length".to_string(),
+        ));
+    }
+
+    let num_var = polynomials[0].num_vars();
+    for poly in polynomials.iter().skip(1) {
+        if poly.num_vars() != num_var {
+            return Err(PCSError::InvalidParameters(
+                "polynomials do not have same num_vars".to_string(),
+            ));
+        }
+    }
+    for point in points.iter() {
+        if point.len() != num_var {
+            return Err(PCSError::InvalidParameters(
+                "points do not have same num_vars".to_string(),
+            ));
+        }
+    }
+
+    let domain = get_uni_domain::<E::Fr>(points_len)?;
+
+    // 1. build `l(points)` which is a list of univariate polynomials that goes
+    // through the points
+    let uni_polys = build_l(points, &domain, true)?;
+
+    // 2. build MLE `w` which is the merge of all MLEs.
+    let merge_poly = merge_polynomials(polynomials)?;
+
+    // 3. build `q(x)` which is a univariate polynomial `W circ l`
+    let q_x = compute_w_circ_l(&merge_poly, &uni_polys, points.len(), true)?;
+
+    // 4. commit to q(x) and sample r from transcript
+    // transcript contains: w commitment, points, q(x)'s commitment
+    let mut transcript = IOPTranscript::new(b"ml kzg");
+    transcript.append_serializable_element(b"w", multi_commitment)?;
+    for point in points {
+        transcript.append_serializable_element(b"w", point)?;
+    }
+
+    let q_x_commit = UnivariateKzgPCS::<E>::commit(uni_prover_param, &q_x)?;
+    transcript.append_serializable_element(b"q(x)", &q_x_commit)?;
+    let r = transcript.get_and_append_challenge(b"r")?;
+    // 5. build q(omega^i) and their openings
+    let mut q_x_opens = vec![];
+    let mut q_x_evals = vec![];
+    for i in 0..points_len {
+        let (q_x_open, q_x_eval) =
+            UnivariateKzgPCS::<E>::open(uni_prover_param, &q_x, &domain.element(i))?;
+        q_x_opens.push(q_x_open);
+        q_x_evals.push(q_x_eval);
+        #[cfg(feature = "extensive_sanity_checks")]
+        {
+            // sanity check
+            let point: Vec<E::Fr> = uni_polys
+                .iter()
+                .map(|poly| poly.evaluate(&domain.element(i)))
+                .collect();
+            let mle_eval = merge_poly.evaluate(&point).unwrap();
+            if mle_eval != q_x_eval {
+                return Err(PCSError::InvalidProver(
+                    "Q(omega) does not match W(l(omega))".to_string(),
+                ));
+            }
+        }
+    }
+
+    // 6. build q(r) and its opening
+    let (q_x_open, q_r_value) = UnivariateKzgPCS::<E>::open(uni_prover_param, &q_x, &r)?;
+    q_x_opens.push(q_x_open);
+    q_x_evals.push(q_r_value);
+
+    // 7. get a point `p := l(r)`
+    let point: Vec<E::Fr> = uni_polys.iter().map(|poly| poly.evaluate(&r)).collect();
+    // 8. output an opening of `w` over point `p`
+    let (mle_opening, mle_eval) = open_internal(ml_prover_param, &merge_poly, &point)?;
+
+    // 9. output value that is `w` evaluated at `p` (which should match `q(r)`)
+    if mle_eval != q_r_value {
+        return Err(PCSError::InvalidProver(
+            "Q(r) does not match W(l(r))".to_string(),
+        ));
+    }
+    end_timer!(open_timer);
+    Ok((
+        MultilinearKzgBatchProof {
+            proof: mle_opening,
+            q_x_commit,
+            q_x_opens,
+        },
+        q_x_evals,
+    ))
+}
+
+/// Verifies that the `multi_commitment` is a valid commitment
+/// to a list of MLEs for the given openings and evaluations in
+/// the batch_proof.
+///
+/// steps:
+///
+/// 1. push w, points and q_com into transcript
+/// 2. sample `r` from transcript
+/// 3. check `q(r) == batch_proof.q_x_value.last` and
+/// `q(omega^i) == batch_proof.q_x_value[i]`
+/// 4. build `l(points)` which is a list of univariate
+/// polynomials that goes through the points
+/// 5. get a point `p := l(r)`
+/// 6. verifies `p` is valid against multilinear KZG proof
+pub(crate) fn batch_verify_internal<E: PairingEngine>(
+    uni_verifier_param: &UnivariateVerifierParam<E>,
+    ml_verifier_param: &MultilinearVerifierParam<E>,
+    multi_commitment: &Commitment<E>,
+    points: &[Vec<E::Fr>],
+    values: &[E::Fr],
+    batch_proof: &MultilinearKzgBatchProof<E>,
+) -> Result<bool, PCSError> {
+    let verify_timer = start_timer!(|| "batch verify");
+
+    // ===================================
+    // Sanity checks on inputs
+    // ===================================
+    let points_len = points.len();
+    if points_len == 0 {
+        return Err(PCSError::InvalidParameters("points is empty".to_string()));
+    }
+
+    // add one here because we also have q(r) and its opening
+    if points_len + 1 != batch_proof.q_x_opens.len() {
+        return Err(PCSError::InvalidParameters(format!(
+            "openings length {} does not match point length {}",
+            points_len + 1,
+            batch_proof.q_x_opens.len()
+        )));
+    }
+
+    if points_len + 1 != values.len() {
+        return Err(PCSError::InvalidParameters(format!(
+            "values length {} does not match point length {}",
+            points_len + 1,
+            values.len()
+        )));
+    }
+
+    let num_var = points[0].len();
+    for point in points.iter().skip(1) {
+        if point.len() != num_var {
+            return Err(PCSError::InvalidParameters(format!(
+                "points do not have same num_vars ({} vs {})",
+                point.len(),
+                num_var,
+            )));
+        }
+    }
+
+    let domain = get_uni_domain::<E::Fr>(points_len)?;
+    // 1. push w, points and q_com into transcript
+    let mut transcript = IOPTranscript::new(b"ml kzg");
+    transcript.append_serializable_element(b"w", multi_commitment)?;
+
+    for point in points {
+        transcript.append_serializable_element(b"w", point)?;
+    }
+
+    transcript.append_serializable_element(b"q(x)", &batch_proof.q_x_commit)?;
+    // 2. sample `r` from transcript
+    let r = transcript.get_and_append_challenge(b"r")?;
+    // 3. check `q(r) == batch_proof.q_x_value.last` and `q(omega^i) =
+    // batch_proof.q_x_value[i]`
+    for (i, value) in values.iter().enumerate().take(points_len) {
+        if !UnivariateKzgPCS::verify(
+            uni_verifier_param,
+            &batch_proof.q_x_commit,
+            &domain.element(i),
+            value,
+            &batch_proof.q_x_opens[i],
+        )? {
+            #[cfg(debug_assertion)]
+            println!("q(omega^{}) verification failed", i);
+            return Ok(false);
+        }
+    }
+
+    if !UnivariateKzgPCS::verify(
+        uni_verifier_param,
+        &batch_proof.q_x_commit,
+        &r,
+        &values[points_len],
+        &batch_proof.q_x_opens[points_len],
+    )? {
+        #[cfg(debug_assertion)]
+        println!("q(r) verification failed");
+        return Ok(false);
+    }
+    // 4. build `l(points)` which is a list of univariate polynomials that goes
+    // through the points
+    let uni_polys = build_l(points, &domain, true)?;
+
+    // 5. get a point `p := l(r)`
+    let point: Vec<E::Fr> = uni_polys.iter().map(|x| x.evaluate(&r)).collect();
+    // 6. verifies `p` is valid against multilinear KZG proof
+    let res = verify_internal(
+        ml_verifier_param,
+        multi_commitment,
+        &point,
+        &values[points_len],
+        &batch_proof.proof,
+    )?;
+    #[cfg(debug_assertion)]
+    if !res {
+        println!("multilinear KZG verification failed");
+    }
+
+    end_timer!(verify_timer);
+
+    Ok(res)
+}
+#[cfg(test)]
+mod tests {
+    use super::*;
+    use crate::{
+        multilinear_kzg::{
+            srs::MultilinearUniversalParams,
+            util::{compute_qx_degree, generate_evaluations_multi_poly},
+            MultilinearKzgPCS, MultilinearKzgProof,
+        },
+        prelude::UnivariateUniversalParams,
+        StructuredReferenceString,
+    };
+    use arithmetic::get_batched_nv;
+    use ark_bls12_381::Bls12_381 as E;
+    use ark_ec::PairingEngine;
+    use ark_poly::{DenseMultilinearExtension, MultilinearExtension};
+    use ark_std::{
+        log2,
+        rand::{CryptoRng, RngCore},
+        test_rng,
+        vec::Vec,
+        UniformRand,
+    };
+
+    type Fr = <E as PairingEngine>::Fr;
+
+    fn test_multi_open_helper<R: RngCore + CryptoRng>(
+        uni_params: &UnivariateUniversalParams<E>,
+        ml_params: &MultilinearUniversalParams<E>,
+        polys: &[Rc<DenseMultilinearExtension<Fr>>],
+        rng: &mut R,
+    ) -> Result<(), PCSError> {
+        let merged_nv = get_batched_nv(polys[0].num_vars(), polys.len());
+        let qx_degree = compute_qx_degree(merged_nv, polys.len());
+        let padded_qx_degree = 1usize << log2(qx_degree);
+
+        let (uni_ck, uni_vk) = uni_params.trim(padded_qx_degree)?;
+        let (ml_ck, ml_vk) = ml_params.trim(merged_nv)?;
+
+        let mut points = Vec::new();
+        for poly in polys.iter() {
+            let point = (0..poly.num_vars())
+                .map(|_| Fr::rand(rng))
+                .collect::<Vec<Fr>>();
+            points.push(point);
+        }
+
+        let evals = generate_evaluations_multi_poly(polys, &points)?;
+
+        let com = MultilinearKzgPCS::multi_commit(&(ml_ck.clone(), uni_ck.clone()), polys)?;
+        let (batch_proof, evaluations) =
+            multi_open_internal(&uni_ck, &ml_ck, polys, &com, &points)?;
+
+        for (a, b) in evals.iter().zip(evaluations.iter()) {
+            assert_eq!(a, b)
+        }
+
+        // good path
+        assert!(batch_verify_internal(
+            &uni_vk,
+            &ml_vk,
+            &com,
+            &points,
+            &evaluations,
+            &batch_proof,
+        )?);
+
+        // bad commitment
+        assert!(!batch_verify_internal(
+            &uni_vk,
+            &ml_vk,
+            &Commitment(<E as PairingEngine>::G1Affine::default()),
+            &points,
+            &evaluations,
+            &batch_proof,
+        )?);
+
+        // bad points
+        assert!(
+            batch_verify_internal(&uni_vk, &ml_vk, &com, &points[1..], &[], &batch_proof,).is_err()
+        );
+
+        // bad proof
+        assert!(batch_verify_internal(
+            &uni_vk,
+            &ml_vk,
+            &com,
+            &points,
+            &evaluations,
+            &MultilinearKzgBatchProof {
+                proof: MultilinearKzgProof { proofs: Vec::new() },
+                q_x_commit: Commitment(<E as PairingEngine>::G1Affine::default()),
+                q_x_opens: vec![],
+            },
+        )
+        .is_err());
+
+        // bad value
+        let mut wrong_evals = evaluations.clone();
+        wrong_evals[0] = Fr::default();
+        assert!(!batch_verify_internal(
+            &uni_vk,
+            &ml_vk,
+            &com,
+            &points,
+            &wrong_evals,
+            &batch_proof
+        )?);
+
+        // bad q(x) commit
+        let mut wrong_proof = batch_proof;
+        wrong_proof.q_x_commit = Commitment(<E as PairingEngine>::G1Affine::default());
+        assert!(!batch_verify_internal(
+            &uni_vk,
+            &ml_vk,
+            &com,
+            &points,
+            &evaluations,
+            &wrong_proof,
+        )?);
+        Ok(())
+    }
+
+    #[test]
+    fn test_multi_open_internal() -> Result<(), PCSError> {
+        let mut rng = test_rng();
+
+        let uni_params =
+            UnivariateUniversalParams::<E>::gen_srs_for_testing(&mut rng, 1usize << 15)?;
+        let ml_params = MultilinearUniversalParams::<E>::gen_srs_for_testing(&mut rng, 15)?;
+        for num_poly in 1..10 {
+            for nv in 1..5 {
+                let polys1: Vec<_> = (0..num_poly)
+                    .map(|_| Rc::new(DenseMultilinearExtension::rand(nv, &mut rng)))
+                    .collect();
+                test_multi_open_helper(&uni_params, &ml_params, &polys1, &mut rng)?;
+            }
+        }
+
+        Ok(())
+    }
+}

pcs/src/multilinear_kzg/batching/single_poly.rs

@@ -0,0 +1,368 @@
+// Copyright (c) 2022 Espresso Systems (espressosys.com)
+// This file is part of the Jellyfish library.
+
+// You should have received a copy of the MIT License
+// along with the Jellyfish library. If not, see <https://mit-license.org/>.
+
+use crate::{
+    multilinear_kzg::{
+        open_internal,
+        srs::{MultilinearProverParam, MultilinearVerifierParam},
+        util::compute_w_circ_l,
+        verify_internal, MultilinearKzgBatchProof,
+    },
+    prelude::{Commitment, UnivariateProverParam, UnivariateVerifierParam},
+    univariate_kzg::UnivariateKzgPCS,
+    PCSError, PolynomialCommitmentScheme,
+};
+use arithmetic::{build_l, get_uni_domain};
+use ark_ec::PairingEngine;
+use ark_poly::{DenseMultilinearExtension, EvaluationDomain, MultilinearExtension, Polynomial};
+use ark_std::{end_timer, format, rc::Rc, start_timer, string::ToString, vec, vec::Vec};
+use rayon::prelude::{IntoParallelIterator, ParallelIterator};
+use transcript::IOPTranscript;
+
+/// Input
+/// - the prover parameters for univariate KZG,
+/// - the prover parameters for multilinear KZG,
+/// - a single MLE,
+/// - a commitment to the MLE
+/// - and a list of points,
+/// compute a multi-opening for this polynomial.
+///
+/// For simplicity, this API requires each MLE to have only one point. If
+/// the caller wish to use more than one points per MLE, it should be
+/// handled at the caller layer.
+///
+///
+/// Returns the proof, consists of
+/// - the multilinear KZG opening
+/// - the univariate KZG commitment to q(x)
+/// - the openings and evaluations of q(x) at omega^i and r
+///
+/// Steps:
+/// 1. build `l(points)` which is a list of univariate polynomials that goes
+/// through the points
+/// 3. build `q(x)` which is a univariate polynomial `W circ l`
+/// 4. commit to q(x) and sample r from transcript
+/// transcript contains: w commitment, points, q(x)'s commitment
+/// 5. build q(omega^i) and their openings
+/// 6. build q(r) and its opening
+/// 7. get a point `p := l(r)`
+/// 8. output an opening of `w` over point `p`
+/// 9. output `w(p)`
+pub(crate) fn multi_open_same_poly_internal<E: PairingEngine>(
+    uni_prover_param: &UnivariateProverParam<E::G1Affine>,
+    ml_prover_param: &MultilinearProverParam<E>,
+    polynomial: &Rc<DenseMultilinearExtension<E::Fr>>,
+    commitment: &Commitment<E>,
+    points: &[Vec<E::Fr>],
+) -> Result<(MultilinearKzgBatchProof<E>, Vec<E::Fr>), PCSError> {
+    let open_timer = start_timer!(|| "multi open");
+
+    // ===================================
+    // Sanity checks on inputs
+    // ===================================
+    let points_len = points.len();
+    if points_len == 0 {
+        return Err(PCSError::InvalidParameters("points is empty".to_string()));
+    }
+
+    let num_var = polynomial.num_vars();
+    for point in points.iter() {
+        if point.len() != num_var {
+            return Err(PCSError::InvalidParameters(
+                "points do not have same num_vars".to_string(),
+            ));
+        }
+    }
+
+    let domain = get_uni_domain::<E::Fr>(points_len)?;
+
+    // 1. build `l(points)` which is a list of univariate polynomials that goes
+    // through the points
+    let uni_polys = build_l(points, &domain, false)?;
+
+    // 3. build `q(x)` which is a univariate polynomial `W circ l`
+    let q_x = compute_w_circ_l(polynomial, &uni_polys, points.len(), false)?;
+
+    // 4. commit to q(x) and sample r from transcript
+    // transcript contains: w commitment, points, q(x)'s commitment
+    let mut transcript = IOPTranscript::new(b"ml kzg");
+    transcript.append_serializable_element(b"w", commitment)?;
+    for point in points {
+        transcript.append_serializable_element(b"w", point)?;
+    }
+
+    let q_x_commit = UnivariateKzgPCS::<E>::commit(uni_prover_param, &q_x)?;
+    transcript.append_serializable_element(b"q(x)", &q_x_commit)?;
+    let r = transcript.get_and_append_challenge(b"r")?;
+    // 5. build q(omega^i) and their openings
+    let mut q_x_opens = vec![];
+    let mut q_x_evals = vec![];
+    for i in 0..points_len {
+        let (q_x_open, q_x_eval) =
+            UnivariateKzgPCS::<E>::open(uni_prover_param, &q_x, &domain.element(i))?;
+        q_x_opens.push(q_x_open);
+        q_x_evals.push(q_x_eval);
+
+        #[cfg(feature = "extensive_sanity_checks")]
+        {
+            // sanity check
+            let point: Vec<E::Fr> = uni_polys
+                .iter()
+                .map(|poly| poly.evaluate(&domain.element(i)))
+                .collect();
+            let mle_eval = polynomial.evaluate(&point).unwrap();
+            if mle_eval != q_x_eval {
+                return Err(PCSError::InvalidProver(
+                    "Q(omega) does not match W(l(omega))".to_string(),
+                ));
+            }
+        }
+    }
+
+    // 6. build q(r) and its opening
+    let (q_x_open, q_r_value) = UnivariateKzgPCS::<E>::open(uni_prover_param, &q_x, &r)?;
+    q_x_opens.push(q_x_open);
+    q_x_evals.push(q_r_value);
+
+    // 7. get a point `p := l(r)`
+    let point: Vec<E::Fr> = uni_polys
+        .into_par_iter()
+        .map(|poly| poly.evaluate(&r))
+        .collect();
+    // 8. output an opening of `w` over point `p`
+    let (mle_opening, mle_eval) = open_internal(ml_prover_param, polynomial, &point)?;
+
+    // 9. output value that is `w` evaluated at `p` (which should match `q(r)`)
+    if mle_eval != q_r_value {
+        return Err(PCSError::InvalidProver(
+            "Q(r) does not match W(l(r))".to_string(),
+        ));
+    }
+    end_timer!(open_timer);
+    Ok((
+        MultilinearKzgBatchProof {
+            proof: mle_opening,
+            q_x_commit,
+            q_x_opens,
+        },
+        q_x_evals,
+    ))
+}
+
+/// Verifies that the `multi_commitment` is a valid commitment
+/// to a list of MLEs for the given openings and evaluations in
+/// the batch_proof.
+///
+/// steps:
+///
+/// 1. push w, points and q_com into transcript
+/// 2. sample `r` from transcript
+/// 3. check `q(r) == batch_proof.q_x_value.last` and
+/// `q(omega^i) == batch_proof.q_x_value[i]`
+/// 4. build `l(points)` which is a list of univariate
+/// polynomials that goes through the points
+/// 5. get a point `p := l(r)`
+/// 6. verifies `p` is valid against multilinear KZG proof
+#[allow(dead_code)]
+pub(crate) fn batch_verify_same_poly_internal<E: PairingEngine>(
+    uni_verifier_param: &UnivariateVerifierParam<E>,
+    ml_verifier_param: &MultilinearVerifierParam<E>,
+    multi_commitment: &Commitment<E>,
+    points: &[Vec<E::Fr>],
+    values: &[E::Fr],
+    batch_proof: &MultilinearKzgBatchProof<E>,
+) -> Result<bool, PCSError> {
+    let verify_timer = start_timer!(|| "batch verify");
+
+    // ===================================
+    // Sanity checks on inputs
+    // ===================================
+    let points_len = points.len();
+    if points_len == 0 {
+        return Err(PCSError::InvalidParameters("points is empty".to_string()));
+    }
+
+    // add one here because we also have q(r) and its opening
+    if points_len + 1 != batch_proof.q_x_opens.len() {
+        return Err(PCSError::InvalidParameters(format!(
+            "openings length {} does not match point length {}",
+            points_len + 1,
+            batch_proof.q_x_opens.len()
+        )));
+    }
+
+    if points_len + 1 != values.len() {
+        return Err(PCSError::InvalidParameters(format!(
+            "values length {} does not match point length {}",
+            points_len + 1,
+            values.len()
+        )));
+    }
+
+    let num_var = points[0].len();
+    for point in points.iter().skip(1) {
+        if point.len() != num_var {
+            return Err(PCSError::InvalidParameters(format!(
+                "points do not have same num_vars ({} vs {})",
+                point.len(),
+                num_var,
+            )));
+        }
+    }
+
+    let domain = get_uni_domain::<E::Fr>(points_len)?;
+    // 1. push w, points and q_com into transcript
+    let mut transcript = IOPTranscript::new(b"ml kzg");
+    transcript.append_serializable_element(b"w", multi_commitment)?;
+
+    for point in points {
+        transcript.append_serializable_element(b"w", point)?;
+    }
+
+    transcript.append_serializable_element(b"q(x)", &batch_proof.q_x_commit)?;
+    // 2. sample `r` from transcript
+    let r = transcript.get_and_append_challenge(b"r")?;
+    // 3. check `q(r) == batch_proof.q_x_value.last` and `q(omega^i) =
+    // batch_proof.q_x_value[i]`
+    for (i, value) in values.iter().enumerate().take(points_len) {
+        if !UnivariateKzgPCS::verify(
+            uni_verifier_param,
+            &batch_proof.q_x_commit,
+            &domain.element(i),
+            value,
+            &batch_proof.q_x_opens[i],
+        )? {
+            #[cfg(debug_assertion)]
+            println!("q(omega^{}) verification failed", i);
+            return Ok(false);
+        }
+    }
+
+    if !UnivariateKzgPCS::verify(
+        uni_verifier_param,
+        &batch_proof.q_x_commit,
+        &r,
+        &values[points_len],
+        &batch_proof.q_x_opens[points_len],
+    )? {
+        #[cfg(debug_assertion)]
+        println!("q(r) verification failed");
+        return Ok(false);
+    }
+    // 4. build `l(points)` which is a list of univariate polynomials that goes
+    // through the points
+    let uni_polys = build_l(points, &domain, false)?;
+
+    // 5. get a point `p := l(r)`
+    let point: Vec<E::Fr> = uni_polys.iter().map(|x| x.evaluate(&r)).collect();
+    // 6. verifies `p` is valid against multilinear KZG proof
+    let res = verify_internal(
+        ml_verifier_param,
+        multi_commitment,
+        &point,
+        &values[points_len],
+        &batch_proof.proof,
+    )?;
+    #[cfg(debug_assertion)]
+    if !res {
+        println!("multilinear KZG verification failed");
+    }
+
+    end_timer!(verify_timer);
+
+    Ok(res)
+}
+#[cfg(test)]
+mod tests {
+    use super::*;
+    use crate::{
+        multilinear_kzg::{
+            srs::MultilinearUniversalParams,
+            util::{compute_qx_degree, generate_evaluations_single_poly},
+            MultilinearKzgPCS,
+        },
+        prelude::UnivariateUniversalParams,
+        StructuredReferenceString,
+    };
+    use ark_bls12_381::Bls12_381 as E;
+    use ark_ec::PairingEngine;
+    use ark_poly::{DenseMultilinearExtension, MultilinearExtension};
+    use ark_std::{
+        log2,
+        rand::{CryptoRng, RngCore},
+        test_rng,
+        vec::Vec,
+        UniformRand,
+    };
+    type Fr = <E as PairingEngine>::Fr;
+
+    fn test_same_poly_multi_open_internal_helper<R: RngCore + CryptoRng>(
+        uni_params: &UnivariateUniversalParams<E>,
+        ml_params: &MultilinearUniversalParams<E>,
+        poly: &Rc<DenseMultilinearExtension<Fr>>,
+        point_len: usize,
+        rng: &mut R,
+    ) -> Result<(), PCSError> {
+        let nv = poly.num_vars;
+        let qx_degree = compute_qx_degree(nv, point_len);
+        let padded_qx_degree = 1usize << log2(qx_degree);
+
+        let (uni_ck, uni_vk) = uni_params.trim(padded_qx_degree)?;
+        let (ml_ck, ml_vk) = ml_params.trim(nv)?;
+
+        let mut points = Vec::new();
+        let mut eval = Vec::new();
+        for _ in 0..point_len {
+            let point = (0..nv).map(|_| Fr::rand(rng)).collect::<Vec<Fr>>();
+            eval.push(poly.evaluate(&point).unwrap());
+            points.push(point);
+        }
+
+        let evals = generate_evaluations_single_poly(poly, &points)?;
+        let com = MultilinearKzgPCS::commit(&(ml_ck.clone(), uni_ck.clone()), poly)?;
+        let (batch_proof, evaluations) =
+            multi_open_same_poly_internal(&uni_ck, &ml_ck, poly, &com, &points)?;
+
+        for (a, b) in evals.iter().zip(evaluations.iter()) {
+            assert_eq!(a, b)
+        }
+
+        // good path
+        assert!(batch_verify_same_poly_internal(
+            &uni_vk,
+            &ml_vk,
+            &com,
+            &points,
+            &evaluations,
+            &batch_proof,
+        )?);
+
+        Ok(())
+    }
+
+    #[test]
+    fn test_same_poly_multi_open_internal() -> Result<(), PCSError> {
+        let mut rng = test_rng();
+
+        let uni_params =
+            UnivariateUniversalParams::<E>::gen_srs_for_testing(&mut rng, 1usize << 15)?;
+        let ml_params = MultilinearUniversalParams::<E>::gen_srs_for_testing(&mut rng, 15)?;
+        for nv in 1..10 {
+            for point_len in 1..10 {
+                // normal polynomials
+                let polys1 = Rc::new(DenseMultilinearExtension::rand(nv, &mut rng));
+                test_same_poly_multi_open_internal_helper(
+                    &uni_params,
+                    &ml_params,
+                    &polys1,
+                    point_len,
+                    &mut rng,
+                )?;
+            }
+        }
+        Ok(())
+    }
+}

pcs/src/multilinear_kzg/mod.rs

@@ -0,0 +1,644 @@
+// Copyright (c) 2022 Espresso Systems (espressosys.com)
+// This file is part of the Jellyfish library.
+
+// You should have received a copy of the MIT License
+// along with the Jellyfish library. If not, see <https://mit-license.org/>.
+
+//! Main module for multilinear KZG commitment scheme
+
+mod batching;
+pub(crate) mod srs;
+pub(crate) mod util;
+
+use self::batching::{
+    batch_verify_internal, batch_verify_same_poly_internal, multi_open_internal,
+    multi_open_same_poly_internal,
+};
+use crate::{
+    prelude::{
+        Commitment, UnivariateProverParam, UnivariateUniversalParams, UnivariateVerifierParam,
+    },
+    univariate_kzg::UnivariateKzgProof,
+    PCSError, PolynomialCommitmentScheme, StructuredReferenceString,
+};
+use arithmetic::{evaluate_opt, merge_polynomials};
+use ark_ec::{
+    msm::{FixedBaseMSM, VariableBaseMSM},
+    AffineCurve, PairingEngine, ProjectiveCurve,
+};
+use ark_ff::PrimeField;
+use ark_poly::{DenseMultilinearExtension, MultilinearExtension};
+use ark_serialize::{CanonicalDeserialize, CanonicalSerialize, Read, SerializationError, Write};
+use ark_std::{
+    borrow::Borrow,
+    end_timer, format,
+    marker::PhantomData,
+    rand::{CryptoRng, RngCore},
+    rc::Rc,
+    start_timer,
+    string::ToString,
+    vec,
+    vec::Vec,
+    One, Zero,
+};
+// use batching::{batch_verify_internal, multi_open_internal};
+use srs::{MultilinearProverParam, MultilinearUniversalParams, MultilinearVerifierParam};
+
+/// KZG Polynomial Commitment Scheme on multilinear polynomials.
+pub struct MultilinearKzgPCS<E: PairingEngine> {
+    #[doc(hidden)]
+    phantom: PhantomData<E>,
+}
+
+#[derive(CanonicalSerialize, CanonicalDeserialize, Clone, Debug, PartialEq, Eq)]
+/// proof of opening
+pub struct MultilinearKzgProof<E: PairingEngine> {
+    /// Evaluation of quotients
+    pub proofs: Vec<E::G1Affine>,
+}
+
+#[derive(CanonicalSerialize, CanonicalDeserialize, Clone, Debug, PartialEq, Eq)]
+/// proof of batch opening
+pub struct MultilinearKzgBatchProof<E: PairingEngine> {
+    /// The actual proof
+    pub proof: MultilinearKzgProof<E>,
+    /// Commitment to q(x):= w(l(x)) where
+    /// - `w` is the merged MLE
+    /// - `l` is the list of univariate polys that goes through all points
+    pub q_x_commit: Commitment<E>,
+    /// openings of q(x) at 1, omega, ..., and r
+    pub q_x_opens: Vec<UnivariateKzgProof<E>>,
+}
+
+impl<E: PairingEngine> PolynomialCommitmentScheme<E> for MultilinearKzgPCS<E> {
+    // Parameters
+    type ProverParam = (
+        MultilinearProverParam<E>,
+        UnivariateProverParam<E::G1Affine>,
+    );
+    type VerifierParam = (MultilinearVerifierParam<E>, UnivariateVerifierParam<E>);
+    type SRS = (MultilinearUniversalParams<E>, UnivariateUniversalParams<E>);
+    // Polynomial and its associated types
+    type Polynomial = Rc<DenseMultilinearExtension<E::Fr>>;
+    type Point = Vec<E::Fr>;
+    type Evaluation = E::Fr;
+    // Commitments and proofs
+    type Commitment = Commitment<E>;
+    type BatchCommitment = Commitment<E>;
+    type Proof = MultilinearKzgProof<E>;
+    type BatchProof = MultilinearKzgBatchProof<E>;
+
+    /// Build SRS for testing.
+    ///
+    /// - For univariate polynomials, `log_size` is the log of maximum degree.
+    /// - For multilinear polynomials, `log_size` is the number of variables.
+    ///
+    /// WARNING: THIS FUNCTION IS FOR TESTING PURPOSE ONLY.
+    /// THE OUTPUT SRS SHOULD NOT BE USED IN PRODUCTION.
+    fn gen_srs_for_testing<R: RngCore + CryptoRng>(
+        rng: &mut R,
+        log_size: usize,
+    ) -> Result<Self::SRS, PCSError> {
+        Ok((
+            MultilinearUniversalParams::<E>::gen_srs_for_testing(rng, log_size)?,
+            UnivariateUniversalParams::<E>::gen_srs_for_testing(rng, 1 << log_size)?,
+        ))
+    }
+
+    /// Trim the universal parameters to specialize the public parameters.
+    /// Input both `supported_log_degree` for univariate and
+    /// `supported_num_vars` for multilinear.
+    fn trim(
+        srs: impl Borrow<Self::SRS>,
+        supported_degree: usize,
+        supported_num_vars: Option<usize>,
+    ) -> Result<(Self::ProverParam, Self::VerifierParam), PCSError> {
+        let supported_num_vars = match supported_num_vars {
+            Some(p) => p,
+            None => {
+                return Err(PCSError::InvalidParameters(
+                    "multilinear should receive a num_var param".to_string(),
+                ))
+            },
+        };
+        let (uni_ck, uni_vk) = srs.borrow().1.trim(supported_degree)?;
+        let (ml_ck, ml_vk) = srs.borrow().0.trim(supported_num_vars)?;
+
+        Ok(((ml_ck, uni_ck), (ml_vk, uni_vk)))
+    }
+
+    /// Generate a commitment for a polynomial.
+    ///
+    /// This function takes `2^num_vars` number of scalar multiplications over
+    /// G1.
+    fn commit(
+        prover_param: impl Borrow<Self::ProverParam>,
+        poly: &Self::Polynomial,
+    ) -> Result<Self::Commitment, PCSError> {
+        let prover_param = prover_param.borrow();
+        let commit_timer = start_timer!(|| "commit");
+        if prover_param.0.num_vars < poly.num_vars {
+            return Err(PCSError::InvalidParameters(format!(
+                "MlE length ({}) exceeds param limit ({})",
+                poly.num_vars, prover_param.0.num_vars
+            )));
+        }
+        let ignored = prover_param.0.num_vars - poly.num_vars;
+        let scalars: Vec<_> = poly
+            .to_evaluations()
+            .into_iter()
+            .map(|x| x.into_repr())
+            .collect();
+        let commitment = VariableBaseMSM::multi_scalar_mul(
+            &prover_param.0.powers_of_g[ignored].evals,
+            scalars.as_slice(),
+        )
+        .into_affine();
+
+        end_timer!(commit_timer);
+        Ok(Commitment(commitment))
+    }
+
+    /// Generate a commitment for a list of polynomials.
+    ///
+    /// This function takes `2^(num_vars + log(polys.len())` number of scalar
+    /// multiplications over G1.
+    fn multi_commit(
+        prover_param: impl Borrow<Self::ProverParam>,
+        polys: &[Self::Polynomial],
+    ) -> Result<Self::Commitment, PCSError> {
+        let prover_param = prover_param.borrow();
+        let commit_timer = start_timer!(|| "multi commit");
+        let poly = merge_polynomials(polys)?;
+
+        let scalars: Vec<_> = poly
+            .to_evaluations()
+            .iter()
+            .map(|x| x.into_repr())
+            .collect();
+
+        let commitment = VariableBaseMSM::multi_scalar_mul(
+            &prover_param.0.powers_of_g[0].evals,
+            scalars.as_slice(),
+        )
+        .into_affine();
+
+        end_timer!(commit_timer);
+        Ok(Commitment(commitment))
+    }
+
+    /// On input a polynomial `p` and a point `point`, outputs a proof for the
+    /// same. This function does not need to take the evaluation value as an
+    /// input.
+    ///
+    /// This function takes 2^{num_var +1} number of scalar multiplications over
+    /// G1:
+    /// - it prodceeds with `num_var` number of rounds,
+    /// - at round i, we compute an MSM for `2^{num_var - i + 1}` number of G2
+    ///   elements.
+    fn open(
+        prover_param: impl Borrow<Self::ProverParam>,
+        polynomial: &Self::Polynomial,
+        point: &Self::Point,
+    ) -> Result<(Self::Proof, Self::Evaluation), PCSError> {
+        open_internal(&prover_param.borrow().0, polynomial, point)
+    }
+
+    /// Input
+    /// - the prover parameters for univariate KZG,
+    /// - the prover parameters for multilinear KZG,
+    /// - a list of multilinear extensions (MLEs),
+    /// - a commitment to all multilinear extensions,
+    /// - and a same number of points,
+    /// compute a multi-opening for all the polynomials.
+    ///
+    /// For simplicity, this API requires each MLE to have only one point. If
+    /// the caller wish to use more than one points per MLE, it should be
+    /// handled at the caller layer.
+    ///
+    /// Returns an error if the lengths do not match.
+    ///
+    /// Returns the proof, consists of
+    /// - the multilinear KZG opening
+    /// - the univariate KZG commitment to q(x)
+    /// - the openings and evaluations of q(x) at omega^i and r
+    ///
+    /// Steps:
+    /// 1. build `l(points)` which is a list of univariate polynomials that goes
+    /// through the points
+    /// 2. build MLE `w` which is the merge of all MLEs.
+    /// 3. build `q(x)` which is a univariate polynomial `W circ l`
+    /// 4. commit to q(x) and sample r from transcript
+    /// transcript contains: w commitment, points, q(x)'s commitment
+    /// 5. build q(omega^i) and their openings
+    /// 6. build q(r) and its opening
+    /// 7. get a point `p := l(r)`
+    /// 8. output an opening of `w` over point `p`
+    /// 9. output `w(p)`
+    fn multi_open(
+        prover_param: impl Borrow<Self::ProverParam>,
+        multi_commitment: &Self::BatchCommitment,
+        polynomials: &[Self::Polynomial],
+        points: &[Self::Point],
+    ) -> Result<(Self::BatchProof, Vec<Self::Evaluation>), PCSError> {
+        multi_open_internal::<E>(
+            &prover_param.borrow().1,
+            &prover_param.borrow().0,
+            polynomials,
+            multi_commitment,
+            points,
+        )
+    }
+
+    /// Input a multilinear extension, and a number of points, and
+    /// a transcript, compute a multi-opening for all the polynomials.
+    fn multi_open_single_poly(
+        prover_param: impl Borrow<Self::ProverParam>,
+        commitment: &Self::Commitment,
+        polynomial: &Self::Polynomial,
+        points: &[Self::Point],
+    ) -> Result<(Self::BatchProof, Vec<Self::Evaluation>), PCSError> {
+        multi_open_same_poly_internal::<E>(
+            &prover_param.borrow().1,
+            &prover_param.borrow().0,
+            polynomial,
+            commitment,
+            points,
+        )
+    }
+
+    /// Verifies that `value` is the evaluation at `x` of the polynomial
+    /// committed inside `comm`.
+    ///
+    /// This function takes
+    /// - num_var number of pairing product.
+    /// - num_var number of MSM
+    fn verify(
+        verifier_param: &Self::VerifierParam,
+        commitment: &Self::Commitment,
+        point: &Self::Point,
+        value: &E::Fr,
+        proof: &Self::Proof,
+    ) -> Result<bool, PCSError> {
+        verify_internal(&verifier_param.0, commitment, point, value, proof)
+    }
+
+    /// Verifies that `value` is the evaluation at `x_i` of the polynomial
+    /// `poly_i` committed inside `comm`.
+    /// steps:
+    ///
+    /// 1. put `q(x)`'s evaluations over `(1, omega,...)` into transcript
+    /// 2. sample `r` from transcript
+    /// 3. check `q(r) == value`
+    /// 4. build `l(points)` which is a list of univariate polynomials that goes
+    /// through the points
+    /// 5. get a point `p := l(r)`
+    /// 6. verifies `p` is verifies against proof
+    fn batch_verify<R: RngCore + CryptoRng>(
+        verifier_param: &Self::VerifierParam,
+        multi_commitment: &Self::BatchCommitment,
+        points: &[Self::Point],
+        values: &[E::Fr],
+        batch_proof: &Self::BatchProof,
+        _rng: &mut R,
+    ) -> Result<bool, PCSError> {
+        batch_verify_internal(
+            &verifier_param.1,
+            &verifier_param.0,
+            multi_commitment,
+            points,
+            values,
+            batch_proof,
+        )
+    }
+
+    /// Verifies that `value_i` is the evaluation at `x_i` of the polynomial
+    /// `poly` committed inside `comm`.
+    fn batch_verify_single_poly(
+        verifier_param: &Self::VerifierParam,
+        commitment: &Self::Commitment,
+        points: &[Self::Point],
+        values: &[E::Fr],
+        batch_proof: &Self::BatchProof,
+    ) -> Result<bool, PCSError> {
+        batch_verify_same_poly_internal(
+            &verifier_param.1,
+            &verifier_param.0,
+            commitment,
+            points,
+            values,
+            batch_proof,
+        )
+    }
+}
+
+/// On input a polynomial `p` and a point `point`, outputs a proof for the
+/// same. This function does not need to take the evaluation value as an
+/// input.
+///
+/// This function takes 2^{num_var +1} number of scalar multiplications over
+/// G1:
+/// - it proceeds with `num_var` number of rounds,
+/// - at round i, we compute an MSM for `2^{num_var - i + 1}` number of G2
+///   elements.
+fn open_internal<E: PairingEngine>(
+    prover_param: &MultilinearProverParam<E>,
+    polynomial: &DenseMultilinearExtension<E::Fr>,
+    point: &[E::Fr],
+) -> Result<(MultilinearKzgProof<E>, E::Fr), PCSError> {
+    let open_timer = start_timer!(|| format!("open mle with {} variable", polynomial.num_vars));
+
+    if polynomial.num_vars() > prover_param.num_vars {
+        return Err(PCSError::InvalidParameters(format!(
+            "Polynomial num_vars {} exceed the limit {}",
+            polynomial.num_vars, prover_param.num_vars
+        )));
+    }
+
+    if polynomial.num_vars() != point.len() {
+        return Err(PCSError::InvalidParameters(format!(
+            "Polynomial num_vars {} does not match point len {}",
+            polynomial.num_vars,
+            point.len()
+        )));
+    }
+
+    let nv = polynomial.num_vars();
+    let ignored = prover_param.num_vars - nv;
+    let mut r: Vec<Vec<E::Fr>> = (0..nv + 1).map(|_| Vec::new()).collect();
+    let mut q: Vec<Vec<E::Fr>> = (0..nv + 1).map(|_| Vec::new()).collect();
+
+    r[nv] = polynomial.to_evaluations();
+
+    let mut proofs = Vec::new();
+
+    for (i, (&point_at_k, gi)) in point
+        .iter()
+        .zip(prover_param.powers_of_g[ignored..].iter())
+        .take(nv)
+        .enumerate()
+    {
+        let ith_round = start_timer!(|| format!("{}-th round", i));
+
+        let k = nv - i;
+        let cur_dim = 1 << (k - 1);
+        let mut cur_q = vec![E::Fr::zero(); cur_dim];
+        let mut cur_r = vec![E::Fr::zero(); cur_dim];
+        let one_minus_point_at_k = E::Fr::one() - point_at_k;
+
+        let ith_round_eval = start_timer!(|| format!("{}-th round eval", i));
+        for b in 0..(1 << (k - 1)) {
+            // q_b = pre_r [2^b + 1] - pre_r [2^b]
+            cur_q[b] = r[k][(b << 1) + 1] - r[k][b << 1];
+
+            // r_b = pre_r [2^b]*(1-p) + pre_r [2^b + 1] * p
+            cur_r[b] = r[k][b << 1] * one_minus_point_at_k + (r[k][(b << 1) + 1] * point_at_k);
+        }
+        end_timer!(ith_round_eval);
+        let scalars: Vec<_> = (0..(1 << k)).map(|x| cur_q[x >> 1].into_repr()).collect();
+
+        q[k] = cur_q;
+        r[k - 1] = cur_r;
+
+        // this is a MSM over G1 and is likely to be the bottleneck
+        proofs.push(VariableBaseMSM::multi_scalar_mul(&gi.evals, &scalars).into_affine());
+        end_timer!(ith_round);
+    }
+    let eval = evaluate_opt(polynomial, point);
+    end_timer!(open_timer);
+    Ok((MultilinearKzgProof { proofs }, eval))
+}
+
+/// Verifies that `value` is the evaluation at `x` of the polynomial
+/// committed inside `comm`.
+///
+/// This function takes
+/// - num_var number of pairing product.
+/// - num_var number of MSM
+fn verify_internal<E: PairingEngine>(
+    verifier_param: &MultilinearVerifierParam<E>,
+    commitment: &Commitment<E>,
+    point: &[E::Fr],
+    value: &E::Fr,
+    proof: &MultilinearKzgProof<E>,
+) -> Result<bool, PCSError> {
+    let verify_timer = start_timer!(|| "verify");
+    let num_var = point.len();
+
+    if num_var > verifier_param.num_vars {
+        return Err(PCSError::InvalidParameters(format!(
+            "point length ({}) exceeds param limit ({})",
+            num_var, verifier_param.num_vars
+        )));
+    }
+
+    let ignored = verifier_param.num_vars - num_var;
+    let prepare_inputs_timer = start_timer!(|| "prepare pairing inputs");
+
+    let scalar_size = E::Fr::size_in_bits();
+    let window_size = FixedBaseMSM::get_mul_window_size(num_var);
+
+    let h_table = FixedBaseMSM::get_window_table(
+        scalar_size,
+        window_size,
+        verifier_param.h.into_projective(),
+    );
+    let h_mul: Vec<E::G2Projective> =
+        FixedBaseMSM::multi_scalar_mul(scalar_size, window_size, &h_table, point);
+
+    let h_vec: Vec<_> = (0..num_var)
+        .map(|i| verifier_param.h_mask[ignored + i].into_projective() - h_mul[i])
+        .collect();
+    let h_vec: Vec<E::G2Affine> = E::G2Projective::batch_normalization_into_affine(&h_vec);
+    end_timer!(prepare_inputs_timer);
+
+    let pairing_product_timer = start_timer!(|| "pairing product");
+
+    let mut pairings: Vec<_> = proof
+        .proofs
+        .iter()
+        .map(|&x| E::G1Prepared::from(x))
+        .zip(h_vec.into_iter().take(num_var).map(E::G2Prepared::from))
+        .collect();
+
+    pairings.push((
+        E::G1Prepared::from(
+            (verifier_param.g.mul(*value) - commitment.0.into_projective()).into_affine(),
+        ),
+        E::G2Prepared::from(verifier_param.h),
+    ));
+
+    let res = E::product_of_pairings(pairings.iter()) == E::Fqk::one();
+
+    end_timer!(pairing_product_timer);
+    end_timer!(verify_timer);
+    Ok(res)
+}
+
+#[cfg(test)]
+mod tests {
+    use super::*;
+    use crate::multilinear_kzg::util::compute_qx_degree;
+    use arithmetic::get_batched_nv;
+    use ark_bls12_381::Bls12_381;
+    use ark_ec::PairingEngine;
+    use ark_poly::{DenseMultilinearExtension, MultilinearExtension};
+    use ark_std::{log2, rand::RngCore, test_rng, vec::Vec, UniformRand};
+
+    type E = Bls12_381;
+    type Fr = <E as PairingEngine>::Fr;
+
+    fn test_single_helper<R: RngCore + CryptoRng>(
+        params: &(MultilinearUniversalParams<E>, UnivariateUniversalParams<E>),
+        poly: &Rc<DenseMultilinearExtension<Fr>>,
+        rng: &mut R,
+    ) -> Result<(), PCSError> {
+        let nv = poly.num_vars();
+        assert_ne!(nv, 0);
+        let uni_degree = 1;
+        let (ck, vk) = MultilinearKzgPCS::trim(params, uni_degree, Some(nv + 1))?;
+        let point: Vec<_> = (0..nv).map(|_| Fr::rand(rng)).collect();
+        let com = MultilinearKzgPCS::commit(&ck, poly)?;
+        let (proof, value) = MultilinearKzgPCS::open(&ck, poly, &point)?;
+
+        assert!(MultilinearKzgPCS::verify(
+            &vk, &com, &point, &value, &proof
+        )?);
+
+        let value = Fr::rand(rng);
+        assert!(!MultilinearKzgPCS::verify(
+            &vk, &com, &point, &value, &proof
+        )?);
+
+        Ok(())
+    }
+
+    #[test]
+    fn test_single_commit() -> Result<(), PCSError> {
+        let mut rng = test_rng();
+
+        let params = MultilinearKzgPCS::<E>::gen_srs_for_testing(&mut rng, 10)?;
+
+        // normal polynomials
+        let poly1 = Rc::new(DenseMultilinearExtension::rand(8, &mut rng));
+        test_single_helper(&params, &poly1, &mut rng)?;
+
+        // single-variate polynomials
+        let poly2 = Rc::new(DenseMultilinearExtension::rand(1, &mut rng));
+        test_single_helper(&params, &poly2, &mut rng)?;
+
+        Ok(())
+    }
+
+    #[test]
+    fn setup_commit_verify_constant_polynomial() {
+        let mut rng = test_rng();
+
+        // normal polynomials
+        assert!(MultilinearKzgPCS::<E>::gen_srs_for_testing(&mut rng, 0).is_err());
+    }
+
+    fn test_multi_open_single_poly_helper<R: RngCore + CryptoRng>(
+        params: &(MultilinearUniversalParams<E>, UnivariateUniversalParams<E>),
+        poly: Rc<DenseMultilinearExtension<Fr>>,
+        num_open: usize,
+        rng: &mut R,
+    ) -> Result<(), PCSError> {
+        let nv = poly.num_vars();
+        assert_ne!(nv, 0);
+        let uni_degree = 1024;
+        let (ck, vk) = MultilinearKzgPCS::trim(params, uni_degree, Some(nv + 1))?;
+        let mut points = vec![];
+        for _ in 0..num_open {
+            let point: Vec<_> = (0..nv).map(|_| Fr::rand(rng)).collect();
+            points.push(point)
+        }
+        let com = MultilinearKzgPCS::commit(&ck, &poly)?;
+        let (proof, mut values) =
+            MultilinearKzgPCS::multi_open_single_poly(&ck, &com, &poly, &points)?;
+        for (a, b) in values.iter().zip(points.iter()) {
+            let p = poly.evaluate(&b).unwrap();
+            assert_eq!(*a, p);
+        }
+
+        assert!(MultilinearKzgPCS::batch_verify_single_poly(
+            &vk, &com, &points, &values, &proof
+        )?);
+
+        values[0] = Fr::rand(rng);
+        assert!(!MultilinearKzgPCS::batch_verify_single_poly(
+            &vk, &com, &points, &values, &proof
+        )?);
+        Ok(())
+    }
+
+    #[test]
+    fn test_multi_open_single_poly() -> Result<(), PCSError> {
+        let mut rng = test_rng();
+
+        let params = MultilinearKzgPCS::<E>::gen_srs_for_testing(&mut rng, 15)?;
+
+        for nv in 1..10 {
+            for num_open in 2..10 {
+                let poly1 = Rc::new(DenseMultilinearExtension::rand(nv, &mut rng));
+                test_multi_open_single_poly_helper(&params, poly1, num_open, &mut rng)?;
+            }
+        }
+
+        Ok(())
+    }
+
+    fn test_multi_open_helper<R: RngCore + CryptoRng>(
+        params: &(MultilinearUniversalParams<E>, UnivariateUniversalParams<E>),
+        polys: &[Rc<DenseMultilinearExtension<Fr>>],
+        num_open: usize,
+        rng: &mut R,
+    ) -> Result<(), PCSError> {
+        let nv = polys[0].num_vars();
+        assert_ne!(nv, 0);
+        let merged_nv = get_batched_nv(nv, polys.len());
+        let qx_degree = compute_qx_degree(merged_nv, polys.len());
+        let padded_qx_degree = 1usize << log2(qx_degree);
+
+        let (ck, vk) = MultilinearKzgPCS::trim(params, padded_qx_degree, Some(merged_nv))?;
+        let mut points = vec![];
+        for _ in 0..num_open {
+            let point: Vec<_> = (0..nv).map(|_| Fr::rand(rng)).collect();
+            points.push(point)
+        }
+        let com = MultilinearKzgPCS::multi_commit(&ck, &polys)?;
+        let (proof, mut values) = MultilinearKzgPCS::multi_open(&ck, &com, polys, &points)?;
+
+        assert!(MultilinearKzgPCS::batch_verify(
+            &vk, &com, &points, &values, &proof, rng
+        )?);
+
+        values[0] = Fr::rand(rng);
+        assert!(!MultilinearKzgPCS::batch_verify_single_poly(
+            &vk, &com, &points, &values, &proof
+        )?);
+
+        Ok(())
+    }
+
+    #[test]
+    fn test_multi_open() -> Result<(), PCSError> {
+        let mut rng = test_rng();
+
+        let params = MultilinearKzgPCS::<E>::gen_srs_for_testing(&mut rng, 15)?;
+
+        // normal polynomials
+        for nv in 1..10 {
+            for num_open in 1..4 {
+                let mut polys = vec![];
+                for _ in 0..num_open {
+                    let poly = Rc::new(DenseMultilinearExtension::rand(nv, &mut rng));
+                    polys.push(poly)
+                }
+
+                test_multi_open_helper(&params, &polys, num_open, &mut rng)?;
+            }
+        }
+        Ok(())
+    }
+}

pcs/src/multilinear_kzg/srs.rs

@@ -0,0 +1,273 @@
+// Copyright (c) 2022 Espresso Systems (espressosys.com)
+// This file is part of the Jellyfish library.
+
+// You should have received a copy of the MIT License
+// along with the Jellyfish library. If not, see <https://mit-license.org/>.
+
+//! Implementing Structured Reference Strings for multilinear polynomial KZG
+use crate::{prelude::PCSError, StructuredReferenceString};
+use ark_ec::{msm::FixedBaseMSM, AffineCurve, PairingEngine, ProjectiveCurve};
+use ark_ff::{Field, PrimeField};
+use ark_poly::DenseMultilinearExtension;
+use ark_serialize::{CanonicalDeserialize, CanonicalSerialize, Read, SerializationError, Write};
+use ark_std::{
+    collections::LinkedList,
+    end_timer, format,
+    rand::{CryptoRng, RngCore},
+    start_timer,
+    string::ToString,
+    vec::Vec,
+    UniformRand,
+};
+use core::iter::FromIterator;
+
+/// Evaluations over {0,1}^n for G1 or G2
+#[derive(CanonicalSerialize, CanonicalDeserialize, Clone, Debug)]
+pub struct Evaluations<C: AffineCurve> {
+    /// The evaluations.
+    pub evals: Vec<C>,
+}
+
+/// Universal Parameter
+#[derive(CanonicalSerialize, CanonicalDeserialize, Clone, Debug)]
+pub struct MultilinearUniversalParams<E: PairingEngine> {
+    /// prover parameters
+    pub prover_param: MultilinearProverParam<E>,
+    /// h^randomness: h^t1, h^t2, ..., **h^{t_nv}**
+    pub h_mask: Vec<E::G2Affine>,
+}
+
+/// Prover Parameters
+#[derive(CanonicalSerialize, CanonicalDeserialize, Clone, Debug)]
+pub struct MultilinearProverParam<E: PairingEngine> {
+    /// number of variables
+    pub num_vars: usize,
+    /// `pp_{num_vars}`, `pp_{num_vars - 1}`, `pp_{num_vars - 2}`, ..., defined
+    /// by XZZPD19
+    pub powers_of_g: Vec<Evaluations<E::G1Affine>>,
+    /// generator for G1
+    pub g: E::G1Affine,
+    /// generator for G2
+    pub h: E::G2Affine,
+}
+
+/// Verifier Parameters
+#[derive(CanonicalSerialize, CanonicalDeserialize, Clone, Debug)]
+pub struct MultilinearVerifierParam<E: PairingEngine> {
+    /// number of variables
+    pub num_vars: usize,
+    /// generator of G1
+    pub g: E::G1Affine,
+    /// generator of G2
+    pub h: E::G2Affine,
+    /// h^randomness: h^t1, h^t2, ..., **h^{t_nv}**
+    pub h_mask: Vec<E::G2Affine>,
+}
+
+impl<E: PairingEngine> StructuredReferenceString<E> for MultilinearUniversalParams<E> {
+    type ProverParam = MultilinearProverParam<E>;
+    type VerifierParam = MultilinearVerifierParam<E>;
+
+    /// Extract the prover parameters from the public parameters.
+    fn extract_prover_param(&self, supported_num_vars: usize) -> Self::ProverParam {
+        let to_reduce = self.prover_param.num_vars - supported_num_vars;
+
+        Self::ProverParam {
+            powers_of_g: self.prover_param.powers_of_g[to_reduce..].to_vec(),
+            g: self.prover_param.g,
+            h: self.prover_param.h,
+            num_vars: supported_num_vars,
+        }
+    }
+
+    /// Extract the verifier parameters from the public parameters.
+    fn extract_verifier_param(&self, supported_num_vars: usize) -> Self::VerifierParam {
+        let to_reduce = self.prover_param.num_vars - supported_num_vars;
+        Self::VerifierParam {
+            num_vars: supported_num_vars,
+            g: self.prover_param.g,
+            h: self.prover_param.h,
+            h_mask: self.h_mask[to_reduce..].to_vec(),
+        }
+    }
+
+    /// Trim the universal parameters to specialize the public parameters
+    /// for multilinear polynomials to the given `supported_num_vars`, and
+    /// returns committer key and verifier key. `supported_num_vars` should
+    /// be in range `1..=params.num_vars`
+    fn trim(
+        &self,
+        supported_num_vars: usize,
+    ) -> Result<(Self::ProverParam, Self::VerifierParam), PCSError> {
+        if supported_num_vars > self.prover_param.num_vars {
+            return Err(PCSError::InvalidParameters(format!(
+                "SRS does not support target number of vars {}",
+                supported_num_vars
+            )));
+        }
+
+        let to_reduce = self.prover_param.num_vars - supported_num_vars;
+        let ck = Self::ProverParam {
+            powers_of_g: self.prover_param.powers_of_g[to_reduce..].to_vec(),
+            g: self.prover_param.g,
+            h: self.prover_param.h,
+            num_vars: supported_num_vars,
+        };
+        let vk = Self::VerifierParam {
+            num_vars: supported_num_vars,
+            g: self.prover_param.g,
+            h: self.prover_param.h,
+            h_mask: self.h_mask[to_reduce..].to_vec(),
+        };
+        Ok((ck, vk))
+    }
+
+    /// Build SRS for testing.
+    /// WARNING: THIS FUNCTION IS FOR TESTING PURPOSE ONLY.
+    /// THE OUTPUT SRS SHOULD NOT BE USED IN PRODUCTION.
+    fn gen_srs_for_testing<R: RngCore + CryptoRng>(
+        rng: &mut R,
+        num_vars: usize,
+    ) -> Result<Self, PCSError> {
+        if num_vars == 0 {
+            return Err(PCSError::InvalidParameters(
+                "constant polynomial not supported".to_string(),
+            ));
+        }
+
+        let total_timer = start_timer!(|| "SRS generation");
+
+        let pp_generation_timer = start_timer!(|| "Prover Param generation");
+
+        let g = E::G1Projective::rand(rng);
+        let h = E::G2Projective::rand(rng);
+
+        let mut powers_of_g = Vec::new();
+
+        let t: Vec<_> = (0..num_vars).map(|_| E::Fr::rand(rng)).collect();
+        let scalar_bits = E::Fr::size_in_bits();
+
+        let mut eq: LinkedList<DenseMultilinearExtension<E::Fr>> =
+            LinkedList::from_iter(eq_extension(&t).into_iter());
+        let mut eq_arr = LinkedList::new();
+        let mut base = eq.pop_back().unwrap().evaluations;
+
+        for i in (0..num_vars).rev() {
+            eq_arr.push_front(remove_dummy_variable(&base, i)?);
+            if i != 0 {
+                let mul = eq.pop_back().unwrap().evaluations;
+                base = base
+                    .into_iter()
+                    .zip(mul.into_iter())
+                    .map(|(a, b)| a * b)
+                    .collect();
+            }
+        }
+
+        let mut pp_powers = Vec::new();
+        let mut total_scalars = 0;
+        for i in 0..num_vars {
+            let eq = eq_arr.pop_front().unwrap();
+            let pp_k_powers = (0..(1 << (num_vars - i))).map(|x| eq[x]);
+            pp_powers.extend(pp_k_powers);
+            total_scalars += 1 << (num_vars - i);
+        }
+        let window_size = FixedBaseMSM::get_mul_window_size(total_scalars);
+        let g_table = FixedBaseMSM::get_window_table(scalar_bits, window_size, g);
+
+        let pp_g = E::G1Projective::batch_normalization_into_affine(
+            &FixedBaseMSM::multi_scalar_mul(scalar_bits, window_size, &g_table, &pp_powers),
+        );
+
+        let mut start = 0;
+        for i in 0..num_vars {
+            let size = 1 << (num_vars - i);
+            let pp_k_g = Evaluations {
+                evals: pp_g[start..(start + size)].to_vec(),
+            };
+            powers_of_g.push(pp_k_g);
+            start += size;
+        }
+
+        let pp = Self::ProverParam {
+            num_vars,
+            g: g.into_affine(),
+            h: h.into_affine(),
+            powers_of_g,
+        };
+
+        end_timer!(pp_generation_timer);
+
+        let vp_generation_timer = start_timer!(|| "VP generation");
+        let h_mask = {
+            let window_size = FixedBaseMSM::get_mul_window_size(num_vars);
+            let h_table = FixedBaseMSM::get_window_table(scalar_bits, window_size, h);
+            E::G2Projective::batch_normalization_into_affine(&FixedBaseMSM::multi_scalar_mul(
+                scalar_bits,
+                window_size,
+                &h_table,
+                &t,
+            ))
+        };
+        end_timer!(vp_generation_timer);
+        end_timer!(total_timer);
+        Ok(Self {
+            prover_param: pp,
+            h_mask,
+        })
+    }
+}
+
+/// fix first `pad` variables of `poly` represented in evaluation form to zero
+fn remove_dummy_variable<F: Field>(poly: &[F], pad: usize) -> Result<Vec<F>, PCSError> {
+    if pad == 0 {
+        return Ok(poly.to_vec());
+    }
+    if !poly.len().is_power_of_two() {
+        return Err(PCSError::InvalidParameters(
+            "Size of polynomial should be power of two.".to_string(),
+        ));
+    }
+    let nv = ark_std::log2(poly.len()) as usize - pad;
+    Ok((0..(1 << nv)).map(|x| poly[x << pad]).collect())
+}
+
+/// Generate eq(t,x), a product of multilinear polynomials with fixed t.
+/// eq(a,b) is takes extensions of a,b in {0,1}^num_vars such that if a and b in
+/// {0,1}^num_vars are equal then this polynomial evaluates to 1.
+fn eq_extension<F: PrimeField>(t: &[F]) -> Vec<DenseMultilinearExtension<F>> {
+    let start = start_timer!(|| "eq extension");
+
+    let dim = t.len();
+    let mut result = Vec::new();
+    for (i, &ti) in t.iter().enumerate().take(dim) {
+        let mut poly = Vec::with_capacity(1 << dim);
+        for x in 0..(1 << dim) {
+            let xi = if x >> i & 1 == 1 { F::one() } else { F::zero() };
+            let ti_xi = ti * xi;
+            poly.push(ti_xi + ti_xi - xi - ti + F::one());
+        }
+        result.push(DenseMultilinearExtension::from_evaluations_vec(dim, poly));
+    }
+
+    end_timer!(start);
+    result
+}
+
+#[cfg(test)]
+mod tests {
+    use super::*;
+    use ark_bls12_381::Bls12_381;
+    use ark_std::test_rng;
+    type E = Bls12_381;
+
+    #[test]
+    fn test_srs_gen() -> Result<(), PCSError> {
+        let mut rng = test_rng();
+        for nv in 4..10 {
+            let _ = MultilinearUniversalParams::<E>::gen_srs_for_testing(&mut rng, nv)?;
+        }
+
+        Ok(())
+    }
+}

pcs/src/multilinear_kzg/util.rs

@@ -0,0 +1,432 @@
+// Copyright (c) 2022 Espresso Systems (espressosys.com)
+// This file is part of the Jellyfish library.
+
+// You should have received a copy of the MIT License
+// along with the Jellyfish library. If not, see <https://mit-license.org/>.
+
+//! Useful utilities for KZG PCS
+use crate::prelude::PCSError;
+use arithmetic::evaluate_no_par;
+use ark_ff::PrimeField;
+use ark_poly::{
+    univariate::DensePolynomial, DenseMultilinearExtension, EvaluationDomain, Evaluations,
+    MultilinearExtension, Polynomial, Radix2EvaluationDomain,
+};
+use ark_std::{end_timer, format, log2, start_timer, string::ToString, vec::Vec};
+use rayon::prelude::{IntoParallelIterator, ParallelIterator};
+
+/// For an MLE w with `mle_num_vars` variables, and `point_len` number of
+/// points, compute the degree of the univariate polynomial `q(x):= w(l(x))`
+/// where l(x) is a list of polynomials that go through all points.
+// uni_degree is computed as `mle_num_vars * point_len`:
+// - each l(x) is of degree `point_len`
+// - mle has degree one
+// - worst case is `\prod_{i=0}^{mle_num_vars-1} l_i(x) < point_len * mle_num_vars`
+#[inline]
+pub fn compute_qx_degree(mle_num_vars: usize, point_len: usize) -> usize {
+    mle_num_vars * ((1 << log2(point_len)) - 1) + 1
+}
+
+/// Compute W \circ l.
+///
+/// Given an MLE W, and a list of univariate polynomials l, generate the
+/// univariate polynomial that composes W with l.
+///
+/// Returns an error if l's length does not matches number of variables in W.
+pub(crate) fn compute_w_circ_l<F: PrimeField>(
+    w: &DenseMultilinearExtension<F>,
+    l: &[DensePolynomial<F>],
+    num_points: usize,
+    with_suffix: bool,
+) -> Result<DensePolynomial<F>, PCSError> {
+    let timer = start_timer!(|| "compute W \\circ l");
+
+    if w.num_vars != l.len() {
+        return Err(PCSError::InvalidParameters(format!(
+            "l's length ({}) does not match num_variables ({})",
+            l.len(),
+            w.num_vars(),
+        )));
+    }
+    let uni_degree = if with_suffix {
+        compute_qx_degree(w.num_vars() + log2(num_points) as usize, num_points)
+    } else {
+        compute_qx_degree(w.num_vars(), num_points)
+    };
+
+    let domain = match Radix2EvaluationDomain::<F>::new(uni_degree) {
+        Some(p) => p,
+        None => {
+            return Err(PCSError::InvalidParameters(
+                "failed to build radix 2 domain".to_string(),
+            ))
+        },
+    };
+
+    let step = start_timer!(|| format!("compute eval {}-dim domain", domain.size()));
+    let res_eval = (0..domain.size())
+        .into_par_iter()
+        .map(|i| {
+            let l_eval: Vec<F> = l.iter().map(|x| x.evaluate(&domain.element(i))).collect();
+            evaluate_no_par(w, &l_eval)
+        })
+        .collect();
+    end_timer!(step);
+
+    let evaluation = Evaluations::from_vec_and_domain(res_eval, domain);
+    let res = evaluation.interpolate();
+
+    end_timer!(timer);
+    Ok(res)
+}
+
+/// Input a list of multilinear polynomials and a list of points,
+/// generate a list of evaluations.
+// Note that this function is only used for testing verifications.
+// In practice verifier does not see polynomials, and the `mle_values`
+// are included in the `batch_proof`.
+#[cfg(test)]
+pub(crate) fn generate_evaluations_multi_poly<F: PrimeField>(
+    polynomials: &[std::rc::Rc<DenseMultilinearExtension<F>>],
+    points: &[Vec<F>],
+) -> Result<Vec<F>, PCSError> {
+    use arithmetic::{build_l, get_uni_domain, merge_polynomials};
+
+    if polynomials.len() != points.len() {
+        return Err(PCSError::InvalidParameters(
+            "polynomial length does not match point length".to_string(),
+        ));
+    }
+    let uni_poly_degree = points.len();
+    let merge_poly = merge_polynomials(polynomials)?;
+
+    let domain = get_uni_domain::<F>(uni_poly_degree)?;
+    let uni_polys = build_l(points, &domain, true)?;
+    let mut mle_values = vec![];
+
+    for i in 0..uni_poly_degree {
+        let point: Vec<F> = uni_polys
+            .iter()
+            .map(|poly| poly.evaluate(&domain.element(i)))
+            .collect();
+
+        let mle_value = merge_poly.evaluate(&point).unwrap();
+        mle_values.push(mle_value)
+    }
+    Ok(mle_values)
+}
+
+/// Input a list of multilinear polynomials and a list of points,
+/// generate a list of evaluations.
+// Note that this function is only used for testing verifications.
+// In practice verifier does not see polynomials, and the `mle_values`
+// are included in the `batch_proof`.
+#[cfg(test)]
+pub(crate) fn generate_evaluations_single_poly<F: PrimeField>(
+    polynomial: &std::rc::Rc<DenseMultilinearExtension<F>>,
+    points: &[Vec<F>],
+) -> Result<Vec<F>, PCSError> {
+    use arithmetic::{build_l, get_uni_domain};
+
+    let uni_poly_degree = points.len();
+
+    let domain = get_uni_domain::<F>(uni_poly_degree)?;
+    let uni_polys = build_l(points, &domain, false)?;
+    let mut mle_values = vec![];
+
+    for i in 0..uni_poly_degree {
+        let point: Vec<F> = uni_polys
+            .iter()
+            .map(|poly| poly.evaluate(&domain.element(i)))
+            .collect();
+
+        let mle_value = polynomial.evaluate(&point).unwrap();
+        mle_values.push(mle_value)
+    }
+    Ok(mle_values)
+}
+
+#[cfg(test)]
+mod test {
+    use super::*;
+    use arithmetic::{build_l, get_uni_domain, merge_polynomials};
+    use ark_bls12_381::Fr;
+    use ark_poly::UVPolynomial;
+    use ark_std::{One, Zero};
+    use std::rc::Rc;
+
+    #[test]
+    fn test_w_circ_l() -> Result<(), PCSError> {
+        test_w_circ_l_helper::<Fr>()
+    }
+
+    fn test_w_circ_l_helper<F: PrimeField>() -> Result<(), PCSError> {
+        {
+            // Example from page 53:
+            // W = 3x1x2 + 2x2 whose evaluations are
+            // 0, 0 |-> 0
+            // 1, 0 |-> 0
+            // 0, 1 |-> 2
+            // 1, 1 |-> 5
+            let w_eval = vec![F::zero(), F::zero(), F::from(2u64), F::from(5u64)];
+            let w = DenseMultilinearExtension::from_evaluations_vec(2, w_eval);
+
+            // l0 =   t + 2
+            // l1 = -2t + 4
+            let l0 = DensePolynomial::from_coefficients_vec(vec![F::from(2u64), F::one()]);
+            let l1 = DensePolynomial::from_coefficients_vec(vec![F::from(4u64), -F::from(2u64)]);
+
+            // res = -6t^2 - 4t + 32
+            let res = compute_w_circ_l(&w, [l0, l1].as_ref(), 4, false)?;
+            let res_rec = DensePolynomial::from_coefficients_vec(vec![
+                F::from(32u64),
+                -F::from(4u64),
+                -F::from(6u64),
+            ]);
+            assert_eq!(res, res_rec);
+        }
+        {
+            // A random example
+            // W = x1x2x3 - 2x1x2 + 3x2x3 - 4x1x3 + 5x1 - 6x2 + 7x3
+            // 0, 0, 0 |->  0
+            // 1, 0, 0 |->  5
+            // 0, 1, 0 |-> -6
+            // 1, 1, 0 |-> -3
+            // 0, 0, 1 |->  7
+            // 1, 0, 1 |->  8
+            // 0, 1, 1 |->  4
+            // 1, 1, 1 |->  4
+            let w_eval = vec![
+                F::zero(),
+                F::from(5u64),
+                -F::from(6u64),
+                -F::from(3u64),
+                F::from(7u64),
+                F::from(8u64),
+                F::from(4u64),
+                F::from(4u64),
+            ];
+            let w = DenseMultilinearExtension::from_evaluations_vec(3, w_eval);
+
+            // l0 =   t + 2
+            // l1 =  3t - 4
+            // l2 = -5t + 6
+            let l0 = DensePolynomial::from_coefficients_vec(vec![F::from(2u64), F::one()]);
+            let l1 = DensePolynomial::from_coefficients_vec(vec![-F::from(4u64), F::from(3u64)]);
+            let l2 = DensePolynomial::from_coefficients_vec(vec![F::from(6u64), -F::from(5u64)]);
+            let res = compute_w_circ_l(&w, [l0, l1, l2].as_ref(), 8, false)?;
+
+            // res = -15t^3 - 23t^2 + 130t - 76
+            let res_rec = DensePolynomial::from_coefficients_vec(vec![
+                -F::from(76u64),
+                F::from(130u64),
+                -F::from(23u64),
+                -F::from(15u64),
+            ]);
+
+            assert_eq!(res, res_rec);
+        }
+        Ok(())
+    }
+
+    #[test]
+    fn test_w_circ_l_with_prefix() -> Result<(), PCSError> {
+        test_w_circ_l_with_prefix_helper::<Fr>()
+    }
+
+    fn test_w_circ_l_with_prefix_helper<F: PrimeField>() -> Result<(), PCSError> {
+        {
+            // Example from page 53:
+            // W = 3x1x2 + 2x2 whose evaluations are
+            // 0, 0 |-> 0
+            // 1, 0 |-> 0
+            // 0, 1 |-> 2
+            // 1, 1 |-> 5
+            let w_eval = vec![F::zero(), F::zero(), F::from(2u64), F::from(5u64)];
+            let w = DenseMultilinearExtension::from_evaluations_vec(2, w_eval);
+
+            // l0 =   t + 2
+            // l1 = -2t + 4
+            let l0 = DensePolynomial::from_coefficients_vec(vec![F::from(2u64), F::one()]);
+            let l1 = DensePolynomial::from_coefficients_vec(vec![F::from(4u64), -F::from(2u64)]);
+
+            // res = -6t^2 - 4t + 32
+            let res = compute_w_circ_l(&w, [l0, l1].as_ref(), 4, true)?;
+            let res_rec = DensePolynomial::from_coefficients_vec(vec![
+                F::from(32u64),
+                -F::from(4u64),
+                -F::from(6u64),
+            ]);
+            assert_eq!(res, res_rec);
+        }
+        {
+            // A random example
+            // W = x1x2x3 - 2x1x2 + 3x2x3 - 4x1x3 + 5x1 - 6x2 + 7x3
+            // 0, 0, 0 |->  0
+            // 1, 0, 0 |->  5
+            // 0, 1, 0 |-> -6
+            // 1, 1, 0 |-> -3
+            // 0, 0, 1 |->  7
+            // 1, 0, 1 |->  8
+            // 0, 1, 1 |->  4
+            // 1, 1, 1 |->  4
+            let w_eval = vec![
+                F::zero(),
+                F::from(5u64),
+                -F::from(6u64),
+                -F::from(3u64),
+                F::from(7u64),
+                F::from(8u64),
+                F::from(4u64),
+                F::from(4u64),
+            ];
+            let w = DenseMultilinearExtension::from_evaluations_vec(3, w_eval);
+
+            // l0 =   t + 2
+            // l1 =  3t - 4
+            // l2 = -5t + 6
+            let l0 = DensePolynomial::from_coefficients_vec(vec![F::from(2u64), F::one()]);
+            let l1 = DensePolynomial::from_coefficients_vec(vec![-F::from(4u64), F::from(3u64)]);
+            let l2 = DensePolynomial::from_coefficients_vec(vec![F::from(6u64), -F::from(5u64)]);
+            let res = compute_w_circ_l(&w, [l0, l1, l2].as_ref(), 8, true)?;
+
+            // res = -15t^3 - 23t^2 + 130t - 76
+            let res_rec = DensePolynomial::from_coefficients_vec(vec![
+                -F::from(76u64),
+                F::from(130u64),
+                -F::from(23u64),
+                -F::from(15u64),
+            ]);
+
+            assert_eq!(res, res_rec);
+        }
+        Ok(())
+    }
+
+    #[test]
+    fn test_qx() -> Result<(), PCSError> {
+        // Example from page 53:
+        // W1 = 3x1x2 + 2x2
+        let w_eval = vec![Fr::zero(), Fr::from(2u64), Fr::zero(), Fr::from(5u64)];
+        let w = Rc::new(DenseMultilinearExtension::from_evaluations_vec(2, w_eval));
+
+        let r = Fr::from(42u64);
+
+        // point 1 is [1, 2]
+        let point1 = vec![Fr::from(1u64), Fr::from(2u64)];
+
+        // point 2 is [3, 4]
+        let point2 = vec![Fr::from(3u64), Fr::from(4u64)];
+
+        // point 3 is [5, 6]
+        let point3 = vec![Fr::from(5u64), Fr::from(6u64)];
+
+        {
+            let domain = get_uni_domain::<Fr>(2)?;
+            let l = build_l(&[point1.clone(), point2.clone()], &domain, false)?;
+
+            let q_x = compute_w_circ_l(&w, &l, 2, false)?;
+
+            let point: Vec<Fr> = l.iter().map(|poly| poly.evaluate(&r)).collect();
+
+            assert_eq!(
+                q_x.evaluate(&r),
+                w.evaluate(&point).unwrap(),
+                "q(r) != w(l(r))"
+            );
+        }
+
+        {
+            let domain = get_uni_domain::<Fr>(3)?;
+
+            let l = build_l(&[point1, point2, point3], &domain, false)?;
+            let q_x = compute_w_circ_l(&w, &l, 3, false)?;
+
+            let point: Vec<Fr> = vec![l[0].evaluate(&r), l[1].evaluate(&r)];
+
+            assert_eq!(
+                q_x.evaluate(&r),
+                w.evaluate(&point).unwrap(),
+                "q(r) != w(l(r))"
+            );
+        }
+        Ok(())
+    }
+
+    #[test]
+    fn test_qx_with_prefix() -> Result<(), PCSError> {
+        // Example from page 53:
+        // W1 = 3x1x2 + 2x2
+        let w_eval = vec![Fr::zero(), Fr::from(2u64), Fr::zero(), Fr::from(5u64)];
+        let w1 = Rc::new(DenseMultilinearExtension::from_evaluations_vec(2, w_eval));
+
+        // W2 = x1x2 + x1
+        let w_eval = vec![Fr::zero(), Fr::zero(), Fr::from(1u64), Fr::from(2u64)];
+        let w2 = Rc::new(DenseMultilinearExtension::from_evaluations_vec(2, w_eval));
+
+        // W3 = x1 + x2
+        let w_eval = vec![Fr::zero(), Fr::one(), Fr::from(1u64), Fr::from(2u64)];
+        let w3 = Rc::new(DenseMultilinearExtension::from_evaluations_vec(2, w_eval));
+
+        let r = Fr::from(42u64);
+
+        // point 1 is [1, 2]
+        let point1 = vec![Fr::from(1u64), Fr::from(2u64)];
+
+        // point 2 is [3, 4]
+        let point2 = vec![Fr::from(3u64), Fr::from(4u64)];
+
+        // point 3 is [5, 6]
+        let point3 = vec![Fr::from(5u64), Fr::from(6u64)];
+
+        {
+            let domain = get_uni_domain::<Fr>(2)?;
+            // w = (3x1x2 + 2x2)(1-x0) + (x1x2 + x1)x0
+            // with evaluations: [0,2,0,5,0,0,1,2]
+            let w = merge_polynomials(&[w1.clone(), w2.clone()])?;
+
+            let l = build_l(&[point1.clone(), point2.clone()], &domain, true)?;
+
+            // sage: P.<x> = PolynomialRing(ZZ)
+            // sage: l0 = -1/2 * x + 1/2
+            // sage: l1 = -x + 2
+            // sage: l2 = -x + 3
+            // sage: w = (3 * l1 * l2 + 2 * l2) * (1-l0) + (l1 * l2 + l1) * l0
+            // sage: w
+            // x^3 - 7/2*x^2 - 7/2*x + 16
+            //
+            // q(x) = x^3 - 7/2*x^2 - 7/2*x + 16
+            let q_x = compute_w_circ_l(&w, &l, 2, true)?;
+
+            let point: Vec<Fr> = l.iter().map(|poly| poly.evaluate(&r)).collect();
+
+            assert_eq!(
+                q_x.evaluate(&r),
+                w.evaluate(&point).unwrap(),
+                "q(r) != w(l(r))"
+            );
+        }
+
+        {
+            let domain = get_uni_domain::<Fr>(3)?;
+            let w = merge_polynomials(&[w1, w2, w3])?;
+
+            let l = build_l(&[point1, point2, point3], &domain, true)?;
+            let q_x = compute_w_circ_l(&w, &l, 3, true)?;
+
+            let point: Vec<Fr> = vec![
+                l[0].evaluate(&r),
+                l[1].evaluate(&r),
+                l[2].evaluate(&r),
+                l[3].evaluate(&r),
+            ];
+
+            assert_eq!(
+                q_x.evaluate(&r),
+                w.evaluate(&point).unwrap(),
+                "q(r) != w(l(r))"
+            );
+        }
+        Ok(())
+    }
+}

pcs/src/prelude.rs

@@ -0,0 +1,21 @@
+// Copyright (c) 2022 Espresso Systems (espressosys.com)
+// This file is part of the Jellyfish library.
+
+// You should have received a copy of the MIT License
+// along with the Jellyfish library. If not, see <https://mit-license.org/>.
+
+//! Prelude
+pub use crate::{
+    errors::PCSError,
+    multilinear_kzg::{
+        srs::{MultilinearProverParam, MultilinearUniversalParams, MultilinearVerifierParam},
+        util::compute_qx_degree,
+        MultilinearKzgBatchProof, MultilinearKzgPCS, MultilinearKzgProof,
+    },
+    structs::Commitment,
+    univariate_kzg::{
+        srs::{UnivariateProverParam, UnivariateUniversalParams, UnivariateVerifierParam},
+        UnivariateKzgBatchProof, UnivariateKzgPCS, UnivariateKzgProof,
+    },
+    PolynomialCommitmentScheme, StructuredReferenceString,
+};

pcs/src/structs.rs

@@ -0,0 +1,25 @@
+// Copyright (c) 2022 Espresso Systems (espressosys.com)
+// This file is part of the Jellyfish library.
+
+// You should have received a copy of the MIT License
+// along with the Jellyfish library. If not, see <https://mit-license.org/>.
+
+use ark_ec::PairingEngine;
+use ark_serialize::{CanonicalDeserialize, CanonicalSerialize, Read, SerializationError, Write};
+use derivative::Derivative;
+
+#[derive(Derivative, CanonicalSerialize, CanonicalDeserialize)]
+#[derivative(
+    Default(bound = ""),
+    Hash(bound = ""),
+    Clone(bound = ""),
+    Copy(bound = ""),
+    Debug(bound = ""),
+    PartialEq(bound = ""),
+    Eq(bound = "")
+)]
+/// A commitment is an Affine point.
+pub struct Commitment<E: PairingEngine>(
+    /// the actual commitment is an affine point.
+    pub E::G1Affine,
+);

pcs/src/univariate_kzg/mod.rs

@@ -0,0 +1,438 @@
+// Copyright (c) 2022 Espresso Systems (espressosys.com)
+// This file is part of the Jellyfish library.
+
+// You should have received a copy of the MIT License
+// along with the Jellyfish library. If not, see <https://mit-license.org/>.
+
+//! Main module for univariate KZG commitment scheme
+
+use crate::{prelude::Commitment, PCSError, PolynomialCommitmentScheme, StructuredReferenceString};
+use ark_ec::{msm::VariableBaseMSM, AffineCurve, PairingEngine, ProjectiveCurve};
+use ark_ff::PrimeField;
+use ark_poly::{univariate::DensePolynomial, Polynomial, UVPolynomial};
+use ark_serialize::{CanonicalDeserialize, CanonicalSerialize, Read, SerializationError, Write};
+use ark_std::{
+    borrow::Borrow,
+    end_timer, format,
+    marker::PhantomData,
+    rand::{CryptoRng, RngCore},
+    start_timer,
+    string::ToString,
+    vec,
+    vec::Vec,
+    One, UniformRand, Zero,
+};
+#[cfg(feature = "parallel")]
+use rayon::prelude::*;
+use srs::{UnivariateProverParam, UnivariateUniversalParams, UnivariateVerifierParam};
+use util::parallelizable_slice_iter;
+
+pub(crate) mod srs;
+
+/// KZG Polynomial Commitment Scheme on univariate polynomial.
+pub struct UnivariateKzgPCS<E: PairingEngine> {
+    #[doc(hidden)]
+    phantom: PhantomData<E>,
+}
+
+#[derive(CanonicalSerialize, CanonicalDeserialize, Clone, Debug, PartialEq, Eq)]
+/// proof of opening
+pub struct UnivariateKzgProof<E: PairingEngine> {
+    /// Evaluation of quotients
+    pub proof: E::G1Affine,
+}
+/// batch proof
+pub type UnivariateKzgBatchProof<E> = Vec<UnivariateKzgProof<E>>;
+
+impl<E: PairingEngine> PolynomialCommitmentScheme<E> for UnivariateKzgPCS<E> {
+    // Parameters
+    type ProverParam = UnivariateProverParam<E::G1Affine>;
+    type VerifierParam = UnivariateVerifierParam<E>;
+    type SRS = UnivariateUniversalParams<E>;
+    // Polynomial and its associated types
+    type Polynomial = DensePolynomial<E::Fr>;
+    type Point = E::Fr;
+    type Evaluation = E::Fr;
+    // Polynomial and its associated types
+    type Commitment = Commitment<E>;
+    type BatchCommitment = Vec<Self::Commitment>;
+    type Proof = UnivariateKzgProof<E>;
+    type BatchProof = UnivariateKzgBatchProof<E>;
+
+    /// Build SRS for testing.
+    ///
+    /// - For univariate polynomials, `supported_size` is the maximum degree.
+    ///
+    /// WARNING: THIS FUNCTION IS FOR TESTING PURPOSE ONLY.
+    /// THE OUTPUT SRS SHOULD NOT BE USED IN PRODUCTION.
+    fn gen_srs_for_testing<R: RngCore + CryptoRng>(
+        rng: &mut R,
+        supported_size: usize,
+    ) -> Result<Self::SRS, PCSError> {
+        Self::SRS::gen_srs_for_testing(rng, supported_size)
+    }
+
+    /// Trim the universal parameters to specialize the public parameters.
+    /// Input `max_degree` for univariate.
+    /// `supported_num_vars` must be None or an error is returned.
+    fn trim(
+        srs: impl Borrow<Self::SRS>,
+        supported_degree: usize,
+        supported_num_vars: Option<usize>,
+    ) -> Result<(Self::ProverParam, Self::VerifierParam), PCSError> {
+        if supported_num_vars.is_some() {
+            return Err(PCSError::InvalidParameters(
+                "univariate should not receive a num_var param".to_string(),
+            ));
+        }
+        srs.borrow().trim(supported_degree)
+    }
+
+    /// Generate a commitment for a polynomial
+    /// Note that the scheme is not hidding
+    fn commit(
+        prover_param: impl Borrow<Self::ProverParam>,
+        poly: &Self::Polynomial,
+    ) -> Result<Self::Commitment, PCSError> {
+        let prover_param = prover_param.borrow();
+        let commit_time =
+            start_timer!(|| format!("Committing to polynomial of degree {} ", poly.degree()));
+
+        if poly.degree() >= prover_param.powers_of_g.len() {
+            return Err(PCSError::InvalidParameters(format!(
+                "uni poly degree {} is larger than allowed {}",
+                poly.degree(),
+                prover_param.powers_of_g.len()
+            )));
+        }
+
+        let (num_leading_zeros, plain_coeffs) = skip_leading_zeros_and_convert_to_bigints(poly);
+
+        let msm_time = start_timer!(|| "MSM to compute commitment to plaintext poly");
+        let commitment = VariableBaseMSM::multi_scalar_mul(
+            &prover_param.powers_of_g[num_leading_zeros..],
+            &plain_coeffs,
+        )
+        .into_affine();
+        end_timer!(msm_time);
+
+        end_timer!(commit_time);
+        Ok(Commitment(commitment))
+    }
+
+    /// Generate a commitment for a list of polynomials
+    fn multi_commit(
+        prover_param: impl Borrow<Self::ProverParam>,
+        polys: &[Self::Polynomial],
+    ) -> Result<Self::BatchCommitment, PCSError> {
+        let prover_param = prover_param.borrow();
+        let commit_time = start_timer!(|| format!("batch commit {} polynomials", polys.len()));
+        let res = parallelizable_slice_iter(polys)
+            .map(|poly| Self::commit(prover_param, poly))
+            .collect::<Result<Vec<Self::Commitment>, PCSError>>()?;
+
+        end_timer!(commit_time);
+        Ok(res)
+    }
+
+    /// On input a polynomial `p` and a point `point`, outputs a proof for the
+    /// same.
+    fn open(
+        prover_param: impl Borrow<Self::ProverParam>,
+        polynomial: &Self::Polynomial,
+        point: &Self::Point,
+    ) -> Result<(Self::Proof, Self::Evaluation), PCSError> {
+        let open_time =
+            start_timer!(|| format!("Opening polynomial of degree {}", polynomial.degree()));
+        let divisor = Self::Polynomial::from_coefficients_vec(vec![-*point, E::Fr::one()]);
+
+        let witness_time = start_timer!(|| "Computing witness polynomial");
+        let witness_polynomial = polynomial / &divisor;
+        end_timer!(witness_time);
+
+        let (num_leading_zeros, witness_coeffs) =
+            skip_leading_zeros_and_convert_to_bigints(&witness_polynomial);
+
+        let proof = VariableBaseMSM::multi_scalar_mul(
+            &prover_param.borrow().powers_of_g[num_leading_zeros..],
+            &witness_coeffs,
+        )
+        .into_affine();
+
+        let eval = polynomial.evaluate(point);
+
+        end_timer!(open_time);
+        Ok((Self::Proof { proof }, eval))
+    }
+
+    /// Input a list of polynomials, and a same number of points,
+    /// compute a multi-opening for all the polynomials.
+    // This is a naive approach
+    // TODO: to implement the more efficient batch opening algorithm
+    // (e.g., the appendix C.4 in https://eprint.iacr.org/2020/1536.pdf)
+    fn multi_open(
+        prover_param: impl Borrow<Self::ProverParam>,
+        _multi_commitment: &Self::BatchCommitment,
+        polynomials: &[Self::Polynomial],
+        points: &[Self::Point],
+    ) -> Result<(Self::BatchProof, Vec<Self::Evaluation>), PCSError> {
+        let open_time = start_timer!(|| format!("batch opening {} polynomials", polynomials.len()));
+        if polynomials.len() != points.len() {
+            return Err(PCSError::InvalidParameters(format!(
+                "poly length {} is different from points length {}",
+                polynomials.len(),
+                points.len()
+            )));
+        }
+        let mut batch_proof = vec![];
+        let mut evals = vec![];
+        for (poly, point) in polynomials.iter().zip(points.iter()) {
+            let (proof, eval) = Self::open(prover_param.borrow(), poly, point)?;
+            batch_proof.push(proof);
+            evals.push(eval);
+        }
+
+        end_timer!(open_time);
+        Ok((batch_proof, evals))
+    }
+
+    /// Input a multilinear extension, and a number of points, and
+    /// a transcript, compute a multi-opening for all the polynomials.
+    fn multi_open_single_poly(
+        _prover_param: impl Borrow<Self::ProverParam>,
+        _commitment: &Self::Commitment,
+        _polynomials: &Self::Polynomial,
+        _points: &[Self::Point],
+    ) -> Result<(Self::BatchProof, Vec<Self::Evaluation>), PCSError> {
+        unimplemented!()
+    }
+
+    /// Verifies that `value` is the evaluation at `x` of the polynomial
+    /// committed inside `comm`.
+    fn verify(
+        verifier_param: &Self::VerifierParam,
+        commitment: &Self::Commitment,
+        point: &Self::Point,
+        value: &E::Fr,
+        proof: &Self::Proof,
+    ) -> Result<bool, PCSError> {
+        let check_time = start_timer!(|| "Checking evaluation");
+        let pairing_inputs: Vec<(E::G1Prepared, E::G2Prepared)> = vec![
+            (
+                (verifier_param.g.mul(value.into_repr())
+                    - proof.proof.mul(point.into_repr())
+                    - commitment.0.into_projective())
+                .into_affine()
+                .into(),
+                verifier_param.h.into(),
+            ),
+            (proof.proof.into(), verifier_param.beta_h.into()),
+        ];
+
+        let res = E::product_of_pairings(pairing_inputs.iter()).is_one();
+
+        end_timer!(check_time, || format!("Result: {}", res));
+        Ok(res)
+    }
+
+    /// Verifies that `value_i` is the evaluation at `x_i` of the polynomial
+    /// `poly_i` committed inside `comm`.
+    // This is a naive approach
+    // TODO: to implement the more efficient batch verification algorithm
+    // (e.g., the appendix C.4 in https://eprint.iacr.org/2020/1536.pdf)
+    fn batch_verify<R: RngCore + CryptoRng>(
+        verifier_param: &Self::VerifierParam,
+        multi_commitment: &Self::BatchCommitment,
+        points: &[Self::Point],
+        values: &[E::Fr],
+        batch_proof: &Self::BatchProof,
+        rng: &mut R,
+    ) -> Result<bool, PCSError> {
+        let check_time =
+            start_timer!(|| format!("Checking {} evaluation proofs", multi_commitment.len()));
+
+        let mut total_c = <E::G1Projective>::zero();
+        let mut total_w = <E::G1Projective>::zero();
+
+        let combination_time = start_timer!(|| "Combining commitments and proofs");
+        let mut randomizer = E::Fr::one();
+        // Instead of multiplying g and gamma_g in each turn, we simply accumulate
+        // their coefficients and perform a final multiplication at the end.
+        let mut g_multiplier = E::Fr::zero();
+        for (((c, z), v), proof) in multi_commitment
+            .iter()
+            .zip(points)
+            .zip(values)
+            .zip(batch_proof)
+        {
+            let w = proof.proof;
+            let mut temp = w.mul(*z);
+            temp.add_assign_mixed(&c.0);
+            let c = temp;
+            g_multiplier += &(randomizer * v);
+            total_c += &c.mul(randomizer.into_repr());
+            total_w += &w.mul(randomizer.into_repr());
+            // We don't need to sample randomizers from the full field,
+            // only from 128-bit strings.
+            randomizer = u128::rand(rng).into();
+        }
+        total_c -= &verifier_param.g.mul(g_multiplier);
+        end_timer!(combination_time);
+
+        let to_affine_time = start_timer!(|| "Converting results to affine for pairing");
+        let affine_points = E::G1Projective::batch_normalization_into_affine(&[-total_w, total_c]);
+        let (total_w, total_c) = (affine_points[0], affine_points[1]);
+        end_timer!(to_affine_time);
+
+        let pairing_time = start_timer!(|| "Performing product of pairings");
+        let result = E::product_of_pairings(&[
+            (total_w.into(), verifier_param.beta_h.into()),
+            (total_c.into(), verifier_param.h.into()),
+        ])
+        .is_one();
+        end_timer!(pairing_time);
+        end_timer!(check_time, || format!("Result: {}", result));
+        Ok(result)
+    }
+
+    /// Verifies that `value_i` is the evaluation at `x_i` of the polynomial
+    /// `poly` committed inside `comm`.
+    fn batch_verify_single_poly(
+        _verifier_param: &Self::VerifierParam,
+        _commitment: &Self::Commitment,
+        _points: &[Self::Point],
+        _values: &[E::Fr],
+        _batch_proof: &Self::BatchProof,
+    ) -> Result<bool, PCSError> {
+        unimplemented!()
+    }
+}
+
+fn skip_leading_zeros_and_convert_to_bigints<F: PrimeField, P: UVPolynomial<F>>(
+    p: &P,
+) -> (usize, Vec<F::BigInt>) {
+    let mut num_leading_zeros = 0;
+    while num_leading_zeros < p.coeffs().len() && p.coeffs()[num_leading_zeros].is_zero() {
+        num_leading_zeros += 1;
+    }
+    let coeffs = convert_to_bigints(&p.coeffs()[num_leading_zeros..]);
+    (num_leading_zeros, coeffs)
+}
+
+fn convert_to_bigints<F: PrimeField>(p: &[F]) -> Vec<F::BigInt> {
+    let to_bigint_time = start_timer!(|| "Converting polynomial coeffs to bigints");
+    let coeffs = p.iter().map(|s| s.into_repr()).collect::<Vec<_>>();
+    end_timer!(to_bigint_time);
+    coeffs
+}
+
+#[cfg(test)]
+mod tests {
+    use super::*;
+    use crate::StructuredReferenceString;
+    use ark_bls12_381::Bls12_381;
+    use ark_ec::PairingEngine;
+    use ark_poly::univariate::DensePolynomial;
+    use ark_std::{test_rng, UniformRand};
+
+    fn end_to_end_test_template<E>() -> Result<(), PCSError>
+    where
+        E: PairingEngine,
+    {
+        let rng = &mut test_rng();
+        for _ in 0..100 {
+            let mut degree = 0;
+            while degree <= 1 {
+                degree = usize::rand(rng) % 20;
+            }
+            let pp = UnivariateKzgPCS::<E>::gen_srs_for_testing(rng, degree)?;
+            let (ck, vk) = pp.trim(degree)?;
+            let p = <DensePolynomial<E::Fr> as UVPolynomial<E::Fr>>::rand(degree, rng);
+            let comm = UnivariateKzgPCS::<E>::commit(&ck, &p)?;
+            let point = E::Fr::rand(rng);
+            let (proof, value) = UnivariateKzgPCS::<E>::open(&ck, &p, &point)?;
+            assert!(
+                UnivariateKzgPCS::<E>::verify(&vk, &comm, &point, &value, &proof)?,
+                "proof was incorrect for max_degree = {}, polynomial_degree = {}",
+                degree,
+                p.degree(),
+            );
+        }
+        Ok(())
+    }
+
+    fn linear_polynomial_test_template<E>() -> Result<(), PCSError>
+    where
+        E: PairingEngine,
+    {
+        let rng = &mut test_rng();
+        for _ in 0..100 {
+            let degree = 50;
+
+            let pp = UnivariateKzgPCS::<E>::gen_srs_for_testing(rng, degree)?;
+            let (ck, vk) = pp.trim(degree)?;
+            let p = <DensePolynomial<E::Fr> as UVPolynomial<E::Fr>>::rand(degree, rng);
+            let comm = UnivariateKzgPCS::<E>::commit(&ck, &p)?;
+            let point = E::Fr::rand(rng);
+            let (proof, value) = UnivariateKzgPCS::<E>::open(&ck, &p, &point)?;
+            assert!(
+                UnivariateKzgPCS::<E>::verify(&vk, &comm, &point, &value, &proof)?,
+                "proof was incorrect for max_degree = {}, polynomial_degree = {}",
+                degree,
+                p.degree(),
+            );
+        }
+        Ok(())
+    }
+
+    fn batch_check_test_template<E>() -> Result<(), PCSError>
+    where
+        E: PairingEngine,
+    {
+        let rng = &mut test_rng();
+        for _ in 0..10 {
+            let mut degree = 0;
+            while degree <= 1 {
+                degree = usize::rand(rng) % 20;
+            }
+            let pp = UnivariateKzgPCS::<E>::gen_srs_for_testing(rng, degree)?;
+            let (ck, vk) = UnivariateKzgPCS::<E>::trim(&pp, degree, None)?;
+            let mut comms = Vec::new();
+            let mut values = Vec::new();
+            let mut points = Vec::new();
+            let mut proofs = Vec::new();
+            for _ in 0..10 {
+                let p = <DensePolynomial<E::Fr> as UVPolynomial<E::Fr>>::rand(degree, rng);
+                let comm = UnivariateKzgPCS::<E>::commit(&ck, &p)?;
+                let point = E::Fr::rand(rng);
+                let (proof, value) = UnivariateKzgPCS::<E>::open(&ck, &p, &point)?;
+
+                assert!(UnivariateKzgPCS::<E>::verify(
+                    &vk, &comm, &point, &value, &proof
+                )?);
+                comms.push(comm);
+                values.push(value);
+                points.push(point);
+                proofs.push(proof);
+            }
+            assert!(UnivariateKzgPCS::<E>::batch_verify(
+                &vk, &comms, &points, &values, &proofs, rng
+            )?);
+        }
+        Ok(())
+    }
+
+    #[test]
+    fn end_to_end_test() {
+        end_to_end_test_template::<Bls12_381>().expect("test failed for bls12-381");
+    }
+
+    #[test]
+    fn linear_polynomial_test() {
+        linear_polynomial_test_template::<Bls12_381>().expect("test failed for bls12-381");
+    }
+    #[test]
+    fn batch_check_test() {
+        batch_check_test_template::<Bls12_381>().expect("test failed for bls12-381");
+    }
+}

pcs/src/univariate_kzg/srs.rs

@@ -0,0 +1,156 @@
+// Copyright (c) 2022 Espresso Systems (espressosys.com)
+// This file is part of the Jellyfish library.
+
+// You should have received a copy of the MIT License
+// along with the Jellyfish library. If not, see <https://mit-license.org/>.
+
+//! Implementing Structured Reference Strings for univariate polynomial KZG
+
+use crate::{PCSError, StructuredReferenceString};
+use ark_ec::{msm::FixedBaseMSM, AffineCurve, PairingEngine, ProjectiveCurve};
+use ark_ff::PrimeField;
+use ark_serialize::{CanonicalDeserialize, CanonicalSerialize, Read, SerializationError, Write};
+use ark_std::{
+    end_timer,
+    rand::{CryptoRng, RngCore},
+    start_timer, vec,
+    vec::Vec,
+    One, UniformRand,
+};
+use derivative::Derivative;
+
+/// `UniversalParams` are the universal parameters for the KZG10 scheme.
+// Adapted from
+// https://github.com/arkworks-rs/poly-commit/blob/master/src/kzg10/data_structures.rs#L20
+#[derive(Debug, Clone, Eq, PartialEq, CanonicalSerialize, CanonicalDeserialize, Default)]
+pub struct UnivariateUniversalParams<E: PairingEngine> {
+    /// Group elements of the form `{ \beta^i G }`, where `i` ranges from 0 to
+    /// `degree`.
+    pub powers_of_g: Vec<E::G1Affine>,
+    /// The generator of G2.
+    pub h: E::G2Affine,
+    /// \beta times the above generator of G2.
+    pub beta_h: E::G2Affine,
+}
+
+impl<E: PairingEngine> UnivariateUniversalParams<E> {
+    /// Returns the maximum supported degree
+    pub fn max_degree(&self) -> usize {
+        self.powers_of_g.len()
+    }
+}
+
+/// `UnivariateProverParam` is used to generate a proof
+#[derive(CanonicalSerialize, CanonicalDeserialize, Clone, Debug, Eq, PartialEq, Default)]
+pub struct UnivariateProverParam<C: AffineCurve> {
+    /// Parameters
+    pub powers_of_g: Vec<C>,
+}
+
+/// `UnivariateVerifierParam` is used to check evaluation proofs for a given
+/// commitment.
+#[derive(Derivative, CanonicalSerialize, CanonicalDeserialize)]
+#[derivative(
+    Default(bound = ""),
+    Clone(bound = ""),
+    Copy(bound = ""),
+    Debug(bound = ""),
+    PartialEq(bound = ""),
+    Eq(bound = "")
+)]
+pub struct UnivariateVerifierParam<E: PairingEngine> {
+    /// The generator of G1.
+    pub g: E::G1Affine,
+    /// The generator of G2.
+    pub h: E::G2Affine,
+    /// \beta times the above generator of G2.
+    pub beta_h: E::G2Affine,
+}
+
+impl<E: PairingEngine> StructuredReferenceString<E> for UnivariateUniversalParams<E> {
+    type ProverParam = UnivariateProverParam<E::G1Affine>;
+    type VerifierParam = UnivariateVerifierParam<E>;
+
+    /// Extract the prover parameters from the public parameters.
+    fn extract_prover_param(&self, supported_size: usize) -> Self::ProverParam {
+        let powers_of_g = self.powers_of_g[..=supported_size].to_vec();
+
+        Self::ProverParam { powers_of_g }
+    }
+
+    /// Extract the verifier parameters from the public parameters.
+    fn extract_verifier_param(&self, _supported_size: usize) -> Self::VerifierParam {
+        Self::VerifierParam {
+            g: self.powers_of_g[0],
+            h: self.h,
+            beta_h: self.beta_h,
+        }
+    }
+
+    /// Trim the universal parameters to specialize the public parameters
+    /// for univariate polynomials to the given `supported_size`, and
+    /// returns committer key and verifier key. `supported_size` should
+    /// be in range `1..params.len()`
+    fn trim(
+        &self,
+        supported_size: usize,
+    ) -> Result<(Self::ProverParam, Self::VerifierParam), PCSError> {
+        let powers_of_g = self.powers_of_g[..=supported_size].to_vec();
+
+        let pk = Self::ProverParam { powers_of_g };
+        let vk = Self::VerifierParam {
+            g: self.powers_of_g[0],
+            h: self.h,
+            beta_h: self.beta_h,
+        };
+        Ok((pk, vk))
+    }
+
+    /// Build SRS for testing.
+    /// WARNING: THIS FUNCTION IS FOR TESTING PURPOSE ONLY.
+    /// THE OUTPUT SRS SHOULD NOT BE USED IN PRODUCTION.
+    fn gen_srs_for_testing<R: RngCore + CryptoRng>(
+        rng: &mut R,
+        max_degree: usize,
+    ) -> Result<Self, PCSError> {
+        let setup_time = start_timer!(|| format!("KZG10::Setup with degree {}", max_degree));
+        let beta = E::Fr::rand(rng);
+        let g = E::G1Projective::rand(rng);
+        let h = E::G2Projective::rand(rng);
+
+        let mut powers_of_beta = vec![E::Fr::one()];
+
+        let mut cur = beta;
+        for _ in 0..max_degree {
+            powers_of_beta.push(cur);
+            cur *= &beta;
+        }
+
+        let window_size = FixedBaseMSM::get_mul_window_size(max_degree + 1);
+
+        let scalar_bits = E::Fr::size_in_bits();
+        let g_time = start_timer!(|| "Generating powers of G");
+        // TODO: parallelization
+        let g_table = FixedBaseMSM::get_window_table(scalar_bits, window_size, g);
+        let powers_of_g = FixedBaseMSM::multi_scalar_mul::<E::G1Projective>(
+            scalar_bits,
+            window_size,
+            &g_table,
+            &powers_of_beta,
+        );
+        end_timer!(g_time);
+
+        let powers_of_g = E::G1Projective::batch_normalization_into_affine(&powers_of_g);
+
+        let h = h.into_affine();
+        let beta_h = h.mul(beta).into_affine();
+
+        let pp = Self {
+            powers_of_g,
+            h,
+            beta_h,
+        };
+        end_timer!(setup_time);
+        Ok(pp)
+    }
+}