import spartan-secq in

packages/Spartan-secq/.cargo/config

@@ -0,0 +1,4 @@
+[build]
+rustflags = [
+  "-C", "target-cpu=native",
+]

packages/Spartan-secq/.github/workflows/rust.yml

@@ -0,0 +1,30 @@
+name: Build and Test Spartan
+
+on:
+  push:
+    branches: [ master ]
+  pull_request:
+    branches: [ master ]
+
+jobs:
+  build_nightly:
+    runs-on: ubuntu-latest
+    steps:
+    - uses: actions/checkout@v2
+    - name: Install
+      run: rustup default nightly
+    - name: Install rustfmt Components
+      run: rustup component add rustfmt
+    - name: Install clippy
+      run: rustup component add clippy
+    - name: Build
+      run: cargo build --verbose
+    - name: Run tests
+      run: cargo test --verbose
+    - name: Build examples
+      run: cargo build --examples --verbose
+    - name: Check Rustfmt Code Style
+      run: cargo fmt --all -- --check
+    - name: Check clippy warnings
+      run: cargo clippy --all-targets --all-features -- -D warnings
+

packages/Spartan-secq/.gitignore

@@ -0,0 +1,12 @@
+# Generated by Cargo
+# will have compiled files and executables
+/target/
+
+# Remove Cargo.lock from gitignore if creating an executable, leave it for libraries
+# More information here https://doc.rust-lang.org/cargo/guide/cargo-toml-vs-cargo-lock.html
+Cargo.lock
+
+# These are backup files generated by rustfmt
+**/*.rs.bk
+
+*.txt

packages/Spartan-secq/.vscode/settings.json

@@ -0,0 +1,3 @@
+{
+    "editor.formatOnSave": true
+}

packages/Spartan-secq/CODE_OF_CONDUCT.md

@@ -0,0 +1,9 @@
+# Microsoft Open Source Code of Conduct
+
+This project has adopted the [Microsoft Open Source Code of Conduct](https://opensource.microsoft.com/codeofconduct/).
+
+Resources:
+
+- [Microsoft Open Source Code of Conduct](https://opensource.microsoft.com/codeofconduct/)
+- [Microsoft Code of Conduct FAQ](https://opensource.microsoft.com/codeofconduct/faq/)
+- Contact [opencode@microsoft.com](mailto:opencode@microsoft.com) with questions or concerns

packages/Spartan-secq/CONTRIBUTING.md

@@ -0,0 +1,12 @@
+This project welcomes contributions and suggestions. Most contributions require you to
+agree to a Contributor License Agreement (CLA) declaring that you have the right to,
+and actually do, grant us the rights to use your contribution. For details, visit
+https://cla.microsoft.com.
+
+When you submit a pull request, a CLA-bot will automatically determine whether you need
+to provide a CLA and decorate the PR appropriately (e.g., label, comment). Simply follow the
+instructions provided by the bot. You will only need to do this once across all repositories using our CLA.
+
+This project has adopted the [Microsoft Open Source Code of Conduct](https://opensource.microsoft.com/codeofconduct/).
+For more information see the [Code of Conduct FAQ](https://opensource.microsoft.com/codeofconduct/faq/)
+or contact [opencode@microsoft.com](mailto:opencode@microsoft.com) with any additional questions or comments.

packages/Spartan-secq/Cargo.toml

@@ -0,0 +1,57 @@
+[package]
+name = "spartan"
+version = "0.7.1"
+authors = ["Srinath Setty <srinath@microsoft.com>"]
+edition = "2021"
+description = "High-speed zkSNARKs without trusted setup"
+documentation = "https://docs.rs/spartan/"
+readme = "README.md"
+repository = "https://github.com/microsoft/Spartan"
+license-file = "LICENSE"
+keywords = ["zkSNARKs", "cryptography", "proofs"]
+
+[dependencies]
+num-bigint-dig = "^0.7"
+secq256k1 = { git = "https://github.com/personaelabs/spartan-ecdsa.git" }
+merlin = "3.0.0"
+rand = "0.7.3"
+digest = "0.8.1"
+sha3 = "0.8.2"
+byteorder = "1.3.4"
+rayon = { version = "1.3.0", optional = true }
+serde = { version = "1.0.106", features = ["derive"] }
+bincode = "1.2.1"
+subtle = { version = "2.4", default-features = false }
+rand_core = { version = "0.6", default-features = false }
+zeroize = { version = "1", default-features = false }
+itertools = "0.10.0"
+colored = "2.0.0"
+flate2 = "1.0.14"
+thiserror = "1.0"
+num-traits = "0.2.15"
+hex-literal = { version = "0.3" }
+multiexp = "0.2.2"
+
+[dev-dependencies]
+criterion = "0.3.1"
+
+[lib]
+name = "libspartan"
+path = "src/lib.rs"
+crate-type = ["cdylib", "rlib"]
+
+[[bin]]
+name = "snark"
+path = "profiler/snark.rs"
+
+[[bin]]
+name = "nizk"
+path = "profiler/nizk.rs"
+
+[[bench]]
+name = "snark"
+harness = false
+
+[[bench]]
+name = "nizk"
+harness = false

packages/Spartan-secq/LICENSE

@@ -0,0 +1,21 @@
+    MIT License
+
+    Copyright (c) Microsoft Corporation.
+
+    Permission is hereby granted, free of charge, to any person obtaining a copy
+    of this software and associated documentation files (the "Software"), to deal
+    in the Software without restriction, including without limitation the rights
+    to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
+    copies of the Software, and to permit persons to whom the Software is
+    furnished to do so, subject to the following conditions:
+
+    The above copyright notice and this permission notice shall be included in all
+    copies or substantial portions of the Software.
+
+    THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
+    IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
+    FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
+    AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
+    LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
+    OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
+    SOFTWARE

packages/Spartan-secq/README.md

@@ -0,0 +1,10 @@
+## Fork of [Spartan](https://github.com/microsoft/Spartan)
+_This fork is still under development._
+
+Modify Spartan to operate over the **base field** of secp256k1.
+
+### Changes from the original Spartan
+- Use the secq256k1 crate instead of curve25519-dalek
+- Modify values in scalar.rs (originally ristretto255.rs) 
+
+Please refer to [spartan-ecdsa](https://github.com/personaelabs/spartan-ecdsa) for development status.

packages/Spartan-secq/SECURITY.md

@@ -0,0 +1,41 @@
+<!-- BEGIN MICROSOFT SECURITY.MD V0.0.3 BLOCK -->
+
+## Security
+
+Microsoft takes the security of our software products and services seriously, which includes all source code repositories managed through our GitHub organizations, which include [Microsoft](https://github.com/Microsoft), [Azure](https://github.com/Azure), [DotNet](https://github.com/dotnet), [AspNet](https://github.com/aspnet), [Xamarin](https://github.com/xamarin), and [our GitHub organizations](https://opensource.microsoft.com/).
+
+If you believe you have found a security vulnerability in any Microsoft-owned repository that meets Microsoft's [Microsoft's definition of a security vulnerability](https://docs.microsoft.com/en-us/previous-versions/tn-archive/cc751383(v=technet.10)) of a security vulnerability, please report it to us as described below.
+
+## Reporting Security Issues
+
+**Please do not report security vulnerabilities through public GitHub issues.**
+
+Instead, please report them to the Microsoft Security Response Center (MSRC) at [https://msrc.microsoft.com/create-report](https://msrc.microsoft.com/create-report).
+
+If you prefer to submit without logging in, send email to [secure@microsoft.com](mailto:secure@microsoft.com).  If possible, encrypt your message with our PGP key; please download it from the the [Microsoft Security Response Center PGP Key page](https://www.microsoft.com/en-us/msrc/pgp-key-msrc).
+
+You should receive a response within 24 hours. If for some reason you do not, please follow up via email to ensure we received your original message. Additional information can be found at [microsoft.com/msrc](https://www.microsoft.com/msrc).
+
+Please include the requested information listed below (as much as you can provide) to help us better understand the nature and scope of the possible issue:
+
+  * Type of issue (e.g. buffer overflow, SQL injection, cross-site scripting, etc.)
+  * Full paths of source file(s) related to the manifestation of the issue
+  * The location of the affected source code (tag/branch/commit or direct URL)
+  * Any special configuration required to reproduce the issue
+  * Step-by-step instructions to reproduce the issue
+  * Proof-of-concept or exploit code (if possible)
+  * Impact of the issue, including how an attacker might exploit the issue
+
+This information will help us triage your report more quickly.
+
+If you are reporting for a bug bounty, more complete reports can contribute to a higher bounty award. Please visit our [Microsoft Bug Bounty Program](https://microsoft.com/msrc/bounty) page for more details about our active programs.
+
+## Preferred Languages
+
+We prefer all communications to be in English.
+
+## Policy
+
+Microsoft follows the principle of [Coordinated Vulnerability Disclosure](https://www.microsoft.com/en-us/msrc/cvd).
+
+<!-- END MICROSOFT SECURITY.MD BLOCK -->

packages/Spartan-secq/benches/nizk.rs

@@ -0,0 +1,92 @@
+#![allow(clippy::assertions_on_result_states)]
+extern crate byteorder;
+extern crate core;
+extern crate criterion;
+extern crate digest;
+extern crate libspartan;
+extern crate merlin;
+extern crate rand;
+extern crate sha3;
+
+use libspartan::{Instance, NIZKGens, NIZK};
+use merlin::Transcript;
+
+use criterion::*;
+
+fn nizk_prove_benchmark(c: &mut Criterion) {
+  for &s in [10, 12, 16].iter() {
+    let plot_config = PlotConfiguration::default().summary_scale(AxisScale::Logarithmic);
+    let mut group = c.benchmark_group("NIZK_prove_benchmark");
+    group.plot_config(plot_config);
+
+    let num_vars = (2_usize).pow(s as u32);
+    let num_cons = num_vars;
+    let num_inputs = 10;
+
+    let (inst, vars, inputs) = Instance::produce_synthetic_r1cs(num_cons, num_vars, num_inputs);
+
+    let gens = NIZKGens::new(num_cons, num_vars, num_inputs);
+
+    let name = format!("NIZK_prove_{}", num_vars);
+    group.bench_function(&name, move |b| {
+      b.iter(|| {
+        let mut prover_transcript = Transcript::new(b"example");
+        NIZK::prove(
+          black_box(&inst),
+          black_box(vars.clone()),
+          black_box(&inputs),
+          black_box(&gens),
+          black_box(&mut prover_transcript),
+        );
+      });
+    });
+    group.finish();
+  }
+}
+
+fn nizk_verify_benchmark(c: &mut Criterion) {
+  for &s in [10, 12, 16].iter() {
+    let plot_config = PlotConfiguration::default().summary_scale(AxisScale::Logarithmic);
+    let mut group = c.benchmark_group("NIZK_verify_benchmark");
+    group.plot_config(plot_config);
+
+    let num_vars = (2_usize).pow(s as u32);
+    let num_cons = num_vars;
+    let num_inputs = 10;
+    let (inst, vars, inputs) = Instance::produce_synthetic_r1cs(num_cons, num_vars, num_inputs);
+
+    let gens = NIZKGens::new(num_cons, num_vars, num_inputs);
+
+    // produce a proof of satisfiability
+    let mut prover_transcript = Transcript::new(b"example");
+    let proof = NIZK::prove(&inst, vars, &inputs, &gens, &mut prover_transcript);
+
+    let name = format!("NIZK_verify_{}", num_cons);
+    group.bench_function(&name, move |b| {
+      b.iter(|| {
+        let mut verifier_transcript = Transcript::new(b"example");
+        assert!(proof
+          .verify(
+            black_box(&inst),
+            black_box(&inputs),
+            black_box(&mut verifier_transcript),
+            black_box(&gens)
+          )
+          .is_ok());
+      });
+    });
+    group.finish();
+  }
+}
+
+fn set_duration() -> Criterion {
+  Criterion::default().sample_size(10)
+}
+
+criterion_group! {
+name = benches_nizk;
+config = set_duration();
+targets = nizk_prove_benchmark, nizk_verify_benchmark
+}
+
+criterion_main!(benches_nizk);

packages/Spartan-secq/benches/snark.rs

@@ -0,0 +1,131 @@
+#![allow(clippy::assertions_on_result_states)]
+extern crate libspartan;
+extern crate merlin;
+
+use libspartan::{Instance, SNARKGens, SNARK};
+use merlin::Transcript;
+
+use criterion::*;
+
+fn snark_encode_benchmark(c: &mut Criterion) {
+  for &s in [10, 12, 16].iter() {
+    let plot_config = PlotConfiguration::default().summary_scale(AxisScale::Logarithmic);
+    let mut group = c.benchmark_group("SNARK_encode_benchmark");
+    group.plot_config(plot_config);
+
+    let num_vars = (2_usize).pow(s as u32);
+    let num_cons = num_vars;
+    let num_inputs = 10;
+    let (inst, _vars, _inputs) = Instance::produce_synthetic_r1cs(num_cons, num_vars, num_inputs);
+
+    // produce public parameters
+    let gens = SNARKGens::new(num_cons, num_vars, num_inputs, num_cons);
+
+    // produce a commitment to R1CS instance
+    let name = format!("SNARK_encode_{}", num_cons);
+    group.bench_function(&name, move |b| {
+      b.iter(|| {
+        SNARK::encode(black_box(&inst), black_box(&gens));
+      });
+    });
+    group.finish();
+  }
+}
+
+fn snark_prove_benchmark(c: &mut Criterion) {
+  for &s in [10, 12, 16].iter() {
+    let plot_config = PlotConfiguration::default().summary_scale(AxisScale::Logarithmic);
+    let mut group = c.benchmark_group("SNARK_prove_benchmark");
+    group.plot_config(plot_config);
+
+    let num_vars = (2_usize).pow(s as u32);
+    let num_cons = num_vars;
+    let num_inputs = 10;
+
+    let (inst, vars, inputs) = Instance::produce_synthetic_r1cs(num_cons, num_vars, num_inputs);
+
+    // produce public parameters
+    let gens = SNARKGens::new(num_cons, num_vars, num_inputs, num_cons);
+
+    // produce a commitment to R1CS instance
+    let (comm, decomm) = SNARK::encode(&inst, &gens);
+
+    // produce a proof
+    let name = format!("SNARK_prove_{}", num_cons);
+    group.bench_function(&name, move |b| {
+      b.iter(|| {
+        let mut prover_transcript = Transcript::new(b"example");
+        SNARK::prove(
+          black_box(&inst),
+          black_box(&comm),
+          black_box(&decomm),
+          black_box(vars.clone()),
+          black_box(&inputs),
+          black_box(&gens),
+          black_box(&mut prover_transcript),
+        );
+      });
+    });
+    group.finish();
+  }
+}
+
+fn snark_verify_benchmark(c: &mut Criterion) {
+  for &s in [10, 12, 16].iter() {
+    let plot_config = PlotConfiguration::default().summary_scale(AxisScale::Logarithmic);
+    let mut group = c.benchmark_group("SNARK_verify_benchmark");
+    group.plot_config(plot_config);
+
+    let num_vars = (2_usize).pow(s as u32);
+    let num_cons = num_vars;
+    let num_inputs = 10;
+    let (inst, vars, inputs) = Instance::produce_synthetic_r1cs(num_cons, num_vars, num_inputs);
+
+    // produce public parameters
+    let gens = SNARKGens::new(num_cons, num_vars, num_inputs, num_cons);
+
+    // produce a commitment to R1CS instance
+    let (comm, decomm) = SNARK::encode(&inst, &gens);
+
+    // produce a proof of satisfiability
+    let mut prover_transcript = Transcript::new(b"example");
+    let proof = SNARK::prove(
+      &inst,
+      &comm,
+      &decomm,
+      vars,
+      &inputs,
+      &gens,
+      &mut prover_transcript,
+    );
+
+    // verify the proof
+    let name = format!("SNARK_verify_{}", num_cons);
+    group.bench_function(&name, move |b| {
+      b.iter(|| {
+        let mut verifier_transcript = Transcript::new(b"example");
+        assert!(proof
+          .verify(
+            black_box(&comm),
+            black_box(&inputs),
+            black_box(&mut verifier_transcript),
+            black_box(&gens)
+          )
+          .is_ok());
+      });
+    });
+    group.finish();
+  }
+}
+
+fn set_duration() -> Criterion {
+  Criterion::default().sample_size(10)
+}
+
+criterion_group! {
+name = benches_snark;
+config = set_duration();
+targets = snark_encode_benchmark, snark_prove_benchmark, snark_verify_benchmark
+}
+
+criterion_main!(benches_snark);

packages/Spartan-secq/examples/cubic.rs

@@ -0,0 +1,147 @@
+//! Demonstrates how to produces a proof for canonical cubic equation: `x^3 + x + 5 = y`.
+//! The example is described in detail [here].
+//!
+//! The R1CS for this problem consists of the following 4 constraints:
+//! `Z0 * Z0 - Z1 = 0`
+//! `Z1 * Z0 - Z2 = 0`
+//! `(Z2 + Z0) * 1 - Z3 = 0`
+//! `(Z3 + 5) * 1 - I0 = 0`
+//!
+//! [here]: https://medium.com/@VitalikButerin/quadratic-arithmetic-programs-from-zero-to-hero-f6d558cea649
+#![allow(clippy::assertions_on_result_states)]
+use libspartan::{InputsAssignment, Instance, SNARKGens, VarsAssignment, SNARK};
+use merlin::Transcript;
+use rand_core::OsRng;
+use secq256k1::elliptic_curve::Field;
+use secq256k1::Scalar;
+
+#[allow(non_snake_case)]
+fn produce_r1cs() -> (
+  usize,
+  usize,
+  usize,
+  usize,
+  Instance,
+  VarsAssignment,
+  InputsAssignment,
+) {
+  // parameters of the R1CS instance
+  let num_cons = 4;
+  let num_vars = 4;
+  let num_inputs = 1;
+  let num_non_zero_entries = 8;
+
+  // We will encode the above constraints into three matrices, where
+  // the coefficients in the matrix are in the little-endian byte order
+  let mut A: Vec<(usize, usize, [u8; 32])> = Vec::new();
+  let mut B: Vec<(usize, usize, [u8; 32])> = Vec::new();
+  let mut C: Vec<(usize, usize, [u8; 32])> = Vec::new();
+
+  let one: [u8; 32] = Scalar::ONE.to_bytes().into();
+
+  // R1CS is a set of three sparse matrices A B C, where is a row for every
+  // constraint and a column for every entry in z = (vars, 1, inputs)
+  // An R1CS instance is satisfiable iff:
+  // Az \circ Bz = Cz, where z = (vars, 1, inputs)
+
+  // constraint 0 entries in (A,B,C)
+  // constraint 0 is Z0 * Z0 - Z1 = 0.
+  A.push((0, 0, one));
+  B.push((0, 0, one));
+  C.push((0, 1, one));
+
+  // constraint 1 entries in (A,B,C)
+  // constraint 1 is Z1 * Z0 - Z2 = 0.
+  A.push((1, 1, one));
+  B.push((1, 0, one));
+  C.push((1, 2, one));
+
+  // constraint 2 entries in (A,B,C)
+  // constraint 2 is (Z2 + Z0) * 1 - Z3 = 0.
+  A.push((2, 2, one));
+  A.push((2, 0, one));
+  B.push((2, num_vars, one));
+  C.push((2, 3, one));
+
+  // constraint 3 entries in (A,B,C)
+  // constraint 3 is (Z3 + 5) * 1 - I0 = 0.
+  A.push((3, 3, one));
+  A.push((3, num_vars, Scalar::from(5u32).to_bytes().into()));
+  B.push((3, num_vars, one));
+  C.push((3, num_vars + 1, one));
+
+  let inst = Instance::new(num_cons, num_vars, num_inputs, &A, &B, &C).unwrap();
+
+  // compute a satisfying assignment
+  let mut csprng: OsRng = OsRng;
+  let z0 = Scalar::random(&mut csprng);
+  let z1 = z0 * z0; // constraint 0
+  let z2 = z1 * z0; // constraint 1
+  let z3 = z2 + z0; // constraint 2
+  let i0 = z3 + Scalar::from(5u32); // constraint 3
+
+  // create a VarsAssignment
+  let mut vars: Vec<[u8; 32]> = vec![Scalar::ZERO.to_bytes().into(); num_vars];
+  vars[0] = z0.to_bytes().into();
+  vars[1] = z1.to_bytes().into();
+  vars[2] = z2.to_bytes().into();
+  vars[3] = z3.to_bytes().into();
+  let assignment_vars = VarsAssignment::new(&vars).unwrap();
+
+  // create an InputsAssignment
+  let mut inputs: Vec<[u8; 32]> = vec![Scalar::ZERO.to_bytes().into(); num_inputs];
+  inputs[0] = i0.to_bytes().into();
+  let assignment_inputs = InputsAssignment::new(&inputs).unwrap();
+
+  // check if the instance we created is satisfiable
+  let res = inst.is_sat(&assignment_vars, &assignment_inputs);
+  assert!(res.unwrap(), "should be satisfied");
+
+  (
+    num_cons,
+    num_vars,
+    num_inputs,
+    num_non_zero_entries,
+    inst,
+    assignment_vars,
+    assignment_inputs,
+  )
+}
+
+fn main() {
+  // produce an R1CS instance
+  let (
+    num_cons,
+    num_vars,
+    num_inputs,
+    num_non_zero_entries,
+    inst,
+    assignment_vars,
+    assignment_inputs,
+  ) = produce_r1cs();
+
+  // produce public parameters
+  let gens = SNARKGens::new(num_cons, num_vars, num_inputs, num_non_zero_entries);
+
+  // create a commitment to the R1CS instance
+  let (comm, decomm) = SNARK::encode(&inst, &gens);
+
+  // produce a proof of satisfiability
+  let mut prover_transcript = Transcript::new(b"snark_example");
+  let proof = SNARK::prove(
+    &inst,
+    &comm,
+    &decomm,
+    assignment_vars,
+    &assignment_inputs,
+    &gens,
+    &mut prover_transcript,
+  );
+
+  // verify the proof of satisfiability
+  let mut verifier_transcript = Transcript::new(b"snark_example");
+  assert!(proof
+    .verify(&comm, &assignment_inputs, &mut verifier_transcript, &gens)
+    .is_ok());
+  println!("proof verification successful!");
+}

packages/Spartan-secq/profiler/nizk.rs

@@ -0,0 +1,52 @@
+#![allow(non_snake_case)]
+#![allow(clippy::assertions_on_result_states)]
+
+extern crate flate2;
+extern crate libspartan;
+extern crate merlin;
+extern crate rand;
+
+use flate2::{write::ZlibEncoder, Compression};
+use libspartan::{Instance, NIZKGens, NIZK};
+use merlin::Transcript;
+
+fn print(msg: &str) {
+  let star = "* ";
+  println!("{:indent$}{}{}", "", star, msg, indent = 2);
+}
+
+pub fn main() {
+  // the list of number of variables (and constraints) in an R1CS instance
+  let inst_sizes = vec![10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20];
+
+  println!("Profiler:: NIZK");
+  for &s in inst_sizes.iter() {
+    let num_vars = (2_usize).pow(s as u32);
+    let num_cons = num_vars;
+    let num_inputs = 10;
+
+    // produce a synthetic R1CSInstance
+    let (inst, vars, inputs) = Instance::produce_synthetic_r1cs(num_cons, num_vars, num_inputs);
+
+    // produce public generators
+    let gens = NIZKGens::new(num_cons, num_vars, num_inputs);
+
+    // produce a proof of satisfiability
+    let mut prover_transcript = Transcript::new(b"nizk_example");
+    let proof = NIZK::prove(&inst, vars, &inputs, &gens, &mut prover_transcript);
+
+    let mut encoder = ZlibEncoder::new(Vec::new(), Compression::default());
+    bincode::serialize_into(&mut encoder, &proof).unwrap();
+    let proof_encoded = encoder.finish().unwrap();
+    let msg_proof_len = format!("NIZK::proof_compressed_len {:?}", proof_encoded.len());
+    print(&msg_proof_len);
+
+    // verify the proof of satisfiability
+    let mut verifier_transcript = Transcript::new(b"nizk_example");
+    assert!(proof
+      .verify(&inst, &inputs, &mut verifier_transcript, &gens)
+      .is_ok());
+
+    println!();
+  }
+}

packages/Spartan-secq/profiler/snark.rs

@@ -0,0 +1,62 @@
+#![allow(non_snake_case)]
+#![allow(clippy::assertions_on_result_states)]
+
+extern crate flate2;
+extern crate libspartan;
+extern crate merlin;
+
+use flate2::{write::ZlibEncoder, Compression};
+use libspartan::{Instance, SNARKGens, SNARK};
+use merlin::Transcript;
+
+fn print(msg: &str) {
+  let star = "* ";
+  println!("{:indent$}{}{}", "", star, msg, indent = 2);
+}
+
+pub fn main() {
+  // the list of number of variables (and constraints) in an R1CS instance
+  let inst_sizes = vec![10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20];
+
+  println!("Profiler:: SNARK");
+  for &s in inst_sizes.iter() {
+    let num_vars = (2_usize).pow(s as u32);
+    let num_cons = num_vars;
+    let num_inputs = 10;
+
+    // produce a synthetic R1CSInstance
+    let (inst, vars, inputs) = Instance::produce_synthetic_r1cs(num_cons, num_vars, num_inputs);
+
+    // produce public generators
+    let gens = SNARKGens::new(num_cons, num_vars, num_inputs, num_cons);
+
+    // create a commitment to R1CSInstance
+    let (comm, decomm) = SNARK::encode(&inst, &gens);
+
+    // produce a proof of satisfiability
+    let mut prover_transcript = Transcript::new(b"snark_example");
+    let proof = SNARK::prove(
+      &inst,
+      &comm,
+      &decomm,
+      vars,
+      &inputs,
+      &gens,
+      &mut prover_transcript,
+    );
+
+    let mut encoder = ZlibEncoder::new(Vec::new(), Compression::default());
+    bincode::serialize_into(&mut encoder, &proof).unwrap();
+    let proof_encoded = encoder.finish().unwrap();
+    let msg_proof_len = format!("SNARK::proof_compressed_len {:?}", proof_encoded.len());
+    print(&msg_proof_len);
+
+    // verify the proof of satisfiability
+    let mut verifier_transcript = Transcript::new(b"snark_example");
+    assert!(proof
+      .verify(&comm, &inputs, &mut verifier_transcript, &gens)
+      .is_ok());
+
+    println!();
+  }
+}

packages/Spartan-secq/rustfmt.toml

@@ -0,0 +1,4 @@
+edition = "2018"
+tab_spaces = 2
+newline_style = "Unix"
+use_try_shorthand = true

packages/Spartan-secq/src/bin/mont_params.rs

@@ -0,0 +1,54 @@
+use hex_literal::hex;
+use num_bigint_dig::{BigInt, BigUint, ModInverse, ToBigInt};
+use num_traits::{FromPrimitive, ToPrimitive};
+use std::ops::Neg;
+
+fn get_words(n: &BigUint) -> [u64; 4] {
+  let mut words = [0u64; 4];
+  for i in 0..4 {
+    let word = n.clone() >> (64 * i) & BigUint::from(0xffffffffffffffffu64);
+    words[i] = word.to_u64().unwrap();
+  }
+  words
+}
+
+fn render_hex(label: String, words: &[u64; 4]) {
+  println!("// {}", label);
+  for word in words {
+    println!("0x{:016x},", word);
+  }
+}
+
+fn main() {
+  let modulus = BigUint::from_bytes_be(&hex!(
+    "fffffffffffffffffffffffffffffffffffffffffffffffffffffffefffffc2f"
+  ));
+
+  let r = BigUint::from_u8(2)
+    .unwrap()
+    .modpow(&BigUint::from_u64(256).unwrap(), &modulus);
+
+  let r2 = BigUint::from_u8(2)
+    .unwrap()
+    .modpow(&BigUint::from_u64(512).unwrap(), &modulus);
+
+  let r3 = BigUint::from_u8(2)
+    .unwrap()
+    .modpow(&BigUint::from_u64(768).unwrap(), &modulus);
+
+  let two_pow_64 = BigUint::from_u128(18446744073709551616u128).unwrap();
+  let one = BigInt::from_u8(1).unwrap();
+
+  let inv = modulus
+    .clone()
+    .mod_inverse(&two_pow_64)
+    .unwrap()
+    .neg()
+    .modpow(&one, &two_pow_64.to_bigint().unwrap());
+
+  render_hex("Modulus".to_string(), &get_words(&modulus));
+  render_hex("R".to_string(), &get_words(&r));
+  render_hex("R2".to_string(), &get_words(&r2));
+  render_hex("R3".to_string(), &get_words(&r3));
+  render_hex("INV".to_string(), &get_words(&inv.to_biguint().unwrap()));
+}

packages/Spartan-secq/src/commitments.rs

@@ -0,0 +1,88 @@
+use super::group::{GroupElement, VartimeMultiscalarMul};
+use super::scalar::Scalar;
+use digest::{ExtendableOutput, Input};
+use secq256k1::AffinePoint;
+use sha3::Shake256;
+use std::io::Read;
+
+#[derive(Debug)]
+pub struct MultiCommitGens {
+  pub n: usize,
+  pub G: Vec<GroupElement>,
+  pub h: GroupElement,
+}
+
+impl MultiCommitGens {
+  pub fn new(n: usize, label: &[u8]) -> Self {
+    let mut shake = Shake256::default();
+    shake.input(label);
+    shake.input(AffinePoint::generator().compress().as_bytes());
+
+    let mut reader = shake.xof_result();
+    let mut gens: Vec<GroupElement> = Vec::new();
+    let mut uniform_bytes = [0u8; 128];
+    for _ in 0..n + 1 {
+      reader.read_exact(&mut uniform_bytes).unwrap();
+      gens.push(AffinePoint::from_uniform_bytes(&uniform_bytes));
+    }
+
+    MultiCommitGens {
+      n,
+      G: gens[..n].to_vec(),
+      h: gens[n],
+    }
+  }
+
+  pub fn clone(&self) -> MultiCommitGens {
+    MultiCommitGens {
+      n: self.n,
+      h: self.h,
+      G: self.G.clone(),
+    }
+  }
+
+  pub fn split_at(&self, mid: usize) -> (MultiCommitGens, MultiCommitGens) {
+    let (G1, G2) = self.G.split_at(mid);
+
+    (
+      MultiCommitGens {
+        n: G1.len(),
+        G: G1.to_vec(),
+        h: self.h,
+      },
+      MultiCommitGens {
+        n: G2.len(),
+        G: G2.to_vec(),
+        h: self.h,
+      },
+    )
+  }
+}
+
+pub trait Commitments {
+  fn commit(&self, blind: &Scalar, gens_n: &MultiCommitGens) -> GroupElement;
+}
+
+impl Commitments for Scalar {
+  fn commit(&self, blind: &Scalar, gens_n: &MultiCommitGens) -> GroupElement {
+    assert_eq!(gens_n.n, 1);
+    GroupElement::vartime_multiscalar_mul(
+      [*self, *blind].to_vec(),
+      [gens_n.G[0], gens_n.h].to_vec(),
+    )
+  }
+}
+
+impl Commitments for Vec<Scalar> {
+  fn commit(&self, blind: &Scalar, gens_n: &MultiCommitGens) -> GroupElement {
+    assert_eq!(gens_n.n, self.len());
+    GroupElement::vartime_multiscalar_mul((*self).clone(), gens_n.G.clone()) + blind * gens_n.h
+  }
+}
+
+impl Commitments for [Scalar] {
+  fn commit(&self, blind: &Scalar, gens_n: &MultiCommitGens) -> GroupElement {
+    assert_eq!(gens_n.n, self.len());
+    GroupElement::vartime_multiscalar_mul(self.to_vec(), gens_n.G.clone()) + blind * gens_n.h
+  }
+}

packages/Spartan-secq/src/dense_mlpoly.rs

@@ -0,0 +1,602 @@
+#![allow(clippy::too_many_arguments)]
+use super::commitments::{Commitments, MultiCommitGens};
+use super::errors::ProofVerifyError;
+use super::group::{CompressedGroup, GroupElement, VartimeMultiscalarMul};
+use super::math::Math;
+use super::nizk::{DotProductProofGens, DotProductProofLog};
+use super::random::RandomTape;
+use super::scalar::Scalar;
+use super::transcript::{AppendToTranscript, ProofTranscript};
+use crate::group::DecompressEncodedPoint;
+use core::ops::Index;
+use merlin::Transcript;
+use serde::{Deserialize, Serialize};
+
+#[cfg(feature = "multicore")]
+use rayon::prelude::*;
+
+#[derive(Debug)]
+pub struct DensePolynomial {
+  num_vars: usize, // the number of variables in the multilinear polynomial
+  len: usize,
+  Z: Vec<Scalar>, // evaluations of the polynomial in all the 2^num_vars Boolean inputs
+}
+
+pub struct PolyCommitmentGens {
+  pub gens: DotProductProofGens,
+}
+
+impl PolyCommitmentGens {
+  // the number of variables in the multilinear polynomial
+  pub fn new(num_vars: usize, label: &'static [u8]) -> PolyCommitmentGens {
+    let (_left, right) = EqPolynomial::compute_factored_lens(num_vars);
+    let gens = DotProductProofGens::new(right.pow2(), label);
+    PolyCommitmentGens { gens }
+  }
+}
+
+pub struct PolyCommitmentBlinds {
+  blinds: Vec<Scalar>,
+}
+
+#[derive(Debug, Serialize, Deserialize)]
+pub struct PolyCommitment {
+  C: Vec<CompressedGroup>,
+}
+
+#[derive(Debug, Serialize, Deserialize)]
+pub struct ConstPolyCommitment {
+  C: CompressedGroup,
+}
+
+pub struct EqPolynomial {
+  r: Vec<Scalar>,
+}
+
+impl EqPolynomial {
+  pub fn new(r: Vec<Scalar>) -> Self {
+    EqPolynomial { r }
+  }
+
+  pub fn evaluate(&self, rx: &[Scalar]) -> Scalar {
+    assert_eq!(self.r.len(), rx.len());
+    (0..rx.len())
+      .map(|i| self.r[i] * rx[i] + (Scalar::one() - self.r[i]) * (Scalar::one() - rx[i]))
+      .product()
+  }
+
+  pub fn evals(&self) -> Vec<Scalar> {
+    let ell = self.r.len();
+
+    let mut evals: Vec<Scalar> = vec![Scalar::one(); ell.pow2()];
+    let mut size = 1;
+    for j in 0..ell {
+      // in each iteration, we double the size of chis
+      size *= 2;
+      for i in (0..size).rev().step_by(2) {
+        // copy each element from the prior iteration twice
+        let scalar = evals[i / 2];
+        evals[i] = scalar * self.r[j];
+        evals[i - 1] = scalar - evals[i];
+      }
+    }
+    evals
+  }
+
+  pub fn compute_factored_lens(ell: usize) -> (usize, usize) {
+    (ell / 2, ell - ell / 2)
+  }
+
+  pub fn compute_factored_evals(&self) -> (Vec<Scalar>, Vec<Scalar>) {
+    let ell = self.r.len();
+    let (left_num_vars, _right_num_vars) = EqPolynomial::compute_factored_lens(ell);
+
+    let L = EqPolynomial::new(self.r[..left_num_vars].to_vec()).evals();
+    let R = EqPolynomial::new(self.r[left_num_vars..ell].to_vec()).evals();
+
+    (L, R)
+  }
+}
+
+pub struct IdentityPolynomial {
+  size_point: usize,
+}
+
+impl IdentityPolynomial {
+  pub fn new(size_point: usize) -> Self {
+    IdentityPolynomial { size_point }
+  }
+
+  pub fn evaluate(&self, r: &[Scalar]) -> Scalar {
+    let len = r.len();
+    assert_eq!(len, self.size_point);
+    (0..len)
+      .map(|i| Scalar::from((len - i - 1).pow2() as u64) * r[i])
+      .sum()
+  }
+}
+
+impl DensePolynomial {
+  pub fn new(Z: Vec<Scalar>) -> Self {
+    DensePolynomial {
+      num_vars: Z.len().log_2(),
+      len: Z.len(),
+      Z,
+    }
+  }
+
+  pub fn get_num_vars(&self) -> usize {
+    self.num_vars
+  }
+
+  pub fn len(&self) -> usize {
+    self.len
+  }
+
+  pub fn clone(&self) -> DensePolynomial {
+    DensePolynomial::new(self.Z[0..self.len].to_vec())
+  }
+
+  pub fn split(&self, idx: usize) -> (DensePolynomial, DensePolynomial) {
+    assert!(idx < self.len());
+    (
+      DensePolynomial::new(self.Z[..idx].to_vec()),
+      DensePolynomial::new(self.Z[idx..2 * idx].to_vec()),
+    )
+  }
+
+  #[cfg(feature = "multicore")]
+  fn commit_inner(&self, blinds: &[Scalar], gens: &MultiCommitGens) -> PolyCommitment {
+    let L_size = blinds.len();
+    let R_size = self.Z.len() / L_size;
+    assert_eq!(L_size * R_size, self.Z.len());
+    let C = (0..L_size)
+      .into_par_iter()
+      .map(|i| {
+        self.Z[R_size * i..R_size * (i + 1)]
+          .commit(&blinds[i], gens)
+          .compress()
+      })
+      .collect();
+    PolyCommitment { C }
+  }
+
+  #[cfg(not(feature = "multicore"))]
+  fn commit_inner(&self, blinds: &[Scalar], gens: &MultiCommitGens) -> PolyCommitment {
+    let L_size = blinds.len();
+    let R_size = self.Z.len() / L_size;
+    assert_eq!(L_size * R_size, self.Z.len());
+    let C = (0..L_size)
+      .map(|i| {
+        self.Z[R_size * i..R_size * (i + 1)]
+          .commit(&blinds[i], gens)
+          .compress()
+      })
+      .collect();
+    PolyCommitment { C }
+  }
+
+  pub fn commit(
+    &self,
+    gens: &PolyCommitmentGens,
+    random_tape: Option<&mut RandomTape>,
+  ) -> (PolyCommitment, PolyCommitmentBlinds) {
+    let n = self.Z.len();
+    let ell = self.get_num_vars();
+    assert_eq!(n, ell.pow2());
+
+    let (left_num_vars, right_num_vars) = EqPolynomial::compute_factored_lens(ell);
+    let L_size = left_num_vars.pow2();
+    let R_size = right_num_vars.pow2();
+    assert_eq!(L_size * R_size, n);
+
+    let blinds = if let Some(t) = random_tape {
+      PolyCommitmentBlinds {
+        blinds: t.random_vector(b"poly_blinds", L_size),
+      }
+    } else {
+      PolyCommitmentBlinds {
+        blinds: vec![Scalar::zero(); L_size],
+      }
+    };
+
+    (self.commit_inner(&blinds.blinds, &gens.gens.gens_n), blinds)
+  }
+
+  pub fn bound(&self, L: &[Scalar]) -> Vec<Scalar> {
+    let (left_num_vars, right_num_vars) = EqPolynomial::compute_factored_lens(self.get_num_vars());
+    let L_size = left_num_vars.pow2();
+    let R_size = right_num_vars.pow2();
+    (0..R_size)
+      .map(|i| (0..L_size).map(|j| L[j] * self.Z[j * R_size + i]).sum())
+      .collect()
+  }
+
+  pub fn bound_poly_var_top(&mut self, r: &Scalar) {
+    let n = self.len() / 2;
+    for i in 0..n {
+      self.Z[i] = self.Z[i] + r * (self.Z[i + n] - self.Z[i]);
+    }
+    self.num_vars -= 1;
+    self.len = n;
+  }
+
+  pub fn bound_poly_var_bot(&mut self, r: &Scalar) {
+    let n = self.len() / 2;
+    for i in 0..n {
+      self.Z[i] = self.Z[2 * i] + r * (self.Z[2 * i + 1] - self.Z[2 * i]);
+    }
+    self.num_vars -= 1;
+    self.len = n;
+  }
+
+  // returns Z(r) in O(n) time
+  pub fn evaluate(&self, r: &[Scalar]) -> Scalar {
+    // r must have a value for each variable
+    assert_eq!(r.len(), self.get_num_vars());
+    let chis = EqPolynomial::new(r.to_vec()).evals();
+    assert_eq!(chis.len(), self.Z.len());
+    DotProductProofLog::compute_dotproduct(&self.Z, &chis)
+  }
+
+  fn vec(&self) -> &Vec<Scalar> {
+    &self.Z
+  }
+
+  pub fn extend(&mut self, other: &DensePolynomial) {
+    // TODO: allow extension even when some vars are bound
+    assert_eq!(self.Z.len(), self.len);
+    let other_vec = other.vec();
+    assert_eq!(other_vec.len(), self.len);
+    self.Z.extend(other_vec);
+    self.num_vars += 1;
+    self.len *= 2;
+    assert_eq!(self.Z.len(), self.len);
+  }
+
+  pub fn merge<'a, I>(polys: I) -> DensePolynomial
+  where
+    I: IntoIterator<Item = &'a DensePolynomial>,
+  {
+    let mut Z: Vec<Scalar> = Vec::new();
+    for poly in polys.into_iter() {
+      Z.extend(poly.vec());
+    }
+
+    // pad the polynomial with zero polynomial at the end
+    Z.resize(Z.len().next_power_of_two(), Scalar::zero());
+
+    DensePolynomial::new(Z)
+  }
+
+  pub fn from_usize(Z: &[usize]) -> Self {
+    DensePolynomial::new(
+      (0..Z.len())
+        .map(|i| Scalar::from(Z[i] as u64))
+        .collect::<Vec<Scalar>>(),
+    )
+  }
+}
+
+impl Index<usize> for DensePolynomial {
+  type Output = Scalar;
+
+  #[inline(always)]
+  fn index(&self, _index: usize) -> &Scalar {
+    &(self.Z[_index])
+  }
+}
+
+impl AppendToTranscript for PolyCommitment {
+  fn append_to_transcript(&self, label: &'static [u8], transcript: &mut Transcript) {
+    transcript.append_message(label, b"poly_commitment_begin");
+    for i in 0..self.C.len() {
+      transcript.append_point(b"poly_commitment_share", &self.C[i]);
+    }
+    transcript.append_message(label, b"poly_commitment_end");
+  }
+}
+
+#[derive(Debug, Serialize, Deserialize)]
+pub struct PolyEvalProof {
+  proof: DotProductProofLog,
+}
+
+impl PolyEvalProof {
+  fn protocol_name() -> &'static [u8] {
+    b"polynomial evaluation proof"
+  }
+
+  pub fn prove(
+    poly: &DensePolynomial,
+    blinds_opt: Option<&PolyCommitmentBlinds>,
+    r: &[Scalar],                  // point at which the polynomial is evaluated
+    Zr: &Scalar,                   // evaluation of \widetilde{Z}(r)
+    blind_Zr_opt: Option<&Scalar>, // specifies a blind for Zr
+    gens: &PolyCommitmentGens,
+    transcript: &mut Transcript,
+    random_tape: &mut RandomTape,
+  ) -> (PolyEvalProof, CompressedGroup) {
+    transcript.append_protocol_name(PolyEvalProof::protocol_name());
+
+    // assert vectors are of the right size
+    assert_eq!(poly.get_num_vars(), r.len());
+
+    let (left_num_vars, right_num_vars) = EqPolynomial::compute_factored_lens(r.len());
+    let L_size = left_num_vars.pow2();
+    let R_size = right_num_vars.pow2();
+
+    let default_blinds = PolyCommitmentBlinds {
+      blinds: vec![Scalar::zero(); L_size],
+    };
+    let blinds = blinds_opt.map_or(&default_blinds, |p| p);
+
+    assert_eq!(blinds.blinds.len(), L_size);
+
+    let zero = Scalar::zero();
+    let blind_Zr = blind_Zr_opt.map_or(&zero, |p| p);
+
+    // compute the L and R vectors
+    let eq = EqPolynomial::new(r.to_vec());
+    let (L, R) = eq.compute_factored_evals();
+    assert_eq!(L.len(), L_size);
+    assert_eq!(R.len(), R_size);
+
+    // compute the vector underneath L*Z and the L*blinds
+    // compute vector-matrix product between L and Z viewed as a matrix
+    let LZ = poly.bound(&L);
+    let LZ_blind: Scalar = (0..L.len()).map(|i| blinds.blinds[i] * L[i]).sum();
+
+    // a dot product proof of size R_size
+    let (proof, _C_LR, C_Zr_prime) = DotProductProofLog::prove(
+      &gens.gens,
+      transcript,
+      random_tape,
+      &LZ,
+      &LZ_blind,
+      &R,
+      Zr,
+      blind_Zr,
+    );
+
+    (PolyEvalProof { proof }, C_Zr_prime)
+  }
+
+  pub fn verify(
+    &self,
+    gens: &PolyCommitmentGens,
+    transcript: &mut Transcript,
+    r: &[Scalar],           // point at which the polynomial is evaluated
+    C_Zr: &CompressedGroup, // commitment to \widetilde{Z}(r)
+    comm: &PolyCommitment,
+  ) -> Result<(), ProofVerifyError> {
+    transcript.append_protocol_name(PolyEvalProof::protocol_name());
+
+    // compute L and R
+    let eq = EqPolynomial::new(r.to_vec());
+    let (L, R) = eq.compute_factored_evals();
+
+    // compute a weighted sum of commitments and L
+    let C_decompressed = comm.C.iter().map(|pt| pt.decompress().unwrap());
+
+    let C_LZ = GroupElement::vartime_multiscalar_mul(L, C_decompressed.collect()).compress();
+
+    self
+      .proof
+      .verify(R.len(), &gens.gens, transcript, &R, &C_LZ, C_Zr)
+  }
+
+  pub fn verify_plain(
+    &self,
+    gens: &PolyCommitmentGens,
+    transcript: &mut Transcript,
+    r: &[Scalar], // point at which the polynomial is evaluated
+    Zr: &Scalar,  // evaluation \widetilde{Z}(r)
+    comm: &PolyCommitment,
+  ) -> Result<(), ProofVerifyError> {
+    // compute a commitment to Zr with a blind of zero
+    let C_Zr = Zr.commit(&Scalar::zero(), &gens.gens.gens_1).compress();
+
+    self.verify(gens, transcript, r, &C_Zr, comm)
+  }
+}
+
+#[cfg(test)]
+mod tests {
+  use super::super::scalar::ScalarFromPrimitives;
+  use super::*;
+  use rand_core::OsRng;
+
+  fn evaluate_with_LR(Z: &[Scalar], r: &[Scalar]) -> Scalar {
+    let eq = EqPolynomial::new(r.to_vec());
+    let (L, R) = eq.compute_factored_evals();
+
+    let ell = r.len();
+    // ensure ell is even
+    assert!(ell % 2 == 0);
+    // compute n = 2^\ell
+    let n = ell.pow2();
+    // compute m = sqrt(n) = 2^{\ell/2}
+    let m = n.square_root();
+
+    // compute vector-matrix product between L and Z viewed as a matrix
+    let LZ = (0..m)
+      .map(|i| (0..m).map(|j| L[j] * Z[j * m + i]).sum())
+      .collect::<Vec<Scalar>>();
+
+    // compute dot product between LZ and R
+    DotProductProofLog::compute_dotproduct(&LZ, &R)
+  }
+
+  #[test]
+  fn check_polynomial_evaluation() {
+    // Z = [1, 2, 1, 4]
+    let Z = vec![
+      Scalar::one(),
+      (2_usize).to_scalar(),
+      (1_usize).to_scalar(),
+      (4_usize).to_scalar(),
+    ];
+
+    // r = [4,3]
+    let r = vec![(4_usize).to_scalar(), (3_usize).to_scalar()];
+
+    let eval_with_LR = evaluate_with_LR(&Z, &r);
+    let poly = DensePolynomial::new(Z);
+
+    let eval = poly.evaluate(&r);
+    assert_eq!(eval, (28_usize).to_scalar());
+    assert_eq!(eval_with_LR, eval);
+  }
+
+  pub fn compute_factored_chis_at_r(r: &[Scalar]) -> (Vec<Scalar>, Vec<Scalar>) {
+    let mut L: Vec<Scalar> = Vec::new();
+    let mut R: Vec<Scalar> = Vec::new();
+
+    let ell = r.len();
+    assert!(ell % 2 == 0); // ensure ell is even
+    let n = ell.pow2();
+    let m = n.square_root();
+
+    // compute row vector L
+    for i in 0..m {
+      let mut chi_i = Scalar::one();
+      for j in 0..ell / 2 {
+        let bit_j = ((m * i) & (1 << (r.len() - j - 1))) > 0;
+        if bit_j {
+          chi_i *= r[j];
+        } else {
+          chi_i *= Scalar::one() - r[j];
+        }
+      }
+      L.push(chi_i);
+    }
+
+    // compute column vector R
+    for i in 0..m {
+      let mut chi_i = Scalar::one();
+      for j in ell / 2..ell {
+        let bit_j = (i & (1 << (r.len() - j - 1))) > 0;
+        if bit_j {
+          chi_i *= r[j];
+        } else {
+          chi_i *= Scalar::one() - r[j];
+        }
+      }
+      R.push(chi_i);
+    }
+    (L, R)
+  }
+
+  pub fn compute_chis_at_r(r: &[Scalar]) -> Vec<Scalar> {
+    let ell = r.len();
+    let n = ell.pow2();
+    let mut chis: Vec<Scalar> = Vec::new();
+    for i in 0..n {
+      let mut chi_i = Scalar::one();
+      for j in 0..r.len() {
+        let bit_j = (i & (1 << (r.len() - j - 1))) > 0;
+        if bit_j {
+          chi_i *= r[j];
+        } else {
+          chi_i *= Scalar::one() - r[j];
+        }
+      }
+      chis.push(chi_i);
+    }
+    chis
+  }
+
+  pub fn compute_outerproduct(L: Vec<Scalar>, R: Vec<Scalar>) -> Vec<Scalar> {
+    assert_eq!(L.len(), R.len());
+    (0..L.len())
+      .map(|i| (0..R.len()).map(|j| L[i] * R[j]).collect::<Vec<Scalar>>())
+      .collect::<Vec<Vec<Scalar>>>()
+      .into_iter()
+      .flatten()
+      .collect::<Vec<Scalar>>()
+  }
+
+  #[test]
+  fn check_memoized_chis() {
+    let mut csprng: OsRng = OsRng;
+
+    let s = 10;
+    let mut r: Vec<Scalar> = Vec::new();
+    for _i in 0..s {
+      r.push(Scalar::random(&mut csprng));
+    }
+    let chis = tests::compute_chis_at_r(&r);
+    let chis_m = EqPolynomial::new(r).evals();
+    assert_eq!(chis, chis_m);
+  }
+
+  #[test]
+  fn check_factored_chis() {
+    let mut csprng: OsRng = OsRng;
+
+    let s = 10;
+    let mut r: Vec<Scalar> = Vec::new();
+    for _i in 0..s {
+      r.push(Scalar::random(&mut csprng));
+    }
+    let chis = EqPolynomial::new(r.clone()).evals();
+    let (L, R) = EqPolynomial::new(r).compute_factored_evals();
+    let O = compute_outerproduct(L, R);
+    assert_eq!(chis, O);
+  }
+
+  #[test]
+  fn check_memoized_factored_chis() {
+    let mut csprng: OsRng = OsRng;
+
+    let s = 10;
+    let mut r: Vec<Scalar> = Vec::new();
+    for _i in 0..s {
+      r.push(Scalar::random(&mut csprng));
+    }
+    let (L, R) = tests::compute_factored_chis_at_r(&r);
+    let eq = EqPolynomial::new(r);
+    let (L2, R2) = eq.compute_factored_evals();
+    assert_eq!(L, L2);
+    assert_eq!(R, R2);
+  }
+
+  #[test]
+  fn check_polynomial_commit() {
+    let Z = vec![
+      (1_usize).to_scalar(),
+      (2_usize).to_scalar(),
+      (1_usize).to_scalar(),
+      (4_usize).to_scalar(),
+    ];
+    let poly = DensePolynomial::new(Z);
+
+    // r = [4,3]
+    let r = vec![(4_usize).to_scalar(), (3_usize).to_scalar()];
+    let eval = poly.evaluate(&r);
+    assert_eq!(eval, (28_usize).to_scalar());
+
+    let gens = PolyCommitmentGens::new(poly.get_num_vars(), b"test-two");
+    let (poly_commitment, blinds) = poly.commit(&gens, None);
+
+    let mut random_tape = RandomTape::new(b"proof");
+    let mut prover_transcript = Transcript::new(b"example");
+    let (proof, C_Zr) = PolyEvalProof::prove(
+      &poly,
+      Some(&blinds),
+      &r,
+      &eval,
+      None,
+      &gens,
+      &mut prover_transcript,
+      &mut random_tape,
+    );
+
+    let mut verifier_transcript = Transcript::new(b"example");
+
+    assert!(proof
+      .verify(&gens, &mut verifier_transcript, &r, &C_Zr, &poly_commitment)
+      .is_ok());
+  }
+}

packages/Spartan-secq/src/errors.rs

@@ -0,0 +1,32 @@
+use core::fmt::Debug;
+use thiserror::Error;
+
+#[derive(Error, Debug)]
+pub enum ProofVerifyError {
+  #[error("Proof verification failed")]
+  InternalError,
+  #[error("Compressed group element failed to decompress: {0:?}")]
+  DecompressionError([u8; 32]),
+}
+
+impl Default for ProofVerifyError {
+  fn default() -> Self {
+    ProofVerifyError::InternalError
+  }
+}
+
+#[derive(Clone, Debug, Eq, PartialEq)]
+pub enum R1CSError {
+  /// returned if the number of constraints is not a power of 2
+  NonPowerOfTwoCons,
+  /// returned if the number of variables is not a power of 2
+  NonPowerOfTwoVars,
+  /// returned if a wrong number of inputs in an assignment are supplied
+  InvalidNumberOfInputs,
+  /// returned if a wrong number of variables in an assignment are supplied
+  InvalidNumberOfVars,
+  /// returned if a [u8;32] does not parse into a valid Scalar in the field of secq256k1
+  InvalidScalar,
+  /// returned if the supplied row or col in (row,col,val) tuple is out of range
+  InvalidIndex,
+}

packages/Spartan-secq/src/group.rs

@@ -0,0 +1,138 @@
+use secq256k1::{AffinePoint, ProjectivePoint};
+
+use super::errors::ProofVerifyError;
+use super::scalar::{Scalar, ScalarBytes, ScalarBytesFromScalar};
+use core::ops::{Mul, MulAssign};
+use multiexp::multiexp;
+
+pub type GroupElement = secq256k1::AffinePoint;
+pub type CompressedGroup = secq256k1::EncodedPoint;
+pub trait CompressedGroupExt {
+  type Group;
+  fn unpack(&self) -> Result<Self::Group, ProofVerifyError>;
+}
+
+impl CompressedGroupExt for CompressedGroup {
+  type Group = secq256k1::AffinePoint;
+  fn unpack(&self) -> Result<Self::Group, ProofVerifyError> {
+    let result = AffinePoint::decompress(*self);
+    if result.is_some().into() {
+      return Ok(result.unwrap());
+    } else {
+      Err(ProofVerifyError::DecompressionError(
+        (*self.to_bytes()).try_into().unwrap(),
+      ))
+    }
+  }
+}
+
+pub trait DecompressEncodedPoint {
+  fn decompress(&self) -> Option<GroupElement>;
+}
+
+impl DecompressEncodedPoint for CompressedGroup {
+  fn decompress(&self) -> Option<GroupElement> {
+    Some(self.unpack().unwrap())
+  }
+}
+
+impl<'b> MulAssign<&'b Scalar> for GroupElement {
+  fn mul_assign(&mut self, scalar: &'b Scalar) {
+    let result = (self as &GroupElement) * Scalar::decompress_scalar(scalar);
+    *self = result;
+  }
+}
+
+impl<'a, 'b> Mul<&'b Scalar> for &'a GroupElement {
+  type Output = GroupElement;
+  fn mul(self, scalar: &'b Scalar) -> GroupElement {
+    *self * Scalar::decompress_scalar(scalar)
+  }
+}
+
+impl<'a, 'b> Mul<&'b GroupElement> for &'a Scalar {
+  type Output = GroupElement;
+
+  fn mul(self, point: &'b GroupElement) -> GroupElement {
+    (*point * Scalar::decompress_scalar(self)).into()
+  }
+}
+
+macro_rules! define_mul_variants {
+  (LHS = $lhs:ty, RHS = $rhs:ty, Output = $out:ty) => {
+    impl<'b> Mul<&'b $rhs> for $lhs {
+      type Output = $out;
+      fn mul(self, rhs: &'b $rhs) -> $out {
+        &self * rhs
+      }
+    }
+
+    impl<'a> Mul<$rhs> for &'a $lhs {
+      type Output = $out;
+      fn mul(self, rhs: $rhs) -> $out {
+        self * &rhs
+      }
+    }
+
+    impl Mul<$rhs> for $lhs {
+      type Output = $out;
+      fn mul(self, rhs: $rhs) -> $out {
+        &self * &rhs
+      }
+    }
+  };
+}
+
+macro_rules! define_mul_assign_variants {
+  (LHS = $lhs:ty, RHS = $rhs:ty) => {
+    impl MulAssign<$rhs> for $lhs {
+      fn mul_assign(&mut self, rhs: $rhs) {
+        *self *= &rhs;
+      }
+    }
+  };
+}
+
+define_mul_assign_variants!(LHS = GroupElement, RHS = Scalar);
+define_mul_variants!(LHS = GroupElement, RHS = Scalar, Output = GroupElement);
+define_mul_variants!(LHS = Scalar, RHS = GroupElement, Output = GroupElement);
+
+pub trait VartimeMultiscalarMul {
+  type Scalar;
+  fn vartime_multiscalar_mul(scalars: Vec<Scalar>, points: Vec<GroupElement>) -> Self;
+}
+
+impl VartimeMultiscalarMul for GroupElement {
+  type Scalar = super::scalar::Scalar;
+  // TODO Borrow the arguments so we don't have to clone them, as it was in the original implementation
+  fn vartime_multiscalar_mul(scalars: Vec<Scalar>, points: Vec<GroupElement>) -> Self {
+    let points: Vec<ProjectivePoint> = points.iter().map(|p| ProjectivePoint::from(p.0)).collect();
+
+    let pairs: Vec<(ScalarBytes, ProjectivePoint)> = scalars
+      .into_iter()
+      .enumerate()
+      .map(|(i, s)| (Scalar::decompress_scalar(&s), points[i]))
+      .collect();
+
+    let result = multiexp::<ProjectivePoint>(pairs.as_slice());
+
+    AffinePoint(result.to_affine())
+  }
+}
+
+#[cfg(test)]
+mod tests {
+  use super::*;
+  #[test]
+  fn msm() {
+    let scalars = vec![Scalar::from(1), Scalar::from(2), Scalar::from(3)];
+    let points = vec![
+      GroupElement::generator(),
+      GroupElement::generator(),
+      GroupElement::generator(),
+    ];
+    let result = GroupElement::vartime_multiscalar_mul(scalars, points);
+
+    assert_eq!(result, GroupElement::generator() * Scalar::from(6));
+  }
+}

packages/Spartan-secq/src/lib.rs

@@ -0,0 +1,751 @@
+#![allow(non_snake_case)]
+#![doc = include_str!("../README.md")]
+#![deny(missing_docs)]
+#![allow(clippy::assertions_on_result_states)]
+
+extern crate byteorder;
+extern crate core;
+extern crate digest;
+extern crate merlin;
+extern crate rand;
+extern crate sha3;
+
+#[cfg(feature = "multicore")]
+extern crate rayon;
+
+mod commitments;
+mod dense_mlpoly;
+mod errors;
+mod group;
+mod math;
+mod nizk;
+mod product_tree;
+mod r1csinstance;
+mod r1csproof;
+mod random;
+mod scalar;
+mod sparse_mlpoly;
+mod sumcheck;
+mod timer;
+mod transcript;
+mod unipoly;
+
+use core::cmp::max;
+use errors::{ProofVerifyError, R1CSError};
+use merlin::Transcript;
+use r1csinstance::{
+  R1CSCommitment, R1CSCommitmentGens, R1CSDecommitment, R1CSEvalProof, R1CSInstance,
+};
+use r1csproof::{R1CSGens, R1CSProof};
+use random::RandomTape;
+use scalar::Scalar;
+use serde::{Deserialize, Serialize};
+use timer::Timer;
+use transcript::{AppendToTranscript, ProofTranscript};
+
+/// `ComputationCommitment` holds a public preprocessed NP statement (e.g., R1CS)
+pub struct ComputationCommitment {
+  comm: R1CSCommitment,
+}
+
+/// `ComputationDecommitment` holds information to decommit `ComputationCommitment`
+pub struct ComputationDecommitment {
+  decomm: R1CSDecommitment,
+}
+
+/// `Assignment` holds an assignment of values to either the inputs or variables in an `Instance`
+#[derive(Serialize, Deserialize, Clone)]
+pub struct Assignment {
+  assignment: Vec<Scalar>,
+}
+
+impl Assignment {
+  /// Constructs a new `Assignment` from a vector
+  pub fn new(assignment: &[[u8; 32]]) -> Result<Assignment, R1CSError> {
+    let bytes_to_scalar = |vec: &[[u8; 32]]| -> Result<Vec<Scalar>, R1CSError> {
+      let mut vec_scalar: Vec<Scalar> = Vec::new();
+      for v in vec {
+        let val = Scalar::from_bytes(v);
+        if val.is_some().unwrap_u8() == 1 {
+          vec_scalar.push(val.unwrap());
+        } else {
+          return Err(R1CSError::InvalidScalar);
+        }
+      }
+      Ok(vec_scalar)
+    };
+
+    let assignment_scalar = bytes_to_scalar(assignment);
+
+    // check for any parsing errors
+    if assignment_scalar.is_err() {
+      return Err(R1CSError::InvalidScalar);
+    }
+
+    Ok(Assignment {
+      assignment: assignment_scalar.unwrap(),
+    })
+  }
+
+  /// pads Assignment to the specified length
+  fn pad(&self, len: usize) -> VarsAssignment {
+    // check that the new length is higher than current length
+    assert!(len > self.assignment.len());
+
+    let padded_assignment = {
+      let mut padded_assignment = self.assignment.clone();
+      padded_assignment.extend(vec![Scalar::zero(); len - self.assignment.len()]);
+      padded_assignment
+    };
+
+    VarsAssignment {
+      assignment: padded_assignment,
+    }
+  }
+}
+
+/// `VarsAssignment` holds an assignment of values to variables in an `Instance`
+pub type VarsAssignment = Assignment;
+
+/// `InputsAssignment` holds an assignment of values to variables in an `Instance`
+pub type InputsAssignment = Assignment;
+
+/// `Instance` holds the description of R1CS matrices and a hash of the matrices
+#[derive(Serialize, Deserialize)]
+pub struct Instance {
+  /// R1CS instance
+  pub inst: R1CSInstance,
+  digest: Vec<u8>,
+}
+
+impl Instance {
+  /// Constructs a new `Instance` and an associated satisfying assignment
+  pub fn new(
+    num_cons: usize,
+    num_vars: usize,
+    num_inputs: usize,
+    A: &[(usize, usize, [u8; 32])],
+    B: &[(usize, usize, [u8; 32])],
+    C: &[(usize, usize, [u8; 32])],
+  ) -> Result<Instance, R1CSError> {
+    let (num_vars_padded, num_cons_padded) = {
+      let num_vars_padded = {
+        let mut num_vars_padded = num_vars;
+
+        // ensure that num_inputs + 1 <= num_vars
+        num_vars_padded = max(num_vars_padded, num_inputs + 1);
+
+        // ensure that num_vars_padded a power of two
+        if num_vars_padded.next_power_of_two() != num_vars_padded {
+          num_vars_padded = num_vars_padded.next_power_of_two();
+        }
+        num_vars_padded
+      };
+
+      let num_cons_padded = {
+        let mut num_cons_padded = num_cons;
+
+        // ensure that num_cons_padded is at least 2
+        if num_cons_padded == 0 || num_cons_padded == 1 {
+          num_cons_padded = 2;
+        }
+
+        // ensure that num_cons_padded is power of 2
+        if num_cons.next_power_of_two() != num_cons {
+          num_cons_padded = num_cons.next_power_of_two();
+        }
+        num_cons_padded
+      };
+
+      (num_vars_padded, num_cons_padded)
+    };
+
+    let bytes_to_scalar =
+      |tups: &[(usize, usize, [u8; 32])]| -> Result<Vec<(usize, usize, Scalar)>, R1CSError> {
+        let mut mat: Vec<(usize, usize, Scalar)> = Vec::new();
+        for &(row, col, val_bytes) in tups {
+          // row must be smaller than num_cons
+          if row >= num_cons {
+            return Err(R1CSError::InvalidIndex);
+          }
+
+          // col must be smaller than num_vars + 1 + num_inputs
+          if col >= num_vars + 1 + num_inputs {
+            return Err(R1CSError::InvalidIndex);
+          }
+
+          let val = Scalar::from_bytes(&val_bytes);
+          if val.is_some().unwrap_u8() == 1 {
+            // if col >= num_vars, it means that it is referencing a 1 or input in the satisfying
+            // assignment
+            if col >= num_vars {
+              mat.push((row, col + num_vars_padded - num_vars, val.unwrap()));
+            } else {
+              mat.push((row, col, val.unwrap()));
+            }
+          } else {
+            return Err(R1CSError::InvalidScalar);
+          }
+        }
+
+        // pad with additional constraints up until num_cons_padded if the original constraints were 0 or 1
+        // we do not need to pad otherwise because the dummy constraints are implicit in the sum-check protocol
+        if num_cons == 0 || num_cons == 1 {
+          for i in tups.len()..num_cons_padded {
+            mat.push((i, num_vars, Scalar::zero()));
+          }
+        }
+
+        Ok(mat)
+      };
+
+    let A_scalar = bytes_to_scalar(A);
+    if A_scalar.is_err() {
+      return Err(A_scalar.err().unwrap());
+    }
+
+    let B_scalar = bytes_to_scalar(B);
+    if B_scalar.is_err() {
+      return Err(B_scalar.err().unwrap());
+    }
+
+    let C_scalar = bytes_to_scalar(C);
+    if C_scalar.is_err() {
+      return Err(C_scalar.err().unwrap());
+    }
+
+    let inst = R1CSInstance::new(
+      num_cons_padded,
+      num_vars_padded,
+      num_inputs,
+      &A_scalar.unwrap(),
+      &B_scalar.unwrap(),
+      &C_scalar.unwrap(),
+    );
+
+    let digest = inst.get_digest();
+
+    Ok(Instance { inst, digest })
+  }
+
+  /// Checks if a given R1CSInstance is satisfiable with a given variables and inputs assignments
+  pub fn is_sat(
+    &self,
+    vars: &VarsAssignment,
+    inputs: &InputsAssignment,
+  ) -> Result<bool, R1CSError> {
+    if vars.assignment.len() > self.inst.get_num_vars() {
+      return Err(R1CSError::InvalidNumberOfInputs);
+    }
+
+    if inputs.assignment.len() != self.inst.get_num_inputs() {
+      return Err(R1CSError::InvalidNumberOfInputs);
+    }
+
+    // we might need to pad variables
+    let padded_vars = {
+      let num_padded_vars = self.inst.get_num_vars();
+      let num_vars = vars.assignment.len();
+      if num_padded_vars > num_vars {
+        vars.pad(num_padded_vars)
+      } else {
+        vars.clone()
+      }
+    };
+
+    Ok(
+      self
+        .inst
+        .is_sat(&padded_vars.assignment, &inputs.assignment),
+    )
+  }
+
+  /// Constructs a new synthetic R1CS `Instance` and an associated satisfying assignment
+  pub fn produce_synthetic_r1cs(
+    num_cons: usize,
+    num_vars: usize,
+    num_inputs: usize,
+  ) -> (Instance, VarsAssignment, InputsAssignment) {
+    let (inst, vars, inputs) = R1CSInstance::produce_synthetic_r1cs(num_cons, num_vars, num_inputs);
+    let digest = inst.get_digest();
+    (
+      Instance { inst, digest },
+      VarsAssignment { assignment: vars },
+      InputsAssignment { assignment: inputs },
+    )
+  }
+}
+
+/// `SNARKGens` holds public parameters for producing and verifying proofs with the Spartan SNARK
+pub struct SNARKGens {
+  gens_r1cs_sat: R1CSGens,
+  gens_r1cs_eval: R1CSCommitmentGens,
+}
+
+impl SNARKGens {
+  /// Constructs a new `SNARKGens` given the size of the R1CS statement
+  /// `num_nz_entries` specifies the maximum number of non-zero entries in any of the three R1CS matrices
+  pub fn new(num_cons: usize, num_vars: usize, num_inputs: usize, num_nz_entries: usize) -> Self {
+    let num_vars_padded = {
+      let mut num_vars_padded = max(num_vars, num_inputs + 1);
+      if num_vars_padded != num_vars_padded.next_power_of_two() {
+        num_vars_padded = num_vars_padded.next_power_of_two();
+      }
+      num_vars_padded
+    };
+
+    let gens_r1cs_sat = R1CSGens::new(b"gens_r1cs_sat", num_cons, num_vars_padded);
+    let gens_r1cs_eval = R1CSCommitmentGens::new(
+      b"gens_r1cs_eval",
+      num_cons,
+      num_vars_padded,
+      num_inputs,
+      num_nz_entries,
+    );
+    SNARKGens {
+      gens_r1cs_sat,
+      gens_r1cs_eval,
+    }
+  }
+}
+
+/// `SNARK` holds a proof produced by Spartan SNARK
+#[derive(Serialize, Deserialize, Debug)]
+pub struct SNARK {
+  r1cs_sat_proof: R1CSProof,
+  inst_evals: (Scalar, Scalar, Scalar),
+  r1cs_eval_proof: R1CSEvalProof,
+}
+
+impl SNARK {
+  fn protocol_name() -> &'static [u8] {
+    b"Spartan SNARK proof"
+  }
+
+  /// A public computation to create a commitment to an R1CS instance
+  pub fn encode(
+    inst: &Instance,
+    gens: &SNARKGens,
+  ) -> (ComputationCommitment, ComputationDecommitment) {
+    let timer_encode = Timer::new("SNARK::encode");
+    let (comm, decomm) = inst.inst.commit(&gens.gens_r1cs_eval);
+    timer_encode.stop();
+    (
+      ComputationCommitment { comm },
+      ComputationDecommitment { decomm },
+    )
+  }
+
+  /// A method to produce a SNARK proof of the satisfiability of an R1CS instance
+  pub fn prove(
+    inst: &Instance,
+    comm: &ComputationCommitment,
+    decomm: &ComputationDecommitment,
+    vars: VarsAssignment,
+    inputs: &InputsAssignment,
+    gens: &SNARKGens,
+    transcript: &mut Transcript,
+  ) -> Self {
+    let timer_prove = Timer::new("SNARK::prove");
+
+    // we create a Transcript object seeded with a random Scalar
+    // to aid the prover produce its randomness
+    let mut random_tape = RandomTape::new(b"proof");
+
+    transcript.append_protocol_name(SNARK::protocol_name());
+    comm.comm.append_to_transcript(b"comm", transcript);
+
+    let (r1cs_sat_proof, rx, ry) = {
+      let (proof, rx, ry) = {
+        // we might need to pad variables
+        let padded_vars = {
+          let num_padded_vars = inst.inst.get_num_vars();
+          let num_vars = vars.assignment.len();
+          if num_padded_vars > num_vars {
+            vars.pad(num_padded_vars)
+          } else {
+            vars
+          }
+        };
+
+        R1CSProof::prove(
+          &inst.inst,
+          padded_vars.assignment,
+          &inputs.assignment,
+          &gens.gens_r1cs_sat,
+          transcript,
+          &mut random_tape,
+        )
+      };
+
+      let proof_encoded: Vec<u8> = bincode::serialize(&proof).unwrap();
+      Timer::print(&format!("len_r1cs_sat_proof {:?}", proof_encoded.len()));
+
+      (proof, rx, ry)
+    };
+
+    // We send evaluations of A, B, C at r = (rx, ry) as claims
+    // to enable the verifier complete the first sum-check
+    let timer_eval = Timer::new("eval_sparse_polys");
+    let inst_evals = {
+      let (Ar, Br, Cr) = inst.inst.evaluate(&rx, &ry);
+      Ar.append_to_transcript(b"Ar_claim", transcript);
+      Br.append_to_transcript(b"Br_claim", transcript);
+      Cr.append_to_transcript(b"Cr_claim", transcript);
+      (Ar, Br, Cr)
+    };
+    timer_eval.stop();
+
+    let r1cs_eval_proof = {
+      let proof = R1CSEvalProof::prove(
+        &decomm.decomm,
+        &rx,
+        &ry,
+        &inst_evals,
+        &gens.gens_r1cs_eval,
+        transcript,
+        &mut random_tape,
+      );
+
+      let proof_encoded: Vec<u8> = bincode::serialize(&proof).unwrap();
+      Timer::print(&format!("len_r1cs_eval_proof {:?}", proof_encoded.len()));
+      proof
+    };
+
+    timer_prove.stop();
+    SNARK {
+      r1cs_sat_proof,
+      inst_evals,
+      r1cs_eval_proof,
+    }
+  }
+
+  /// A method to verify the SNARK proof of the satisfiability of an R1CS instance
+  pub fn verify(
+    &self,
+    comm: &ComputationCommitment,
+    input: &InputsAssignment,
+    transcript: &mut Transcript,
+    gens: &SNARKGens,
+  ) -> Result<(), ProofVerifyError> {
+    let timer_verify = Timer::new("SNARK::verify");
+    transcript.append_protocol_name(SNARK::protocol_name());
+
+    // append a commitment to the computation to the transcript
+    comm.comm.append_to_transcript(b"comm", transcript);
+
+    let timer_sat_proof = Timer::new("verify_sat_proof");
+    assert_eq!(input.assignment.len(), comm.comm.get_num_inputs());
+    let (rx, ry) = self.r1cs_sat_proof.verify(
+      comm.comm.get_num_vars(),
+      comm.comm.get_num_cons(),
+      &input.assignment,
+      &self.inst_evals,
+      transcript,
+      &gens.gens_r1cs_sat,
+    )?;
+    timer_sat_proof.stop();
+
+    let timer_eval_proof = Timer::new("verify_eval_proof");
+    let (Ar, Br, Cr) = &self.inst_evals;
+    Ar.append_to_transcript(b"Ar_claim", transcript);
+    Br.append_to_transcript(b"Br_claim", transcript);
+    Cr.append_to_transcript(b"Cr_claim", transcript);
+    self.r1cs_eval_proof.verify(
+      &comm.comm,
+      &rx,
+      &ry,
+      &self.inst_evals,
+      &gens.gens_r1cs_eval,
+      transcript,
+    )?;
+    timer_eval_proof.stop();
+    timer_verify.stop();
+    Ok(())
+  }
+}
+
+/// `NIZKGens` holds public parameters for producing and verifying proofs with the Spartan NIZK
+pub struct NIZKGens {
+  gens_r1cs_sat: R1CSGens,
+}
+
+impl NIZKGens {
+  /// Constructs a new `NIZKGens` given the size of the R1CS statement
+  pub fn new(num_cons: usize, num_vars: usize, num_inputs: usize) -> Self {
+    let num_vars_padded = {
+      let mut num_vars_padded = max(num_vars, num_inputs + 1);
+      if num_vars_padded != num_vars_padded.next_power_of_two() {
+        num_vars_padded = num_vars_padded.next_power_of_two();
+      }
+      num_vars_padded
+    };
+
+    let gens_r1cs_sat = R1CSGens::new(b"gens_r1cs_sat", num_cons, num_vars_padded);
+    NIZKGens { gens_r1cs_sat }
+  }
+}
+
+/// `NIZK` holds a proof produced by Spartan NIZK
+#[derive(Serialize, Deserialize, Debug)]
+pub struct NIZK {
+  r1cs_sat_proof: R1CSProof,
+  r: (Vec<Scalar>, Vec<Scalar>),
+}
+
+impl NIZK {
+  fn protocol_name() -> &'static [u8] {
+    b"Spartan NIZK proof"
+  }
+
+  /// A method to produce a NIZK proof of the satisfiability of an R1CS instance
+  pub fn prove(
+    inst: &Instance,
+    vars: VarsAssignment,
+    input: &InputsAssignment,
+    gens: &NIZKGens,
+    transcript: &mut Transcript,
+  ) -> Self {
+    let timer_prove = Timer::new("NIZK::prove");
+    // we create a Transcript object seeded with a random Scalar
+    // to aid the prover produce its randomness
+    let mut random_tape = RandomTape::new(b"proof");
+
+    transcript.append_protocol_name(NIZK::protocol_name());
+    transcript.append_message(b"R1CSInstanceDigest", &inst.digest);
+
+    let (r1cs_sat_proof, rx, ry) = {
+      // we might need to pad variables
+      let padded_vars = {
+        let num_padded_vars = inst.inst.get_num_vars();
+        let num_vars = vars.assignment.len();
+        if num_padded_vars > num_vars {
+          vars.pad(num_padded_vars)
+        } else {
+          vars
+        }
+      };
+
+      let (proof, rx, ry) = R1CSProof::prove(
+        &inst.inst,
+        padded_vars.assignment,
+        &input.assignment,
+        &gens.gens_r1cs_sat,
+        transcript,
+        &mut random_tape,
+      );
+      let proof_encoded: Vec<u8> = bincode::serialize(&proof).unwrap();
+      Timer::print(&format!("len_r1cs_sat_proof {:?}", proof_encoded.len()));
+      (proof, rx, ry)
+    };
+
+    timer_prove.stop();
+    NIZK {
+      r1cs_sat_proof,
+      r: (rx, ry),
+    }
+  }
+
+  /// A method to verify a NIZK proof of the satisfiability of an R1CS instance
+  pub fn verify(
+    &self,
+    inst: &Instance,
+    input: &InputsAssignment,
+    transcript: &mut Transcript,
+    gens: &NIZKGens,
+  ) -> Result<(), ProofVerifyError> {
+    let timer_verify = Timer::new("NIZK::verify");
+
+    transcript.append_protocol_name(NIZK::protocol_name());
+    transcript.append_message(b"R1CSInstanceDigest", &inst.digest);
+
+    // We send evaluations of A, B, C at r = (rx, ry) as claims
+    // to enable the verifier complete the first sum-check
+    let timer_eval = Timer::new("eval_sparse_polys");
+    let (claimed_rx, claimed_ry) = &self.r;
+    let inst_evals = inst.inst.evaluate(claimed_rx, claimed_ry);
+    timer_eval.stop();
+
+    let timer_sat_proof = Timer::new("verify_sat_proof");
+    assert_eq!(input.assignment.len(), inst.inst.get_num_inputs());
+    let (rx, ry) = self.r1cs_sat_proof.verify(
+      inst.inst.get_num_vars(),
+      inst.inst.get_num_cons(),
+      &input.assignment,
+      &inst_evals,
+      transcript,
+      &gens.gens_r1cs_sat,
+    )?;
+
+    // verify if claimed rx and ry are correct
+    assert_eq!(rx, *claimed_rx);
+    assert_eq!(ry, *claimed_ry);
+    timer_sat_proof.stop();
+    timer_verify.stop();
+
+    Ok(())
+  }
+}
+
+#[cfg(test)]
+mod tests {
+  use super::*;
+
+  #[test]
+  pub fn check_snark() {
+    let num_vars = 256;
+    let num_cons = num_vars;
+    let num_inputs = 10;
+
+    // produce public generators
+    let gens = SNARKGens::new(num_cons, num_vars, num_inputs, num_cons);
+
+    // produce a synthetic R1CSInstance
+    let (inst, vars, inputs) = Instance::produce_synthetic_r1cs(num_cons, num_vars, num_inputs);
+
+    // create a commitment to R1CSInstance
+    let (comm, decomm) = SNARK::encode(&inst, &gens);
+
+    // produce a proof
+    let mut prover_transcript = Transcript::new(b"example");
+    let proof = SNARK::prove(
+      &inst,
+      &comm,
+      &decomm,
+      vars,
+      &inputs,
+      &gens,
+      &mut prover_transcript,
+    );
+
+    // verify the proof
+    let mut verifier_transcript = Transcript::new(b"example");
+    assert!(proof
+      .verify(&comm, &inputs, &mut verifier_transcript, &gens)
+      .is_ok());
+  }
+
+  #[test]
+  pub fn check_r1cs_invalid_index() {
+    let num_cons = 4;
+    let num_vars = 8;
+    let num_inputs = 1;
+
+    let zero: [u8; 32] = [
+      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+      0,
+    ];
+
+    let A = vec![(0, 0, zero)];
+    let B = vec![(100, 1, zero)];
+    let C = vec![(1, 1, zero)];
+
+    let inst = Instance::new(num_cons, num_vars, num_inputs, &A, &B, &C);
+    assert!(inst.is_err());
+    assert_eq!(inst.err(), Some(R1CSError::InvalidIndex));
+  }
+
+  #[test]
+  pub fn check_r1cs_invalid_scalar() {
+    let num_cons = 4;
+    let num_vars = 8;
+    let num_inputs = 1;
+
+    let zero: [u8; 32] = [
+      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+      0,
+    ];
+
+    let larger_than_mod = [255; 32];
+
+    let A = vec![(0, 0, zero)];
+    let B = vec![(1, 1, larger_than_mod)];
+    let C = vec![(1, 1, zero)];
+
+    let inst = Instance::new(num_cons, num_vars, num_inputs, &A, &B, &C);
+    assert!(inst.is_err());
+    assert_eq!(inst.err(), Some(R1CSError::InvalidScalar));
+  }
+
+  #[test]
+  fn test_padded_constraints() {
+    // parameters of the R1CS instance
+    let num_cons = 1;
+    let num_vars = 0;
+    let num_inputs = 3;
+    let num_non_zero_entries = 3;
+
+    // We will encode the above constraints into three matrices, where
+    // the coefficients in the matrix are in the little-endian byte order
+    let mut A: Vec<(usize, usize, [u8; 32])> = Vec::new();
+    let mut B: Vec<(usize, usize, [u8; 32])> = Vec::new();
+    let mut C: Vec<(usize, usize, [u8; 32])> = Vec::new();
+
+    // Create a^2 + b + 13
+    A.push((0, num_vars + 2, Scalar::one().to_bytes())); // 1*a
+    B.push((0, num_vars + 2, Scalar::one().to_bytes())); // 1*a
+    C.push((0, num_vars + 1, Scalar::one().to_bytes())); // 1*z
+    C.push((0, num_vars, (-Scalar::from(13u64)).to_bytes())); // -13*1
+    C.push((0, num_vars + 3, (-Scalar::one()).to_bytes())); // -1*b
+
+    // Var Assignments (Z_0 = 16 is the only output)
+    let vars = vec![Scalar::zero().to_bytes(); num_vars];
+
+    // create an InputsAssignment (a = 1, b = 2)
+    let mut inputs = vec![Scalar::zero().to_bytes(); num_inputs];
+    inputs[0] = Scalar::from(16u64).to_bytes();
+    inputs[1] = Scalar::from(1u64).to_bytes();
+    inputs[2] = Scalar::from(2u64).to_bytes();
+
+    let assignment_inputs = InputsAssignment::new(&inputs).unwrap();
+    let assignment_vars = VarsAssignment::new(&vars).unwrap();
+
+    // Check if instance is satisfiable
+    let inst = Instance::new(num_cons, num_vars, num_inputs, &A, &B, &C).unwrap();
+    let res = inst.is_sat(&assignment_vars, &assignment_inputs);
+    assert!(res.unwrap(), "should be satisfied");
+
+    // SNARK public params
+    let gens = SNARKGens::new(num_cons, num_vars, num_inputs, num_non_zero_entries);
+
+    // create a commitment to the R1CS instance
+    let (comm, decomm) = SNARK::encode(&inst, &gens);
+
+    // produce a SNARK
+    let mut prover_transcript = Transcript::new(b"snark_example");
+    let proof = SNARK::prove(
+      &inst,
+      &comm,
+      &decomm,
+      assignment_vars.clone(),
+      &assignment_inputs,
+      &gens,
+      &mut prover_transcript,
+    );
+
+    // verify the SNARK
+    let mut verifier_transcript = Transcript::new(b"snark_example");
+    assert!(proof
+      .verify(&comm, &assignment_inputs, &mut verifier_transcript, &gens)
+      .is_ok());
+
+    // NIZK public params
+    let gens = NIZKGens::new(num_cons, num_vars, num_inputs);
+
+    // produce a NIZK
+    let mut prover_transcript = Transcript::new(b"nizk_example");
+    let proof = NIZK::prove(
+      &inst,
+      assignment_vars,
+      &assignment_inputs,
+      &gens,
+      &mut prover_transcript,
+    );
+
+    // verify the NIZK
+    let mut verifier_transcript = Transcript::new(b"nizk_example");
+    assert!(proof
+      .verify(&inst, &assignment_inputs, &mut verifier_transcript, &gens)
+      .is_ok());
+  }
+}

packages/Spartan-secq/src/math.rs

@@ -0,0 +1,36 @@
+pub trait Math {
+  fn square_root(self) -> usize;
+  fn pow2(self) -> usize;
+  fn get_bits(self, num_bits: usize) -> Vec<bool>;
+  fn log_2(self) -> usize;
+}
+
+impl Math for usize {
+  #[inline]
+  fn square_root(self) -> usize {
+    (self as f64).sqrt() as usize
+  }
+
+  #[inline]
+  fn pow2(self) -> usize {
+    let base: usize = 2;
+    base.pow(self as u32)
+  }
+
+  /// Returns the num_bits from n in a canonical order
+  fn get_bits(self, num_bits: usize) -> Vec<bool> {
+    (0..num_bits)
+      .map(|shift_amount| ((self & (1 << (num_bits - shift_amount - 1))) > 0))
+      .collect::<Vec<bool>>()
+  }
+
+  fn log_2(self) -> usize {
+    assert_ne!(self, 0);
+
+    if self.is_power_of_two() {
+      (1usize.leading_zeros() - self.leading_zeros()) as usize
+    } else {
+      (0usize.leading_zeros() - self.leading_zeros()) as usize
+    }
+  }
+}

packages/Spartan-secq/src/nizk/bullet.rs

@@ -0,0 +1,267 @@
+//! This module is an adaptation of code from the bulletproofs crate.
+//! See NOTICE.md for more details
+#![allow(non_snake_case)]
+#![allow(clippy::type_complexity)]
+#![allow(clippy::too_many_arguments)]
+use super::super::errors::ProofVerifyError;
+use super::super::group::{CompressedGroup, GroupElement, VartimeMultiscalarMul};
+use super::super::math::Math;
+use super::super::scalar::Scalar;
+use super::super::transcript::ProofTranscript;
+use crate::group::DecompressEncodedPoint;
+use core::iter;
+use merlin::Transcript;
+use serde::{Deserialize, Serialize};
+
+#[derive(Debug, Serialize, Deserialize)]
+pub struct BulletReductionProof {
+  L_vec: Vec<CompressedGroup>,
+  R_vec: Vec<CompressedGroup>,
+}
+
+impl BulletReductionProof {
+  /// Create an inner-product proof.
+  ///
+  /// The proof is created with respect to the bases \\(G\\).
+  ///
+  /// The `transcript` is passed in as a parameter so that the
+  /// challenges depend on the *entire* transcript (including parent
+  /// protocols).
+  ///
+  /// The lengths of the vectors must all be the same, and must all be
+  /// either 0 or a power of 2.
+  pub fn prove(
+    transcript: &mut Transcript,
+    Q: &GroupElement,
+    G_vec: &[GroupElement],
+    H: &GroupElement,
+    a_vec: &[Scalar],
+    b_vec: &[Scalar],
+    blind: &Scalar,
+    blinds_vec: &[(Scalar, Scalar)],
+  ) -> (
+    BulletReductionProof,
+    GroupElement,
+    Scalar,
+    Scalar,
+    GroupElement,
+    Scalar,
+  ) {
+    // Create slices G, H, a, b backed by their respective
+    // vectors.  This lets us reslice as we compress the lengths
+    // of the vectors in the main loop below.
+    let mut G = &mut G_vec.to_owned()[..];
+    let mut a = &mut a_vec.to_owned()[..];
+    let mut b = &mut b_vec.to_owned()[..];
+
+    // All of the input vectors must have a length that is a power of two.
+    let mut n = G.len();
+    assert!(n.is_power_of_two());
+    let lg_n = n.log_2();
+
+    // All of the input vectors must have the same length.
+    assert_eq!(G.len(), n);
+    assert_eq!(a.len(), n);
+    assert_eq!(b.len(), n);
+    assert_eq!(blinds_vec.len(), 2 * lg_n);
+
+    let mut L_vec = Vec::with_capacity(lg_n);
+    let mut R_vec = Vec::with_capacity(lg_n);
+    let mut blinds_iter = blinds_vec.iter();
+    let mut blind_fin = *blind;
+
+    while n != 1 {
+      n /= 2;
+      let (a_L, a_R) = a.split_at_mut(n);
+      let (b_L, b_R) = b.split_at_mut(n);
+      let (G_L, G_R) = G.split_at_mut(n);
+
+      let c_L = inner_product(a_L, b_R);
+      let c_R = inner_product(a_R, b_L);
+
+      let (blind_L, blind_R) = blinds_iter.next().unwrap();
+
+      let L = GroupElement::vartime_multiscalar_mul(
+        a_L
+          .iter()
+          .chain(iter::once(&c_L))
+          .chain(iter::once(blind_L))
+          .map(|s| *s)
+          .collect(),
+        G_R
+          .iter()
+          .chain(iter::once(Q))
+          .chain(iter::once(H))
+          .map(|s| *s)
+          .collect(),
+      );
+
+      let R = GroupElement::vartime_multiscalar_mul(
+        a_R
+          .iter()
+          .chain(iter::once(&c_R))
+          .chain(iter::once(blind_R))
+          .map(|s| *s)
+          .collect(),
+        G_L
+          .iter()
+          .chain(iter::once(Q))
+          .chain(iter::once(H))
+          .map(|s| *s)
+          .collect(),
+      );
+
+      transcript.append_point(b"L", &L.compress());
+      transcript.append_point(b"R", &R.compress());
+
+      let u = transcript.challenge_scalar(b"u");
+      let u_inv = u.invert().unwrap();
+
+      for i in 0..n {
+        a_L[i] = a_L[i] * u + u_inv * a_R[i];
+        b_L[i] = b_L[i] * u_inv + u * b_R[i];
+        G_L[i] =
+          GroupElement::vartime_multiscalar_mul([u_inv, u].to_vec(), [G_L[i], G_R[i]].to_vec());
+      }
+
+      blind_fin = blind_fin + blind_L * u * u + blind_R * u_inv * u_inv;
+
+      L_vec.push(L.compress());
+      R_vec.push(R.compress());
+
+      a = a_L;
+      b = b_L;
+      G = G_L;
+    }
+
+    let Gamma_hat = GroupElement::vartime_multiscalar_mul(
+      [a[0], a[0] * b[0], blind_fin].to_vec(),
+      [G[0], *Q, *H].to_vec(),
+    );
+
+    (
+      BulletReductionProof { L_vec, R_vec },
+      Gamma_hat,
+      a[0],
+      b[0],
+      G[0],
+      blind_fin,
+    )
+  }
+
+  /// Computes three vectors of verification scalars \\([u\_{i}^{2}]\\), \\([u\_{i}^{-2}]\\) and \\([s\_{i}]\\) for combined multiscalar multiplication
+  /// in a parent protocol. See [inner product protocol notes](index.html#verification-equation) for details.
+  /// The verifier must provide the input length \\(n\\) explicitly to avoid unbounded allocation within the inner product proof.
+  fn verification_scalars(
+    &self,
+    n: usize,
+    transcript: &mut Transcript,
+  ) -> Result<(Vec<Scalar>, Vec<Scalar>, Vec<Scalar>), ProofVerifyError> {
+    let lg_n = self.L_vec.len();
+    if lg_n >= 32 {
+      // 4 billion multiplications should be enough for anyone
+      // and this check prevents overflow in 1<<lg_n below.
+      return Err(ProofVerifyError::InternalError);
+    }
+    if n != (1 << lg_n) {
+      return Err(ProofVerifyError::InternalError);
+    }
+
+    // 1. Recompute x_k,...,x_1 based on the proof transcript
+    let mut challenges = Vec::with_capacity(lg_n);
+    for (L, R) in self.L_vec.iter().zip(self.R_vec.iter()) {
+      transcript.append_point(b"L", L);
+      transcript.append_point(b"R", R);
+      challenges.push(transcript.challenge_scalar(b"u"));
+    }
+
+    // 2. Compute 1/(u_k...u_1) and 1/u_k, ..., 1/u_1
+    let mut challenges_inv = challenges.clone();
+    let allinv = Scalar::batch_invert(&mut challenges_inv);
+
+    // 3. Compute u_i^2 and (1/u_i)^2
+    for i in 0..lg_n {
+      challenges[i] = challenges[i].square();
+      challenges_inv[i] = challenges_inv[i].square();
+    }
+    let challenges_sq = challenges;
+    let challenges_inv_sq = challenges_inv;
+
+    // 4. Compute s values inductively.
+    let mut s = Vec::with_capacity(n);
+    s.push(allinv);
+    for i in 1..n {
+      let lg_i = (32 - 1 - (i as u32).leading_zeros()) as usize;
+      let k = 1 << lg_i;
+      // The challenges are stored in "creation order" as [u_k,...,u_1],
+      // so u_{lg(i)+1} = is indexed by (lg_n-1) - lg_i
+      let u_lg_i_sq = challenges_sq[(lg_n - 1) - lg_i];
+      s.push(s[i - k] * u_lg_i_sq);
+    }
+
+    Ok((challenges_sq, challenges_inv_sq, s))
+  }
+
+  /// This method is for testing that proof generation work,
+  /// but for efficiency the actual protocols would use `verification_scalars`
+  /// method to combine inner product verification with other checks
+  /// in a single multiscalar multiplication.
+  pub fn verify(
+    &self,
+    n: usize,
+    a: &[Scalar],
+    transcript: &mut Transcript,
+    Gamma: &GroupElement,
+    G: &[GroupElement],
+  ) -> Result<(GroupElement, GroupElement, Scalar), ProofVerifyError> {
+    let (u_sq, u_inv_sq, s) = self.verification_scalars(n, transcript)?;
+
+    let Ls = self
+      .L_vec
+      .iter()
+      .map(|p| p.decompress().ok_or(ProofVerifyError::InternalError))
+      .collect::<Result<Vec<_>, _>>()?;
+
+    let Rs = self
+      .R_vec
+      .iter()
+      .map(|p| p.decompress().ok_or(ProofVerifyError::InternalError))
+      .collect::<Result<Vec<_>, _>>()?;
+
+    let G_hat = GroupElement::vartime_multiscalar_mul(s.clone(), G.to_vec());
+    let a_hat = inner_product(a, &s);
+
+    let Gamma_hat = GroupElement::vartime_multiscalar_mul(
+      u_sq
+        .iter()
+        .chain(u_inv_sq.iter())
+        .chain(iter::once(&Scalar::one()))
+        .map(|s| *s)
+        .collect(),
+      Ls.iter()
+        .chain(Rs.iter())
+        .chain(iter::once(Gamma))
+        .map(|p| *p)
+        .collect(),
+    );
+
+    Ok((G_hat, Gamma_hat, a_hat))
+  }
+}
+
+/// Computes an inner product of two vectors
+/// \\[
+///    {\langle {\mathbf{a}}, {\mathbf{b}} \rangle} = \sum\_{i=0}^{n-1} a\_i \cdot b\_i.
+/// \\]
+/// Panics if the lengths of \\(\mathbf{a}\\) and \\(\mathbf{b}\\) are not equal.
+pub fn inner_product(a: &[Scalar], b: &[Scalar]) -> Scalar {
+  assert!(
+    a.len() == b.len(),
+    "inner_product(a,b): lengths of vectors do not match"
+  );
+  let mut out = Scalar::zero();
+  for i in 0..a.len() {
+    out += a[i] * b[i];
+  }
+  out
+}

packages/Spartan-secq/src/nizk/mod.rs

@@ -0,0 +1,725 @@
+#![allow(clippy::too_many_arguments)]
+use super::commitments::{Commitments, MultiCommitGens};
+use super::errors::ProofVerifyError;
+use super::group::{CompressedGroup, CompressedGroupExt};
+use super::math::Math;
+use super::random::RandomTape;
+use super::scalar::Scalar;
+use super::transcript::{AppendToTranscript, ProofTranscript};
+use crate::group::DecompressEncodedPoint;
+use merlin::Transcript;
+use serde::{Deserialize, Serialize};
+
+mod bullet;
+use bullet::BulletReductionProof;
+
+#[derive(Serialize, Deserialize, Debug)]
+pub struct KnowledgeProof {
+  alpha: CompressedGroup,
+  z1: Scalar,
+  z2: Scalar,
+}
+
+impl KnowledgeProof {
+  fn protocol_name() -> &'static [u8] {
+    b"knowledge proof"
+  }
+
+  pub fn prove(
+    gens_n: &MultiCommitGens,
+    transcript: &mut Transcript,
+    random_tape: &mut RandomTape,
+    x: &Scalar,
+    r: &Scalar,
+  ) -> (KnowledgeProof, CompressedGroup) {
+    transcript.append_protocol_name(KnowledgeProof::protocol_name());
+
+    // produce two random Scalars
+    let t1 = random_tape.random_scalar(b"t1");
+    let t2 = random_tape.random_scalar(b"t2");
+
+    let C = x.commit(r, gens_n).compress();
+    C.append_to_transcript(b"C", transcript);
+
+    let alpha = t1.commit(&t2, gens_n).compress();
+    alpha.append_to_transcript(b"alpha", transcript);
+
+    let c = transcript.challenge_scalar(b"c");
+
+    let z1 = x * c + t1;
+    let z2 = r * c + t2;
+
+    (KnowledgeProof { alpha, z1, z2 }, C)
+  }
+
+  pub fn verify(
+    &self,
+    gens_n: &MultiCommitGens,
+    transcript: &mut Transcript,
+    C: &CompressedGroup,
+  ) -> Result<(), ProofVerifyError> {
+    transcript.append_protocol_name(KnowledgeProof::protocol_name());
+    C.append_to_transcript(b"C", transcript);
+    self.alpha.append_to_transcript(b"alpha", transcript);
+
+    let c = transcript.challenge_scalar(b"c");
+
+    let lhs = self.z1.commit(&self.z2, gens_n).compress();
+    let rhs = (c * C.unpack()? + self.alpha.unpack()?).compress();
+
+    if lhs == rhs {
+      Ok(())
+    } else {
+      Err(ProofVerifyError::InternalError)
+    }
+  }
+}
+
+#[derive(Serialize, Deserialize, Debug)]
+pub struct EqualityProof {
+  alpha: CompressedGroup,
+  z: Scalar,
+}
+
+impl EqualityProof {
+  fn protocol_name() -> &'static [u8] {
+    b"equality proof"
+  }
+
+  pub fn prove(
+    gens_n: &MultiCommitGens,
+    transcript: &mut Transcript,
+    random_tape: &mut RandomTape,
+    v1: &Scalar,
+    s1: &Scalar,
+    v2: &Scalar,
+    s2: &Scalar,
+  ) -> (EqualityProof, CompressedGroup, CompressedGroup) {
+    transcript.append_protocol_name(EqualityProof::protocol_name());
+
+    // produce a random Scalar
+    let r = random_tape.random_scalar(b"r");
+
+    let C1 = v1.commit(s1, gens_n).compress();
+    C1.append_to_transcript(b"C1", transcript);
+
+    let C2 = v2.commit(s2, gens_n).compress();
+    C2.append_to_transcript(b"C2", transcript);
+
+    let alpha = (r * gens_n.h).compress();
+    alpha.append_to_transcript(b"alpha", transcript);
+
+    let c = transcript.challenge_scalar(b"c");
+
+    let z = c * (s1 - s2) + r;
+
+    (EqualityProof { alpha, z }, C1, C2)
+  }
+
+  pub fn verify(
+    &self,
+    gens_n: &MultiCommitGens,
+    transcript: &mut Transcript,
+    C1: &CompressedGroup,
+    C2: &CompressedGroup,
+  ) -> Result<(), ProofVerifyError> {
+    transcript.append_protocol_name(EqualityProof::protocol_name());
+    C1.append_to_transcript(b"C1", transcript);
+    C2.append_to_transcript(b"C2", transcript);
+    self.alpha.append_to_transcript(b"alpha", transcript);
+
+    let c = transcript.challenge_scalar(b"c");
+    let rhs = {
+      let C = C1.unpack()? - C2.unpack()?;
+      (c * C + self.alpha.unpack()?).compress()
+    };
+
+    let lhs = (self.z * gens_n.h).compress();
+
+    if lhs == rhs {
+      Ok(())
+    } else {
+      Err(ProofVerifyError::InternalError)
+    }
+  }
+}
+
+#[derive(Serialize, Deserialize, Debug)]
+pub struct ProductProof {
+  alpha: CompressedGroup,
+  beta: CompressedGroup,
+  delta: CompressedGroup,
+  z: [Scalar; 5],
+}
+
+impl ProductProof {
+  fn protocol_name() -> &'static [u8] {
+    b"product proof"
+  }
+
+  pub fn prove(
+    gens_n: &MultiCommitGens,
+    transcript: &mut Transcript,
+    random_tape: &mut RandomTape,
+    x: &Scalar,
+    rX: &Scalar,
+    y: &Scalar,
+    rY: &Scalar,
+    z: &Scalar,
+    rZ: &Scalar,
+  ) -> (
+    ProductProof,
+    CompressedGroup,
+    CompressedGroup,
+    CompressedGroup,
+  ) {
+    transcript.append_protocol_name(ProductProof::protocol_name());
+
+    // produce five random Scalar
+    let b1 = random_tape.random_scalar(b"b1");
+    let b2 = random_tape.random_scalar(b"b2");
+    let b3 = random_tape.random_scalar(b"b3");
+    let b4 = random_tape.random_scalar(b"b4");
+    let b5 = random_tape.random_scalar(b"b5");
+
+    let X = x.commit(rX, gens_n).compress();
+    X.append_to_transcript(b"X", transcript);
+
+    let Y = y.commit(rY, gens_n).compress();
+    Y.append_to_transcript(b"Y", transcript);
+
+    let Z = z.commit(rZ, gens_n).compress();
+    Z.append_to_transcript(b"Z", transcript);
+
+    let alpha = b1.commit(&b2, gens_n).compress();
+    alpha.append_to_transcript(b"alpha", transcript);
+
+    let beta = b3.commit(&b4, gens_n).compress();
+    beta.append_to_transcript(b"beta", transcript);
+
+    let delta = {
+      let gens_X = &MultiCommitGens {
+        n: 1,
+        G: vec![X.decompress().unwrap()],
+        h: gens_n.h,
+      };
+      b3.commit(&b5, gens_X).compress()
+    };
+    delta.append_to_transcript(b"delta", transcript);
+
+    let c = transcript.challenge_scalar(b"c");
+
+    let z1 = b1 + c * x;
+    let z2 = b2 + c * rX;
+    let z3 = b3 + c * y;
+    let z4 = b4 + c * rY;
+    let z5 = b5 + c * (rZ - rX * y);
+    let z = [z1, z2, z3, z4, z5];
+
+    (
+      ProductProof {
+        alpha,
+        beta,
+        delta,
+        z,
+      },
+      X,
+      Y,
+      Z,
+    )
+  }
+
+  fn check_equality(
+    P: &CompressedGroup,
+    X: &CompressedGroup,
+    c: &Scalar,
+    gens_n: &MultiCommitGens,
+    z1: &Scalar,
+    z2: &Scalar,
+  ) -> bool {
+    let lhs = (P.decompress().unwrap() + c * X.decompress().unwrap()).compress();
+    let rhs = z1.commit(z2, gens_n).compress();
+
+    lhs == rhs
+  }
+
+  pub fn verify(
+    &self,
+    gens_n: &MultiCommitGens,
+    transcript: &mut Transcript,
+    X: &CompressedGroup,
+    Y: &CompressedGroup,
+    Z: &CompressedGroup,
+  ) -> Result<(), ProofVerifyError> {
+    transcript.append_protocol_name(ProductProof::protocol_name());
+
+    X.append_to_transcript(b"X", transcript);
+    Y.append_to_transcript(b"Y", transcript);
+    Z.append_to_transcript(b"Z", transcript);
+    self.alpha.append_to_transcript(b"alpha", transcript);
+    self.beta.append_to_transcript(b"beta", transcript);
+    self.delta.append_to_transcript(b"delta", transcript);
+
+    let z1 = self.z[0];
+    let z2 = self.z[1];
+    let z3 = self.z[2];
+    let z4 = self.z[3];
+    let z5 = self.z[4];
+
+    let c = transcript.challenge_scalar(b"c");
+
+    if ProductProof::check_equality(&self.alpha, X, &c, gens_n, &z1, &z2)
+      && ProductProof::check_equality(&self.beta, Y, &c, gens_n, &z3, &z4)
+      && ProductProof::check_equality(
+        &self.delta,
+        Z,
+        &c,
+        &MultiCommitGens {
+          n: 1,
+          G: vec![X.unpack()?],
+          h: gens_n.h,
+        },
+        &z3,
+        &z5,
+      )
+    {
+      Ok(())
+    } else {
+      Err(ProofVerifyError::InternalError)
+    }
+  }
+}
+
+#[derive(Debug, Serialize, Deserialize)]
+pub struct DotProductProof {
+  delta: CompressedGroup,
+  beta: CompressedGroup,
+  z: Vec<Scalar>,
+  z_delta: Scalar,
+  z_beta: Scalar,
+}
+
+impl DotProductProof {
+  fn protocol_name() -> &'static [u8] {
+    b"dot product proof"
+  }
+
+  pub fn compute_dotproduct(a: &[Scalar], b: &[Scalar]) -> Scalar {
+    assert_eq!(a.len(), b.len());
+    (0..a.len()).map(|i| a[i] * b[i]).sum()
+  }
+
+  pub fn prove(
+    gens_1: &MultiCommitGens,
+    gens_n: &MultiCommitGens,
+    transcript: &mut Transcript,
+    random_tape: &mut RandomTape,
+    x_vec: &[Scalar],
+    blind_x: &Scalar,
+    a_vec: &[Scalar],
+    y: &Scalar,
+    blind_y: &Scalar,
+  ) -> (DotProductProof, CompressedGroup, CompressedGroup) {
+    transcript.append_protocol_name(DotProductProof::protocol_name());
+
+    let n = x_vec.len();
+    assert_eq!(x_vec.len(), a_vec.len());
+    assert_eq!(gens_n.n, a_vec.len());
+    assert_eq!(gens_1.n, 1);
+
+    // produce randomness for the proofs
+    let d_vec = random_tape.random_vector(b"d_vec", n);
+    let r_delta = random_tape.random_scalar(b"r_delta");
+    let r_beta = random_tape.random_scalar(b"r_beta");
+
+    let Cx = x_vec.commit(blind_x, gens_n).compress();
+    Cx.append_to_transcript(b"Cx", transcript);
+
+    let Cy = y.commit(blind_y, gens_1).compress();
+    Cy.append_to_transcript(b"Cy", transcript);
+
+    a_vec.append_to_transcript(b"a", transcript);
+
+    let delta = d_vec.commit(&r_delta, gens_n).compress();
+    delta.append_to_transcript(b"delta", transcript);
+
+    let dotproduct_a_d = DotProductProof::compute_dotproduct(a_vec, &d_vec);
+
+    let beta = dotproduct_a_d.commit(&r_beta, gens_1).compress();
+    beta.append_to_transcript(b"beta", transcript);
+
+    let c = transcript.challenge_scalar(b"c");
+
+    let z = (0..d_vec.len())
+      .map(|i| c * x_vec[i] + d_vec[i])
+      .collect::<Vec<Scalar>>();
+
+    let z_delta = c * blind_x + r_delta;
+    let z_beta = c * blind_y + r_beta;
+
+    (
+      DotProductProof {
+        delta,
+        beta,
+        z,
+        z_delta,
+        z_beta,
+      },
+      Cx,
+      Cy,
+    )
+  }
+
+  pub fn verify(
+    &self,
+    gens_1: &MultiCommitGens,
+    gens_n: &MultiCommitGens,
+    transcript: &mut Transcript,
+    a: &[Scalar],
+    Cx: &CompressedGroup,
+    Cy: &CompressedGroup,
+  ) -> Result<(), ProofVerifyError> {
+    assert_eq!(gens_n.n, a.len());
+    assert_eq!(gens_1.n, 1);
+
+    transcript.append_protocol_name(DotProductProof::protocol_name());
+    Cx.append_to_transcript(b"Cx", transcript);
+    Cy.append_to_transcript(b"Cy", transcript);
+    a.append_to_transcript(b"a", transcript);
+    self.delta.append_to_transcript(b"delta", transcript);
+    self.beta.append_to_transcript(b"beta", transcript);
+
+    let c = transcript.challenge_scalar(b"c");
+
+    let mut result =
+      c * Cx.unpack()? + self.delta.unpack()? == self.z.commit(&self.z_delta, gens_n);
+
+    let dotproduct_z_a = DotProductProof::compute_dotproduct(&self.z, a);
+    result &= c * Cy.unpack()? + self.beta.unpack()? == dotproduct_z_a.commit(&self.z_beta, gens_1);
+
+    if result {
+      Ok(())
+    } else {
+      Err(ProofVerifyError::InternalError)
+    }
+  }
+}
+
+pub struct DotProductProofGens {
+  n: usize,
+  pub gens_n: MultiCommitGens,
+  pub gens_1: MultiCommitGens,
+}
+
+impl DotProductProofGens {
+  pub fn new(n: usize, label: &[u8]) -> Self {
+    let (gens_n, gens_1) = MultiCommitGens::new(n + 1, label).split_at(n);
+    DotProductProofGens { n, gens_n, gens_1 }
+  }
+}
+
+#[derive(Debug, Serialize, Deserialize)]
+pub struct DotProductProofLog {
+  bullet_reduction_proof: BulletReductionProof,
+  delta: CompressedGroup,
+  beta: CompressedGroup,
+  z1: Scalar,
+  z2: Scalar,
+}
+
+impl DotProductProofLog {
+  fn protocol_name() -> &'static [u8] {
+    b"dot product proof (log)"
+  }
+
+  pub fn compute_dotproduct(a: &[Scalar], b: &[Scalar]) -> Scalar {
+    assert_eq!(a.len(), b.len());
+    (0..a.len()).map(|i| a[i] * b[i]).sum()
+  }
+
+  pub fn prove(
+    gens: &DotProductProofGens,
+    transcript: &mut Transcript,
+    random_tape: &mut RandomTape,
+    x_vec: &[Scalar],
+    blind_x: &Scalar,
+    a_vec: &[Scalar],
+    y: &Scalar,
+    blind_y: &Scalar,
+  ) -> (DotProductProofLog, CompressedGroup, CompressedGroup) {
+    transcript.append_protocol_name(DotProductProofLog::protocol_name());
+
+    let n = x_vec.len();
+    assert_eq!(x_vec.len(), a_vec.len());
+    assert_eq!(gens.n, n);
+
+    // produce randomness for generating a proof
+    let d = random_tape.random_scalar(b"d");
+    let r_delta = random_tape.random_scalar(b"r_delta");
+    let r_beta = random_tape.random_scalar(b"r_delta");
+    let blinds_vec = {
+      let v1 = random_tape.random_vector(b"blinds_vec_1", 2 * n.log_2());
+      let v2 = random_tape.random_vector(b"blinds_vec_2", 2 * n.log_2());
+      (0..v1.len())
+        .map(|i| (v1[i], v2[i]))
+        .collect::<Vec<(Scalar, Scalar)>>()
+    };
+
+    let Cx = x_vec.commit(blind_x, &gens.gens_n).compress();
+    Cx.append_to_transcript(b"Cx", transcript);
+
+    let Cy = y.commit(blind_y, &gens.gens_1).compress();
+    Cy.append_to_transcript(b"Cy", transcript);
+
+    a_vec.append_to_transcript(b"a", transcript);
+
+    let blind_Gamma = blind_x + blind_y;
+    let (bullet_reduction_proof, _Gamma_hat, x_hat, a_hat, g_hat, rhat_Gamma) =
+      BulletReductionProof::prove(
+        transcript,
+        &gens.gens_1.G[0],
+        &gens.gens_n.G,
+        &gens.gens_n.h,
+        x_vec,
+        a_vec,
+        &blind_Gamma,
+        &blinds_vec,
+      );
+    let y_hat = x_hat * a_hat;
+
+    let delta = {
+      let gens_hat = MultiCommitGens {
+        n: 1,
+        G: vec![g_hat],
+        h: gens.gens_1.h,
+      };
+      d.commit(&r_delta, &gens_hat).compress()
+    };
+    delta.append_to_transcript(b"delta", transcript);
+
+    let beta = d.commit(&r_beta, &gens.gens_1).compress();
+    beta.append_to_transcript(b"beta", transcript);
+
+    let c = transcript.challenge_scalar(b"c");
+
+    let z1 = d + c * y_hat;
+    let z2 = a_hat * (c * rhat_Gamma + r_beta) + r_delta;
+
+    (
+      DotProductProofLog {
+        bullet_reduction_proof,
+        delta,
+        beta,
+        z1,
+        z2,
+      },
+      Cx,
+      Cy,
+    )
+  }
+
+  pub fn verify(
+    &self,
+    n: usize,
+    gens: &DotProductProofGens,
+    transcript: &mut Transcript,
+    a: &[Scalar],
+    Cx: &CompressedGroup,
+    Cy: &CompressedGroup,
+  ) -> Result<(), ProofVerifyError> {
+    assert_eq!(gens.n, n);
+    assert_eq!(a.len(), n);
+
+    transcript.append_protocol_name(DotProductProofLog::protocol_name());
+    Cx.append_to_transcript(b"Cx", transcript);
+    Cy.append_to_transcript(b"Cy", transcript);
+    a.append_to_transcript(b"a", transcript);
+
+    let Gamma = Cx.unpack()? + Cy.unpack()?;
+
+    let (g_hat, Gamma_hat, a_hat) =
+      self
+        .bullet_reduction_proof
+        .verify(n, a, transcript, &Gamma, &gens.gens_n.G)?;
+    self.delta.append_to_transcript(b"delta", transcript);
+    self.beta.append_to_transcript(b"beta", transcript);
+
+    let c = transcript.challenge_scalar(b"c");
+
+    let c_s = &c;
+    let beta_s = self.beta.unpack()?;
+    let a_hat_s = &a_hat;
+    let delta_s = self.delta.unpack()?;
+    let z1_s = &self.z1;
+    let z2_s = &self.z2;
+
+    let lhs = ((Gamma_hat * c_s + beta_s) * a_hat_s + delta_s).compress();
+    let rhs = ((g_hat + gens.gens_1.G[0] * a_hat_s) * z1_s + gens.gens_1.h * z2_s).compress();
+
+    assert_eq!(lhs, rhs);
+
+    if lhs == rhs {
+      Ok(())
+    } else {
+      Err(ProofVerifyError::InternalError)
+    }
+  }
+}
+
+#[cfg(test)]
+mod tests {
+  use super::*;
+  use rand_core::OsRng;
+  #[test]
+  fn check_knowledgeproof() {
+    let mut csprng: OsRng = OsRng;
+
+    let gens_1 = MultiCommitGens::new(1, b"test-knowledgeproof");
+
+    let x = Scalar::random(&mut csprng);
+    let r = Scalar::random(&mut csprng);
+
+    let mut random_tape = RandomTape::new(b"proof");
+    let mut prover_transcript = Transcript::new(b"example");
+    let (proof, committed_value) =
+      KnowledgeProof::prove(&gens_1, &mut prover_transcript, &mut random_tape, &x, &r);
+
+    let mut verifier_transcript = Transcript::new(b"example");
+    assert!(proof
+      .verify(&gens_1, &mut verifier_transcript, &committed_value)
+      .is_ok());
+  }
+
+  #[test]
+  fn check_equalityproof() {
+    let mut csprng: OsRng = OsRng;
+
+    let gens_1 = MultiCommitGens::new(1, b"test-equalityproof");
+    let v1 = Scalar::random(&mut csprng);
+    let v2 = v1;
+    let s1 = Scalar::random(&mut csprng);
+    let s2 = Scalar::random(&mut csprng);
+
+    let mut random_tape = RandomTape::new(b"proof");
+    let mut prover_transcript = Transcript::new(b"example");
+    let (proof, C1, C2) = EqualityProof::prove(
+      &gens_1,
+      &mut prover_transcript,
+      &mut random_tape,
+      &v1,
+      &s1,
+      &v2,
+      &s2,
+    );
+
+    let mut verifier_transcript = Transcript::new(b"example");
+    assert!(proof
+      .verify(&gens_1, &mut verifier_transcript, &C1, &C2)
+      .is_ok());
+  }
+
+  #[test]
+  fn check_productproof() {
+    let mut csprng: OsRng = OsRng;
+
+    let gens_1 = MultiCommitGens::new(1, b"test-productproof");
+    let x = Scalar::random(&mut csprng);
+    let rX = Scalar::random(&mut csprng);
+    let y = Scalar::random(&mut csprng);
+    let rY = Scalar::random(&mut csprng);
+    let z = x * y;
+    let rZ = Scalar::random(&mut csprng);
+
+    let mut random_tape = RandomTape::new(b"proof");
+    let mut prover_transcript = Transcript::new(b"example");
+    let (proof, X, Y, Z) = ProductProof::prove(
+      &gens_1,
+      &mut prover_transcript,
+      &mut random_tape,
+      &x,
+      &rX,
+      &y,
+      &rY,
+      &z,
+      &rZ,
+    );
+
+    let mut verifier_transcript = Transcript::new(b"example");
+    assert!(proof
+      .verify(&gens_1, &mut verifier_transcript, &X, &Y, &Z)
+      .is_ok());
+  }
+
+  #[test]
+  fn check_dotproductproof() {
+    let mut csprng: OsRng = OsRng;
+
+    let n = 1024;
+
+    let gens_1 = MultiCommitGens::new(1, b"test-two");
+    let gens_1024 = MultiCommitGens::new(n, b"test-1024");
+
+    let mut x: Vec<Scalar> = Vec::new();
+    let mut a: Vec<Scalar> = Vec::new();
+    for _ in 0..n {
+      x.push(Scalar::random(&mut csprng));
+      a.push(Scalar::random(&mut csprng));
+    }
+    let y = DotProductProofLog::compute_dotproduct(&x, &a);
+    let r_x = Scalar::random(&mut csprng);
+    let r_y = Scalar::random(&mut csprng);
+
+    let mut random_tape = RandomTape::new(b"proof");
+    let mut prover_transcript = Transcript::new(b"example");
+    let (proof, Cx, Cy) = DotProductProof::prove(
+      &gens_1,
+      &gens_1024,
+      &mut prover_transcript,
+      &mut random_tape,
+      &x,
+      &r_x,
+      &a,
+      &y,
+      &r_y,
+    );
+
+    let mut verifier_transcript = Transcript::new(b"example");
+    assert!(proof
+      .verify(&gens_1, &gens_1024, &mut verifier_transcript, &a, &Cx, &Cy)
+      .is_ok());
+  }
+
+  #[test]
+  fn check_dotproductproof_log() {
+    let mut csprng: OsRng = OsRng;
+
+    let n = 1024;
+
+    let gens = DotProductProofGens::new(n, b"test-1024");
+
+    let x: Vec<Scalar> = (0..n).map(|_i| Scalar::random(&mut csprng)).collect();
+    let a: Vec<Scalar> = (0..n).map(|_i| Scalar::random(&mut csprng)).collect();
+    let y = DotProductProof::compute_dotproduct(&x, &a);
+
+    let r_x = Scalar::random(&mut csprng);
+    let r_y = Scalar::random(&mut csprng);
+
+    let mut random_tape = RandomTape::new(b"proof");
+    let mut prover_transcript = Transcript::new(b"example");
+    let (proof, Cx, Cy) = DotProductProofLog::prove(
+      &gens,
+      &mut prover_transcript,
+      &mut random_tape,
+      &x,
+      &r_x,
+      &a,
+      &y,
+      &r_y,
+    );
+
+    let mut verifier_transcript = Transcript::new(b"example");
+    assert!(proof
+      .verify(n, &gens, &mut verifier_transcript, &a, &Cx, &Cy)
+      .is_ok());
+  }
+}

packages/Spartan-secq/src/product_tree.rs

@@ -0,0 +1,486 @@
+#![allow(dead_code)]
+use super::dense_mlpoly::DensePolynomial;
+use super::dense_mlpoly::EqPolynomial;
+use super::math::Math;
+use super::scalar::Scalar;
+use super::sumcheck::SumcheckInstanceProof;
+use super::transcript::ProofTranscript;
+use merlin::Transcript;
+use serde::{Deserialize, Serialize};
+
+#[derive(Debug)]
+pub struct ProductCircuit {
+  left_vec: Vec<DensePolynomial>,
+  right_vec: Vec<DensePolynomial>,
+}
+
+impl ProductCircuit {
+  fn compute_layer(
+    inp_left: &DensePolynomial,
+    inp_right: &DensePolynomial,
+  ) -> (DensePolynomial, DensePolynomial) {
+    let len = inp_left.len() + inp_right.len();
+    let outp_left = (0..len / 4)
+      .map(|i| inp_left[i] * inp_right[i])
+      .collect::<Vec<Scalar>>();
+    let outp_right = (len / 4..len / 2)
+      .map(|i| inp_left[i] * inp_right[i])
+      .collect::<Vec<Scalar>>();
+
+    (
+      DensePolynomial::new(outp_left),
+      DensePolynomial::new(outp_right),
+    )
+  }
+
+  pub fn new(poly: &DensePolynomial) -> Self {
+    let mut left_vec: Vec<DensePolynomial> = Vec::new();
+    let mut right_vec: Vec<DensePolynomial> = Vec::new();
+
+    let num_layers = poly.len().log_2();
+    let (outp_left, outp_right) = poly.split(poly.len() / 2);
+
+    left_vec.push(outp_left);
+    right_vec.push(outp_right);
+
+    for i in 0..num_layers - 1 {
+      let (outp_left, outp_right) = ProductCircuit::compute_layer(&left_vec[i], &right_vec[i]);
+      left_vec.push(outp_left);
+      right_vec.push(outp_right);
+    }
+
+    ProductCircuit {
+      left_vec,
+      right_vec,
+    }
+  }
+
+  pub fn evaluate(&self) -> Scalar {
+    let len = self.left_vec.len();
+    assert_eq!(self.left_vec[len - 1].get_num_vars(), 0);
+    assert_eq!(self.right_vec[len - 1].get_num_vars(), 0);
+    self.left_vec[len - 1][0] * self.right_vec[len - 1][0]
+  }
+}
+
+pub struct DotProductCircuit {
+  left: DensePolynomial,
+  right: DensePolynomial,
+  weight: DensePolynomial,
+}
+
+impl DotProductCircuit {
+  pub fn new(left: DensePolynomial, right: DensePolynomial, weight: DensePolynomial) -> Self {
+    assert_eq!(left.len(), right.len());
+    assert_eq!(left.len(), weight.len());
+    DotProductCircuit {
+      left,
+      right,
+      weight,
+    }
+  }
+
+  pub fn evaluate(&self) -> Scalar {
+    (0..self.left.len())
+      .map(|i| self.left[i] * self.right[i] * self.weight[i])
+      .sum()
+  }
+
+  pub fn split(&mut self) -> (DotProductCircuit, DotProductCircuit) {
+    let idx = self.left.len() / 2;
+    assert_eq!(idx * 2, self.left.len());
+    let (l1, l2) = self.left.split(idx);
+    let (r1, r2) = self.right.split(idx);
+    let (w1, w2) = self.weight.split(idx);
+    (
+      DotProductCircuit {
+        left: l1,
+        right: r1,
+        weight: w1,
+      },
+      DotProductCircuit {
+        left: l2,
+        right: r2,
+        weight: w2,
+      },
+    )
+  }
+}
+
+#[allow(dead_code)]
+#[derive(Debug, Serialize, Deserialize)]
+pub struct LayerProof {
+  pub proof: SumcheckInstanceProof,
+  pub claims: Vec<Scalar>,
+}
+
+#[allow(dead_code)]
+impl LayerProof {
+  pub fn verify(
+    &self,
+    claim: Scalar,
+    num_rounds: usize,
+    degree_bound: usize,
+    transcript: &mut Transcript,
+  ) -> (Scalar, Vec<Scalar>) {
+    self
+      .proof
+      .verify(claim, num_rounds, degree_bound, transcript)
+      .unwrap()
+  }
+}
+
+#[allow(dead_code)]
+#[derive(Debug, Serialize, Deserialize)]
+pub struct LayerProofBatched {
+  pub proof: SumcheckInstanceProof,
+  pub claims_prod_left: Vec<Scalar>,
+  pub claims_prod_right: Vec<Scalar>,
+}
+
+#[allow(dead_code)]
+impl LayerProofBatched {
+  pub fn verify(
+    &self,
+    claim: Scalar,
+    num_rounds: usize,
+    degree_bound: usize,
+    transcript: &mut Transcript,
+  ) -> (Scalar, Vec<Scalar>) {
+    self
+      .proof
+      .verify(claim, num_rounds, degree_bound, transcript)
+      .unwrap()
+  }
+}
+
+#[derive(Debug, Serialize, Deserialize)]
+pub struct ProductCircuitEvalProof {
+  proof: Vec<LayerProof>,
+}
+
+#[derive(Debug, Serialize, Deserialize)]
+pub struct ProductCircuitEvalProofBatched {
+  proof: Vec<LayerProofBatched>,
+  claims_dotp: (Vec<Scalar>, Vec<Scalar>, Vec<Scalar>),
+}
+
+impl ProductCircuitEvalProof {
+  #![allow(dead_code)]
+  pub fn prove(
+    circuit: &mut ProductCircuit,
+    transcript: &mut Transcript,
+  ) -> (Self, Scalar, Vec<Scalar>) {
+    let mut proof: Vec<LayerProof> = Vec::new();
+    let num_layers = circuit.left_vec.len();
+
+    let mut claim = circuit.evaluate();
+    let mut rand = Vec::new();
+    for layer_id in (0..num_layers).rev() {
+      let len = circuit.left_vec[layer_id].len() + circuit.right_vec[layer_id].len();
+
+      let mut poly_C = DensePolynomial::new(EqPolynomial::new(rand.clone()).evals());
+      assert_eq!(poly_C.len(), len / 2);
+
+      let num_rounds_prod = poly_C.len().log_2();
+      let comb_func_prod = |poly_A_comp: &Scalar,
+                            poly_B_comp: &Scalar,
+                            poly_C_comp: &Scalar|
+       -> Scalar { poly_A_comp * poly_B_comp * poly_C_comp };
+      let (proof_prod, rand_prod, claims_prod) = SumcheckInstanceProof::prove_cubic(
+        &claim,
+        num_rounds_prod,
+        &mut circuit.left_vec[layer_id],
+        &mut circuit.right_vec[layer_id],
+        &mut poly_C,
+        comb_func_prod,
+        transcript,
+      );
+
+      transcript.append_scalar(b"claim_prod_left", &claims_prod[0]);
+      transcript.append_scalar(b"claim_prod_right", &claims_prod[1]);
+
+      // produce a random challenge
+      let r_layer = transcript.challenge_scalar(b"challenge_r_layer");
+      claim = claims_prod[0] + r_layer * (claims_prod[1] - claims_prod[0]);
+
+      let mut ext = vec![r_layer];
+      ext.extend(rand_prod);
+      rand = ext;
+
+      proof.push(LayerProof {
+        proof: proof_prod,
+        claims: claims_prod[0..claims_prod.len() - 1].to_vec(),
+      });
+    }
+
+    (ProductCircuitEvalProof { proof }, claim, rand)
+  }
+
+  pub fn verify(
+    &self,
+    eval: Scalar,
+    len: usize,
+    transcript: &mut Transcript,
+  ) -> (Scalar, Vec<Scalar>) {
+    let num_layers = len.log_2();
+    let mut claim = eval;
+    let mut rand: Vec<Scalar> = Vec::new();
+    //let mut num_rounds = 0;
+    assert_eq!(self.proof.len(), num_layers);
+    for (num_rounds, i) in (0..num_layers).enumerate() {
+      let (claim_last, rand_prod) = self.proof[i].verify(claim, num_rounds, 3, transcript);
+
+      let claims_prod = &self.proof[i].claims;
+      transcript.append_scalar(b"claim_prod_left", &claims_prod[0]);
+      transcript.append_scalar(b"claim_prod_right", &claims_prod[1]);
+
+      assert_eq!(rand.len(), rand_prod.len());
+      let eq: Scalar = (0..rand.len())
+        .map(|i| {
+          rand[i] * rand_prod[i] + (Scalar::one() - rand[i]) * (Scalar::one() - rand_prod[i])
+        })
+        .product();
+      assert_eq!(claims_prod[0] * claims_prod[1] * eq, claim_last);
+
+      // produce a random challenge
+      let r_layer = transcript.challenge_scalar(b"challenge_r_layer");
+      claim = (Scalar::one() - r_layer) * claims_prod[0] + r_layer * claims_prod[1];
+      let mut ext = vec![r_layer];
+      ext.extend(rand_prod);
+      rand = ext;
+    }
+
+    (claim, rand)
+  }
+}
+
+impl ProductCircuitEvalProofBatched {
+  pub fn prove(
+    prod_circuit_vec: &mut Vec<&mut ProductCircuit>,
+    dotp_circuit_vec: &mut Vec<&mut DotProductCircuit>,
+    transcript: &mut Transcript,
+  ) -> (Self, Vec<Scalar>) {
+    assert!(!prod_circuit_vec.is_empty());
+
+    let mut claims_dotp_final = (Vec::new(), Vec::new(), Vec::new());
+
+    let mut proof_layers: Vec<LayerProofBatched> = Vec::new();
+    let num_layers = prod_circuit_vec[0].left_vec.len();
+    let mut claims_to_verify = (0..prod_circuit_vec.len())
+      .map(|i| prod_circuit_vec[i].evaluate())
+      .collect::<Vec<Scalar>>();
+    let mut rand = Vec::new();
+    for layer_id in (0..num_layers).rev() {
+      // prepare paralell instance that share poly_C first
+      let len = prod_circuit_vec[0].left_vec[layer_id].len()
+        + prod_circuit_vec[0].right_vec[layer_id].len();
+
+      let mut poly_C_par = DensePolynomial::new(EqPolynomial::new(rand.clone()).evals());
+      assert_eq!(poly_C_par.len(), len / 2);
+
+      let num_rounds_prod = poly_C_par.len().log_2();
+      let comb_func_prod = |poly_A_comp: &Scalar,
+                            poly_B_comp: &Scalar,
+                            poly_C_comp: &Scalar|
+       -> Scalar { poly_A_comp * poly_B_comp * poly_C_comp };
+
+      let mut poly_A_batched_par: Vec<&mut DensePolynomial> = Vec::new();
+      let mut poly_B_batched_par: Vec<&mut DensePolynomial> = Vec::new();
+      for prod_circuit in prod_circuit_vec.iter_mut() {
+        poly_A_batched_par.push(&mut prod_circuit.left_vec[layer_id]);
+        poly_B_batched_par.push(&mut prod_circuit.right_vec[layer_id])
+      }
+      let poly_vec_par = (
+        &mut poly_A_batched_par,
+        &mut poly_B_batched_par,
+        &mut poly_C_par,
+      );
+
+      // prepare sequential instances that don't share poly_C
+      let mut poly_A_batched_seq: Vec<&mut DensePolynomial> = Vec::new();
+      let mut poly_B_batched_seq: Vec<&mut DensePolynomial> = Vec::new();
+      let mut poly_C_batched_seq: Vec<&mut DensePolynomial> = Vec::new();
+      if layer_id == 0 && !dotp_circuit_vec.is_empty() {
+        // add additional claims
+        for item in dotp_circuit_vec.iter() {
+          claims_to_verify.push(item.evaluate());
+          assert_eq!(len / 2, item.left.len());
+          assert_eq!(len / 2, item.right.len());
+          assert_eq!(len / 2, item.weight.len());
+        }
+
+        for dotp_circuit in dotp_circuit_vec.iter_mut() {
+          poly_A_batched_seq.push(&mut dotp_circuit.left);
+          poly_B_batched_seq.push(&mut dotp_circuit.right);
+          poly_C_batched_seq.push(&mut dotp_circuit.weight);
+        }
+      }
+      let poly_vec_seq = (
+        &mut poly_A_batched_seq,
+        &mut poly_B_batched_seq,
+        &mut poly_C_batched_seq,
+      );
+
+      // produce a fresh set of coeffs and a joint claim
+      let coeff_vec =
+        transcript.challenge_vector(b"rand_coeffs_next_layer", claims_to_verify.len());
+      let claim = (0..claims_to_verify.len())
+        .map(|i| claims_to_verify[i] * coeff_vec[i])
+        .sum();
+
+      let (proof, rand_prod, claims_prod, claims_dotp) = SumcheckInstanceProof::prove_cubic_batched(
+        &claim,
+        num_rounds_prod,
+        poly_vec_par,
+        poly_vec_seq,
+        &coeff_vec,
+        comb_func_prod,
+        transcript,
+      );
+
+      let (claims_prod_left, claims_prod_right, _claims_eq) = claims_prod;
+      for i in 0..prod_circuit_vec.len() {
+        transcript.append_scalar(b"claim_prod_left", &claims_prod_left[i]);
+        transcript.append_scalar(b"claim_prod_right", &claims_prod_right[i]);
+      }
+
+      if layer_id == 0 && !dotp_circuit_vec.is_empty() {
+        let (claims_dotp_left, claims_dotp_right, claims_dotp_weight) = claims_dotp;
+        for i in 0..dotp_circuit_vec.len() {
+          transcript.append_scalar(b"claim_dotp_left", &claims_dotp_left[i]);
+          transcript.append_scalar(b"claim_dotp_right", &claims_dotp_right[i]);
+          transcript.append_scalar(b"claim_dotp_weight", &claims_dotp_weight[i]);
+        }
+        claims_dotp_final = (claims_dotp_left, claims_dotp_right, claims_dotp_weight);
+      }
+
+      // produce a random challenge to condense two claims into a single claim
+      let r_layer = transcript.challenge_scalar(b"challenge_r_layer");
+
+      claims_to_verify = (0..prod_circuit_vec.len())
+        .map(|i| claims_prod_left[i] + r_layer * (claims_prod_right[i] - claims_prod_left[i]))
+        .collect::<Vec<Scalar>>();
+
+      let mut ext = vec![r_layer];
+      ext.extend(rand_prod);
+      rand = ext;
+
+      proof_layers.push(LayerProofBatched {
+        proof,
+        claims_prod_left,
+        claims_prod_right,
+      });
+    }
+
+    (
+      ProductCircuitEvalProofBatched {
+        proof: proof_layers,
+        claims_dotp: claims_dotp_final,
+      },
+      rand,
+    )
+  }
+
+  pub fn verify(
+    &self,
+    claims_prod_vec: &[Scalar],
+    claims_dotp_vec: &[Scalar],
+    len: usize,
+    transcript: &mut Transcript,
+  ) -> (Vec<Scalar>, Vec<Scalar>, Vec<Scalar>) {
+    let num_layers = len.log_2();
+    let mut rand: Vec<Scalar> = Vec::new();
+    //let mut num_rounds = 0;
+    assert_eq!(self.proof.len(), num_layers);
+
+    let mut claims_to_verify = claims_prod_vec.to_owned();
+    let mut claims_to_verify_dotp: Vec<Scalar> = Vec::new();
+    for (num_rounds, i) in (0..num_layers).enumerate() {
+      if i == num_layers - 1 {
+        claims_to_verify.extend(claims_dotp_vec);
+      }
+
+      // produce random coefficients, one for each instance
+      let coeff_vec =
+        transcript.challenge_vector(b"rand_coeffs_next_layer", claims_to_verify.len());
+
+      // produce a joint claim
+      let claim = (0..claims_to_verify.len())
+        .map(|i| claims_to_verify[i] * coeff_vec[i])
+        .sum();
+
+      let (claim_last, rand_prod) = self.proof[i].verify(claim, num_rounds, 3, transcript);
+
+      let claims_prod_left = &self.proof[i].claims_prod_left;
+      let claims_prod_right = &self.proof[i].claims_prod_right;
+      assert_eq!(claims_prod_left.len(), claims_prod_vec.len());
+      assert_eq!(claims_prod_right.len(), claims_prod_vec.len());
+
+      for i in 0..claims_prod_vec.len() {
+        transcript.append_scalar(b"claim_prod_left", &claims_prod_left[i]);
+        transcript.append_scalar(b"claim_prod_right", &claims_prod_right[i]);
+      }
+
+      assert_eq!(rand.len(), rand_prod.len());
+      let eq: Scalar = (0..rand.len())
+        .map(|i| {
+          rand[i] * rand_prod[i] + (Scalar::one() - rand[i]) * (Scalar::one() - rand_prod[i])
+        })
+        .product();
+      let mut claim_expected: Scalar = (0..claims_prod_vec.len())
+        .map(|i| coeff_vec[i] * (claims_prod_left[i] * claims_prod_right[i] * eq))
+        .sum();
+
+      // add claims from the dotp instances
+      if i == num_layers - 1 {
+        let num_prod_instances = claims_prod_vec.len();
+        let (claims_dotp_left, claims_dotp_right, claims_dotp_weight) = &self.claims_dotp;
+        for i in 0..claims_dotp_left.len() {
+          transcript.append_scalar(b"claim_dotp_left", &claims_dotp_left[i]);
+          transcript.append_scalar(b"claim_dotp_right", &claims_dotp_right[i]);
+          transcript.append_scalar(b"claim_dotp_weight", &claims_dotp_weight[i]);
+
+          claim_expected += coeff_vec[i + num_prod_instances]
+            * claims_dotp_left[i]
+            * claims_dotp_right[i]
+            * claims_dotp_weight[i];
+        }
+      }
+
+      assert_eq!(claim_expected, claim_last);
+
+      // produce a random challenge
+      let r_layer = transcript.challenge_scalar(b"challenge_r_layer");
+
+      claims_to_verify = (0..claims_prod_left.len())
+        .map(|i| claims_prod_left[i] + r_layer * (claims_prod_right[i] - claims_prod_left[i]))
+        .collect::<Vec<Scalar>>();
+
+      // add claims to verify for dotp circuit
+      if i == num_layers - 1 {
+        let (claims_dotp_left, claims_dotp_right, claims_dotp_weight) = &self.claims_dotp;
+
+        for i in 0..claims_dotp_vec.len() / 2 {
+          // combine left claims
+          let claim_left = claims_dotp_left[2 * i]
+            + r_layer * (claims_dotp_left[2 * i + 1] - claims_dotp_left[2 * i]);
+
+          let claim_right = claims_dotp_right[2 * i]
+            + r_layer * (claims_dotp_right[2 * i + 1] - claims_dotp_right[2 * i]);
+
+          let claim_weight = claims_dotp_weight[2 * i]
+            + r_layer * (claims_dotp_weight[2 * i + 1] - claims_dotp_weight[2 * i]);
+          claims_to_verify_dotp.push(claim_left);
+          claims_to_verify_dotp.push(claim_right);
+          claims_to_verify_dotp.push(claim_weight);
+        }
+      }
+
+      let mut ext = vec![r_layer];
+      ext.extend(rand_prod);
+      rand = ext;
+    }
+    (claims_to_verify, claims_to_verify_dotp, rand)
+  }
+}

packages/Spartan-secq/src/r1csinstance.rs

@@ -0,0 +1,367 @@
+use crate::transcript::AppendToTranscript;
+
+use super::dense_mlpoly::DensePolynomial;
+use super::errors::ProofVerifyError;
+use super::math::Math;
+use super::random::RandomTape;
+use super::scalar::Scalar;
+use super::sparse_mlpoly::{
+  MultiSparseMatPolynomialAsDense, SparseMatEntry, SparseMatPolyCommitment,
+  SparseMatPolyCommitmentGens, SparseMatPolyEvalProof, SparseMatPolynomial,
+};
+use super::timer::Timer;
+use flate2::{write::ZlibEncoder, Compression};
+use merlin::Transcript;
+use rand_core::OsRng;
+use serde::{Deserialize, Serialize};
+
+#[derive(Debug, Serialize, Deserialize)]
+pub struct R1CSInstance {
+  num_cons: usize,
+  num_vars: usize,
+  num_inputs: usize,
+  A: SparseMatPolynomial,
+  B: SparseMatPolynomial,
+  C: SparseMatPolynomial,
+}
+
+pub struct R1CSCommitmentGens {
+  gens: SparseMatPolyCommitmentGens,
+}
+
+impl R1CSCommitmentGens {
+  pub fn new(
+    label: &'static [u8],
+    num_cons: usize,
+    num_vars: usize,
+    num_inputs: usize,
+    num_nz_entries: usize,
+  ) -> R1CSCommitmentGens {
+    assert!(num_inputs < num_vars);
+    let num_poly_vars_x = num_cons.log_2();
+    let num_poly_vars_y = (2 * num_vars).log_2();
+    let gens =
+      SparseMatPolyCommitmentGens::new(label, num_poly_vars_x, num_poly_vars_y, num_nz_entries, 3);
+    R1CSCommitmentGens { gens }
+  }
+}
+
+#[derive(Debug, Serialize, Deserialize)]
+pub struct R1CSCommitment {
+  num_cons: usize,
+  num_vars: usize,
+  num_inputs: usize,
+  comm: SparseMatPolyCommitment,
+}
+
+impl AppendToTranscript for R1CSCommitment {
+  fn append_to_transcript(&self, _label: &'static [u8], transcript: &mut Transcript) {
+    transcript.append_u64(b"num_cons", self.num_cons as u64);
+    transcript.append_u64(b"num_vars", self.num_vars as u64);
+    transcript.append_u64(b"num_inputs", self.num_inputs as u64);
+    self.comm.append_to_transcript(b"comm", transcript);
+  }
+}
+
+pub struct R1CSDecommitment {
+  dense: MultiSparseMatPolynomialAsDense,
+}
+
+impl R1CSCommitment {
+  pub fn get_num_cons(&self) -> usize {
+    self.num_cons
+  }
+
+  pub fn get_num_vars(&self) -> usize {
+    self.num_vars
+  }
+
+  pub fn get_num_inputs(&self) -> usize {
+    self.num_inputs
+  }
+}
+
+impl R1CSInstance {
+  pub fn new(
+    num_cons: usize,
+    num_vars: usize,
+    num_inputs: usize,
+    A: &[(usize, usize, Scalar)],
+    B: &[(usize, usize, Scalar)],
+    C: &[(usize, usize, Scalar)],
+  ) -> R1CSInstance {
+    Timer::print(&format!("number_of_constraints {}", num_cons));
+    Timer::print(&format!("number_of_variables {}", num_vars));
+    Timer::print(&format!("number_of_inputs {}", num_inputs));
+    Timer::print(&format!("number_non-zero_entries_A {}", A.len()));
+    Timer::print(&format!("number_non-zero_entries_B {}", B.len()));
+    Timer::print(&format!("number_non-zero_entries_C {}", C.len()));
+
+    // check that num_cons is a power of 2
+    assert_eq!(num_cons.next_power_of_two(), num_cons);
+
+    // check that num_vars is a power of 2
+    assert_eq!(num_vars.next_power_of_two(), num_vars);
+
+    // check that number_inputs + 1 <= num_vars
+    assert!(num_inputs < num_vars);
+
+    // no errors, so create polynomials
+    let num_poly_vars_x = num_cons.log_2();
+    let num_poly_vars_y = (2 * num_vars).log_2();
+
+    let mat_A = (0..A.len())
+      .map(|i| SparseMatEntry::new(A[i].0, A[i].1, A[i].2))
+      .collect::<Vec<SparseMatEntry>>();
+    let mat_B = (0..B.len())
+      .map(|i| SparseMatEntry::new(B[i].0, B[i].1, B[i].2))
+      .collect::<Vec<SparseMatEntry>>();
+    let mat_C = (0..C.len())
+      .map(|i| SparseMatEntry::new(C[i].0, C[i].1, C[i].2))
+      .collect::<Vec<SparseMatEntry>>();
+
+    let poly_A = SparseMatPolynomial::new(num_poly_vars_x, num_poly_vars_y, mat_A);
+    let poly_B = SparseMatPolynomial::new(num_poly_vars_x, num_poly_vars_y, mat_B);
+    let poly_C = SparseMatPolynomial::new(num_poly_vars_x, num_poly_vars_y, mat_C);
+
+    R1CSInstance {
+      num_cons,
+      num_vars,
+      num_inputs,
+      A: poly_A,
+      B: poly_B,
+      C: poly_C,
+    }
+  }
+
+  pub fn get_num_vars(&self) -> usize {
+    self.num_vars
+  }
+
+  pub fn get_num_cons(&self) -> usize {
+    self.num_cons
+  }
+
+  pub fn get_num_inputs(&self) -> usize {
+    self.num_inputs
+  }
+
+  pub fn get_digest(&self) -> Vec<u8> {
+    let mut encoder = ZlibEncoder::new(Vec::new(), Compression::default());
+    bincode::serialize_into(&mut encoder, &self).unwrap();
+    encoder.finish().unwrap()
+  }
+
+  pub fn produce_synthetic_r1cs(
+    num_cons: usize,
+    num_vars: usize,
+    num_inputs: usize,
+  ) -> (R1CSInstance, Vec<Scalar>, Vec<Scalar>) {
+    Timer::print(&format!("number_of_constraints {}", num_cons));
+    Timer::print(&format!("number_of_variables {}", num_vars));
+    Timer::print(&format!("number_of_inputs {}", num_inputs));
+
+    let mut csprng: OsRng = OsRng;
+
+    // assert num_cons and num_vars are power of 2
+    assert_eq!((num_cons.log_2()).pow2(), num_cons);
+    assert_eq!((num_vars.log_2()).pow2(), num_vars);
+
+    // num_inputs + 1 <= num_vars
+    assert!(num_inputs < num_vars);
+
+    // z is organized as [vars,1,io]
+    let size_z = num_vars + num_inputs + 1;
+
+    // produce a random satisfying assignment
+    let Z = {
+      let mut Z: Vec<Scalar> = (0..size_z)
+        .map(|_i| Scalar::random(&mut csprng))
+        .collect::<Vec<Scalar>>();
+      Z[num_vars] = Scalar::one(); // set the constant term to 1
+      Z
+    };
+
+    // three sparse matrices
+    let mut A: Vec<SparseMatEntry> = Vec::new();
+    let mut B: Vec<SparseMatEntry> = Vec::new();
+    let mut C: Vec<SparseMatEntry> = Vec::new();
+    let one = Scalar::one();
+    for i in 0..num_cons {
+      let A_idx = i % size_z;
+      let B_idx = (i + 2) % size_z;
+      A.push(SparseMatEntry::new(i, A_idx, one));
+      B.push(SparseMatEntry::new(i, B_idx, one));
+      let AB_val = Z[A_idx] * Z[B_idx];
+
+      let C_idx = (i + 3) % size_z;
+      let C_val = Z[C_idx];
+
+      if C_val == Scalar::zero() {
+        C.push(SparseMatEntry::new(i, num_vars, AB_val));
+      } else {
+        C.push(SparseMatEntry::new(
+          i,
+          C_idx,
+          AB_val * C_val.invert().unwrap(),
+        ));
+      }
+    }
+
+    Timer::print(&format!("number_non-zero_entries_A {}", A.len()));
+    Timer::print(&format!("number_non-zero_entries_B {}", B.len()));
+    Timer::print(&format!("number_non-zero_entries_C {}", C.len()));
+
+    let num_poly_vars_x = num_cons.log_2();
+    let num_poly_vars_y = (2 * num_vars).log_2();
+    let poly_A = SparseMatPolynomial::new(num_poly_vars_x, num_poly_vars_y, A);
+    let poly_B = SparseMatPolynomial::new(num_poly_vars_x, num_poly_vars_y, B);
+    let poly_C = SparseMatPolynomial::new(num_poly_vars_x, num_poly_vars_y, C);
+
+    let inst = R1CSInstance {
+      num_cons,
+      num_vars,
+      num_inputs,
+      A: poly_A,
+      B: poly_B,
+      C: poly_C,
+    };
+
+    assert!(inst.is_sat(&Z[..num_vars], &Z[num_vars + 1..]));
+
+    (inst, Z[..num_vars].to_vec(), Z[num_vars + 1..].to_vec())
+  }
+
+  pub fn is_sat(&self, vars: &[Scalar], input: &[Scalar]) -> bool {
+    assert_eq!(vars.len(), self.num_vars);
+    assert_eq!(input.len(), self.num_inputs);
+
+    let z = {
+      let mut z = vars.to_vec();
+      z.extend(&vec![Scalar::one()]);
+      z.extend(input);
+      z
+    };
+
+    // verify if Az * Bz - Cz = [0...]
+    let Az = self
+      .A
+      .multiply_vec(self.num_cons, self.num_vars + self.num_inputs + 1, &z);
+    let Bz = self
+      .B
+      .multiply_vec(self.num_cons, self.num_vars + self.num_inputs + 1, &z);
+    let Cz = self
+      .C
+      .multiply_vec(self.num_cons, self.num_vars + self.num_inputs + 1, &z);
+
+    assert_eq!(Az.len(), self.num_cons);
+    assert_eq!(Bz.len(), self.num_cons);
+    assert_eq!(Cz.len(), self.num_cons);
+    let res: usize = (0..self.num_cons)
+      .map(|i| usize::from(Az[i] * Bz[i] != Cz[i]))
+      .sum();
+
+    res == 0
+  }
+
+  pub fn multiply_vec(
+    &self,
+    num_rows: usize,
+    num_cols: usize,
+    z: &[Scalar],
+  ) -> (DensePolynomial, DensePolynomial, DensePolynomial) {
+    assert_eq!(num_rows, self.num_cons);
+    assert_eq!(z.len(), num_cols);
+    assert!(num_cols > self.num_vars);
+    (
+      DensePolynomial::new(self.A.multiply_vec(num_rows, num_cols, z)),
+      DensePolynomial::new(self.B.multiply_vec(num_rows, num_cols, z)),
+      DensePolynomial::new(self.C.multiply_vec(num_rows, num_cols, z)),
+    )
+  }
+
+  pub fn compute_eval_table_sparse(
+    &self,
+    num_rows: usize,
+    num_cols: usize,
+    evals: &[Scalar],
+  ) -> (Vec<Scalar>, Vec<Scalar>, Vec<Scalar>) {
+    assert_eq!(num_rows, self.num_cons);
+    assert!(num_cols > self.num_vars);
+
+    let evals_A = self.A.compute_eval_table_sparse(evals, num_rows, num_cols);
+    let evals_B = self.B.compute_eval_table_sparse(evals, num_rows, num_cols);
+    let evals_C = self.C.compute_eval_table_sparse(evals, num_rows, num_cols);
+
+    (evals_A, evals_B, evals_C)
+  }
+
+  pub fn evaluate(&self, rx: &[Scalar], ry: &[Scalar]) -> (Scalar, Scalar, Scalar) {
+    let evals = SparseMatPolynomial::multi_evaluate(&[&self.A, &self.B, &self.C], rx, ry);
+    (evals[0], evals[1], evals[2])
+  }
+
+  pub fn commit(&self, gens: &R1CSCommitmentGens) -> (R1CSCommitment, R1CSDecommitment) {
+    let (comm, dense) = SparseMatPolynomial::multi_commit(&[&self.A, &self.B, &self.C], &gens.gens);
+    let r1cs_comm = R1CSCommitment {
+      num_cons: self.num_cons,
+      num_vars: self.num_vars,
+      num_inputs: self.num_inputs,
+      comm,
+    };
+
+    let r1cs_decomm = R1CSDecommitment { dense };
+
+    (r1cs_comm, r1cs_decomm)
+  }
+}
+
+#[derive(Debug, Serialize, Deserialize)]
+pub struct R1CSEvalProof {
+  proof: SparseMatPolyEvalProof,
+}
+
+impl R1CSEvalProof {
+  pub fn prove(
+    decomm: &R1CSDecommitment,
+    rx: &[Scalar], // point at which the polynomial is evaluated
+    ry: &[Scalar],
+    evals: &(Scalar, Scalar, Scalar),
+    gens: &R1CSCommitmentGens,
+    transcript: &mut Transcript,
+    random_tape: &mut RandomTape,
+  ) -> R1CSEvalProof {
+    let timer = Timer::new("R1CSEvalProof::prove");
+    let proof = SparseMatPolyEvalProof::prove(
+      &decomm.dense,
+      rx,
+      ry,
+      &[evals.0, evals.1, evals.2],
+      &gens.gens,
+      transcript,
+      random_tape,
+    );
+    timer.stop();
+
+    R1CSEvalProof { proof }
+  }
+
+  pub fn verify(
+    &self,
+    comm: &R1CSCommitment,
+    rx: &[Scalar], // point at which the R1CS matrix polynomials are evaluated
+    ry: &[Scalar],
+    evals: &(Scalar, Scalar, Scalar),
+    gens: &R1CSCommitmentGens,
+    transcript: &mut Transcript,
+  ) -> Result<(), ProofVerifyError> {
+    self.proof.verify(
+      &comm.comm,
+      rx,
+      ry,
+      &[evals.0, evals.1, evals.2],
+      &gens.gens,
+      transcript,
+    )
+  }
+}

packages/Spartan-secq/src/r1csproof.rs

@@ -0,0 +1,608 @@
+#![allow(clippy::too_many_arguments)]
+use super::commitments::{Commitments, MultiCommitGens};
+use super::dense_mlpoly::{
+  DensePolynomial, EqPolynomial, PolyCommitment, PolyCommitmentGens, PolyEvalProof,
+};
+use super::errors::ProofVerifyError;
+use super::group::{CompressedGroup, GroupElement, VartimeMultiscalarMul};
+use super::math::Math;
+use super::nizk::{EqualityProof, KnowledgeProof, ProductProof};
+use super::r1csinstance::R1CSInstance;
+use super::random::RandomTape;
+use super::scalar::Scalar;
+use super::sparse_mlpoly::{SparsePolyEntry, SparsePolynomial};
+use super::sumcheck::ZKSumcheckInstanceProof;
+use super::timer::Timer;
+use super::transcript::{AppendToTranscript, ProofTranscript};
+use crate::group::DecompressEncodedPoint;
+use core::iter;
+use merlin::Transcript;
+use serde::{Deserialize, Serialize};
+
+#[derive(Serialize, Deserialize, Debug)]
+pub struct R1CSProof {
+  comm_vars: PolyCommitment,
+  sc_proof_phase1: ZKSumcheckInstanceProof,
+  claims_phase2: (
+    CompressedGroup,
+    CompressedGroup,
+    CompressedGroup,
+    CompressedGroup,
+  ),
+  pok_claims_phase2: (KnowledgeProof, ProductProof),
+  proof_eq_sc_phase1: EqualityProof,
+  sc_proof_phase2: ZKSumcheckInstanceProof,
+  comm_vars_at_ry: CompressedGroup,
+  proof_eval_vars_at_ry: PolyEvalProof,
+  proof_eq_sc_phase2: EqualityProof,
+}
+
+pub struct R1CSSumcheckGens {
+  gens_1: MultiCommitGens,
+  gens_3: MultiCommitGens,
+  gens_4: MultiCommitGens,
+}
+
+// TODO: fix passing gens_1_ref
+impl R1CSSumcheckGens {
+  pub fn new(label: &'static [u8], gens_1_ref: &MultiCommitGens) -> Self {
+    let gens_1 = gens_1_ref.clone();
+    let gens_3 = MultiCommitGens::new(3, label);
+    let gens_4 = MultiCommitGens::new(4, label);
+
+    R1CSSumcheckGens {
+      gens_1,
+      gens_3,
+      gens_4,
+    }
+  }
+}
+
+pub struct R1CSGens {
+  gens_sc: R1CSSumcheckGens,
+  gens_pc: PolyCommitmentGens,
+}
+
+impl R1CSGens {
+  pub fn new(label: &'static [u8], _num_cons: usize, num_vars: usize) -> Self {
+    let num_poly_vars = num_vars.log_2();
+    let gens_pc = PolyCommitmentGens::new(num_poly_vars, label);
+    let gens_sc = R1CSSumcheckGens::new(label, &gens_pc.gens.gens_1);
+    R1CSGens { gens_sc, gens_pc }
+  }
+}
+
+impl R1CSProof {
+  fn prove_phase_one(
+    num_rounds: usize,
+    evals_tau: &mut DensePolynomial,
+    evals_Az: &mut DensePolynomial,
+    evals_Bz: &mut DensePolynomial,
+    evals_Cz: &mut DensePolynomial,
+    gens: &R1CSSumcheckGens,
+    transcript: &mut Transcript,
+    random_tape: &mut RandomTape,
+  ) -> (ZKSumcheckInstanceProof, Vec<Scalar>, Vec<Scalar>, Scalar) {
+    let comb_func = |poly_A_comp: &Scalar,
+                     poly_B_comp: &Scalar,
+                     poly_C_comp: &Scalar,
+                     poly_D_comp: &Scalar|
+     -> Scalar { poly_A_comp * (poly_B_comp * poly_C_comp - poly_D_comp) };
+
+    let (sc_proof_phase_one, r, claims, blind_claim_postsc) =
+      ZKSumcheckInstanceProof::prove_cubic_with_additive_term(
+        &Scalar::zero(), // claim is zero
+        &Scalar::zero(), // blind for claim is also zero
+        num_rounds,
+        evals_tau,
+        evals_Az,
+        evals_Bz,
+        evals_Cz,
+        comb_func,
+        &gens.gens_1,
+        &gens.gens_4,
+        transcript,
+        random_tape,
+      );
+
+    (sc_proof_phase_one, r, claims, blind_claim_postsc)
+  }
+
+  fn prove_phase_two(
+    num_rounds: usize,
+    claim: &Scalar,
+    blind_claim: &Scalar,
+    evals_z: &mut DensePolynomial,
+    evals_ABC: &mut DensePolynomial,
+    gens: &R1CSSumcheckGens,
+    transcript: &mut Transcript,
+    random_tape: &mut RandomTape,
+  ) -> (ZKSumcheckInstanceProof, Vec<Scalar>, Vec<Scalar>, Scalar) {
+    let comb_func =
+      |poly_A_comp: &Scalar, poly_B_comp: &Scalar| -> Scalar { poly_A_comp * poly_B_comp };
+    let (sc_proof_phase_two, r, claims, blind_claim_postsc) = ZKSumcheckInstanceProof::prove_quad(
+      claim,
+      blind_claim,
+      num_rounds,
+      evals_z,
+      evals_ABC,
+      comb_func,
+      &gens.gens_1,
+      &gens.gens_3,
+      transcript,
+      random_tape,
+    );
+
+    (sc_proof_phase_two, r, claims, blind_claim_postsc)
+  }
+
+  fn protocol_name() -> &'static [u8] {
+    b"R1CS proof"
+  }
+
+  pub fn prove(
+    inst: &R1CSInstance,
+    vars: Vec<Scalar>,
+    input: &[Scalar],
+    gens: &R1CSGens,
+    transcript: &mut Transcript,
+    random_tape: &mut RandomTape,
+  ) -> (R1CSProof, Vec<Scalar>, Vec<Scalar>) {
+    let timer_prove = Timer::new("R1CSProof::prove");
+    transcript.append_protocol_name(R1CSProof::protocol_name());
+
+    // we currently require the number of |inputs| + 1 to be at most number of vars
+    assert!(input.len() < vars.len());
+
+    input.append_to_transcript(b"input", transcript);
+
+    let timer_commit = Timer::new("polycommit");
+    let (poly_vars, comm_vars, blinds_vars) = {
+      // create a multilinear polynomial using the supplied assignment for variables
+      let poly_vars = DensePolynomial::new(vars.clone());
+
+      // produce a commitment to the satisfying assignment
+      let (comm_vars, blinds_vars) = poly_vars.commit(&gens.gens_pc, Some(random_tape));
+
+      // add the commitment to the prover's transcript
+      comm_vars.append_to_transcript(b"poly_commitment", transcript);
+      (poly_vars, comm_vars, blinds_vars)
+    };
+    timer_commit.stop();
+
+    let timer_sc_proof_phase1 = Timer::new("prove_sc_phase_one");
+
+    // append input to variables to create a single vector z
+    let z = {
+      let num_inputs = input.len();
+      let num_vars = vars.len();
+      let mut z = vars;
+      z.extend(&vec![Scalar::one()]); // add constant term in z
+      z.extend(input);
+      z.extend(&vec![Scalar::zero(); num_vars - num_inputs - 1]); // we will pad with zeros
+      z
+    };
+
+    // derive the verifier's challenge tau
+    let (num_rounds_x, num_rounds_y) = (inst.get_num_cons().log_2(), z.len().log_2());
+    let tau = transcript.challenge_vector(b"challenge_tau", num_rounds_x);
+    // compute the initial evaluation table for R(\tau, x)
+    let mut poly_tau = DensePolynomial::new(EqPolynomial::new(tau).evals());
+    let (mut poly_Az, mut poly_Bz, mut poly_Cz) =
+      inst.multiply_vec(inst.get_num_cons(), z.len(), &z);
+
+    let (sc_proof_phase1, rx, _claims_phase1, blind_claim_postsc1) = R1CSProof::prove_phase_one(
+      num_rounds_x,
+      &mut poly_tau,
+      &mut poly_Az,
+      &mut poly_Bz,
+      &mut poly_Cz,
+      &gens.gens_sc,
+      transcript,
+      random_tape,
+    );
+    assert_eq!(poly_tau.len(), 1);
+    assert_eq!(poly_Az.len(), 1);
+    assert_eq!(poly_Bz.len(), 1);
+    assert_eq!(poly_Cz.len(), 1);
+    timer_sc_proof_phase1.stop();
+
+    let (tau_claim, Az_claim, Bz_claim, Cz_claim) =
+      (&poly_tau[0], &poly_Az[0], &poly_Bz[0], &poly_Cz[0]);
+    let (Az_blind, Bz_blind, Cz_blind, prod_Az_Bz_blind) = (
+      random_tape.random_scalar(b"Az_blind"),
+      random_tape.random_scalar(b"Bz_blind"),
+      random_tape.random_scalar(b"Cz_blind"),
+      random_tape.random_scalar(b"prod_Az_Bz_blind"),
+    );
+
+    let (pok_Cz_claim, comm_Cz_claim) = {
+      KnowledgeProof::prove(
+        &gens.gens_sc.gens_1,
+        transcript,
+        random_tape,
+        Cz_claim,
+        &Cz_blind,
+      )
+    };
+
+    let (proof_prod, comm_Az_claim, comm_Bz_claim, comm_prod_Az_Bz_claims) = {
+      let prod = Az_claim * Bz_claim;
+      ProductProof::prove(
+        &gens.gens_sc.gens_1,
+        transcript,
+        random_tape,
+        Az_claim,
+        &Az_blind,
+        Bz_claim,
+        &Bz_blind,
+        &prod,
+        &prod_Az_Bz_blind,
+      )
+    };
+
+    comm_Az_claim.append_to_transcript(b"comm_Az_claim", transcript);
+    comm_Bz_claim.append_to_transcript(b"comm_Bz_claim", transcript);
+    comm_Cz_claim.append_to_transcript(b"comm_Cz_claim", transcript);
+    comm_prod_Az_Bz_claims.append_to_transcript(b"comm_prod_Az_Bz_claims", transcript);
+
+    // prove the final step of sum-check #1
+    let taus_bound_rx = tau_claim;
+    let blind_expected_claim_postsc1 = taus_bound_rx * (prod_Az_Bz_blind - Cz_blind);
+    let claim_post_phase1 = (Az_claim * Bz_claim - Cz_claim) * taus_bound_rx;
+    let (proof_eq_sc_phase1, _C1, _C2) = EqualityProof::prove(
+      &gens.gens_sc.gens_1,
+      transcript,
+      random_tape,
+      &claim_post_phase1,
+      &blind_expected_claim_postsc1,
+      &claim_post_phase1,
+      &blind_claim_postsc1,
+    );
+
+    let timer_sc_proof_phase2 = Timer::new("prove_sc_phase_two");
+    // combine the three claims into a single claim
+    let r_A = transcript.challenge_scalar(b"challenege_Az");
+    let r_B = transcript.challenge_scalar(b"challenege_Bz");
+    let r_C = transcript.challenge_scalar(b"challenege_Cz");
+    let claim_phase2 = r_A * Az_claim + r_B * Bz_claim + r_C * Cz_claim;
+    let blind_claim_phase2 = r_A * Az_blind + r_B * Bz_blind + r_C * Cz_blind;
+
+    let evals_ABC = {
+      // compute the initial evaluation table for R(\tau, x)
+      let evals_rx = EqPolynomial::new(rx.clone()).evals();
+      let (evals_A, evals_B, evals_C) =
+        inst.compute_eval_table_sparse(inst.get_num_cons(), z.len(), &evals_rx);
+
+      assert_eq!(evals_A.len(), evals_B.len());
+      assert_eq!(evals_A.len(), evals_C.len());
+      (0..evals_A.len())
+        .map(|i| r_A * evals_A[i] + r_B * evals_B[i] + r_C * evals_C[i])
+        .collect::<Vec<Scalar>>()
+    };
+
+    // another instance of the sum-check protocol
+    let (sc_proof_phase2, ry, claims_phase2, blind_claim_postsc2) = R1CSProof::prove_phase_two(
+      num_rounds_y,
+      &claim_phase2,
+      &blind_claim_phase2,
+      &mut DensePolynomial::new(z),
+      &mut DensePolynomial::new(evals_ABC),
+      &gens.gens_sc,
+      transcript,
+      random_tape,
+    );
+    timer_sc_proof_phase2.stop();
+
+    let timer_polyeval = Timer::new("polyeval");
+    let eval_vars_at_ry = poly_vars.evaluate(&ry[1..]);
+    let blind_eval = random_tape.random_scalar(b"blind_eval");
+    let (proof_eval_vars_at_ry, comm_vars_at_ry) = PolyEvalProof::prove(
+      &poly_vars,
+      Some(&blinds_vars),
+      &ry[1..],
+      &eval_vars_at_ry,
+      Some(&blind_eval),
+      &gens.gens_pc,
+      transcript,
+      random_tape,
+    );
+    timer_polyeval.stop();
+
+    // prove the final step of sum-check #2
+    let blind_eval_Z_at_ry = (Scalar::one() - ry[0]) * blind_eval;
+    let blind_expected_claim_postsc2 = claims_phase2[1] * blind_eval_Z_at_ry;
+    let claim_post_phase2 = claims_phase2[0] * claims_phase2[1];
+    let (proof_eq_sc_phase2, _C1, _C2) = EqualityProof::prove(
+      &gens.gens_pc.gens.gens_1,
+      transcript,
+      random_tape,
+      &claim_post_phase2,
+      &blind_expected_claim_postsc2,
+      &claim_post_phase2,
+      &blind_claim_postsc2,
+    );
+
+    timer_prove.stop();
+
+    (
+      R1CSProof {
+        comm_vars,
+        sc_proof_phase1,
+        claims_phase2: (
+          comm_Az_claim,
+          comm_Bz_claim,
+          comm_Cz_claim,
+          comm_prod_Az_Bz_claims,
+        ),
+        pok_claims_phase2: (pok_Cz_claim, proof_prod),
+        proof_eq_sc_phase1,
+        sc_proof_phase2,
+        comm_vars_at_ry,
+        proof_eval_vars_at_ry,
+        proof_eq_sc_phase2,
+      },
+      rx,
+      ry,
+    )
+  }
+
+  pub fn verify(
+    &self,
+    num_vars: usize,
+    num_cons: usize,
+    input: &[Scalar],
+    evals: &(Scalar, Scalar, Scalar),
+    transcript: &mut Transcript,
+    gens: &R1CSGens,
+  ) -> Result<(Vec<Scalar>, Vec<Scalar>), ProofVerifyError> {
+    transcript.append_protocol_name(R1CSProof::protocol_name());
+
+    input.append_to_transcript(b"input", transcript);
+
+    let n = num_vars;
+    // add the commitment to the verifier's transcript
+    self
+      .comm_vars
+      .append_to_transcript(b"poly_commitment", transcript);
+
+    let (num_rounds_x, num_rounds_y) = (num_cons.log_2(), (2 * num_vars).log_2());
+
+    // derive the verifier's challenge tau
+    let tau = transcript.challenge_vector(b"challenge_tau", num_rounds_x);
+
+    // verify the first sum-check instance
+    let claim_phase1 = Scalar::zero()
+      .commit(&Scalar::zero(), &gens.gens_sc.gens_1)
+      .compress();
+    let (comm_claim_post_phase1, rx) = self.sc_proof_phase1.verify(
+      &claim_phase1,
+      num_rounds_x,
+      3,
+      &gens.gens_sc.gens_1,
+      &gens.gens_sc.gens_4,
+      transcript,
+    )?;
+    // perform the intermediate sum-check test with claimed Az, Bz, and Cz
+    let (comm_Az_claim, comm_Bz_claim, comm_Cz_claim, comm_prod_Az_Bz_claims) = &self.claims_phase2;
+    let (pok_Cz_claim, proof_prod) = &self.pok_claims_phase2;
+
+    pok_Cz_claim.verify(&gens.gens_sc.gens_1, transcript, comm_Cz_claim)?;
+    proof_prod.verify(
+      &gens.gens_sc.gens_1,
+      transcript,
+      comm_Az_claim,
+      comm_Bz_claim,
+      comm_prod_Az_Bz_claims,
+    )?;
+
+    comm_Az_claim.append_to_transcript(b"comm_Az_claim", transcript);
+    comm_Bz_claim.append_to_transcript(b"comm_Bz_claim", transcript);
+    comm_Cz_claim.append_to_transcript(b"comm_Cz_claim", transcript);
+    comm_prod_Az_Bz_claims.append_to_transcript(b"comm_prod_Az_Bz_claims", transcript);
+
+    let taus_bound_rx: Scalar = (0..rx.len())
+      .map(|i| rx[i] * tau[i] + (Scalar::one() - rx[i]) * (Scalar::one() - tau[i]))
+      .product();
+    let expected_claim_post_phase1 = (taus_bound_rx
+      * (comm_prod_Az_Bz_claims.decompress().unwrap() - comm_Cz_claim.decompress().unwrap()))
+    .compress();
+
+    // verify proof that expected_claim_post_phase1 == claim_post_phase1
+    self.proof_eq_sc_phase1.verify(
+      &gens.gens_sc.gens_1,
+      transcript,
+      &expected_claim_post_phase1,
+      &comm_claim_post_phase1,
+    )?;
+
+    // derive three public challenges and then derive a joint claim
+    let r_A = transcript.challenge_scalar(b"challenege_Az");
+    let r_B = transcript.challenge_scalar(b"challenege_Bz");
+    let r_C = transcript.challenge_scalar(b"challenege_Cz");
+
+    // r_A * comm_Az_claim + r_B * comm_Bz_claim + r_C * comm_Cz_claim;
+    let comm_claim_phase2 = GroupElement::vartime_multiscalar_mul(
+      iter::once(r_A)
+        .chain(iter::once(r_B))
+        .chain(iter::once(r_C))
+        .collect(),
+      iter::once(&comm_Az_claim)
+        .chain(iter::once(&comm_Bz_claim))
+        .chain(iter::once(&comm_Cz_claim))
+        .map(|pt| pt.decompress().unwrap())
+        .collect(),
+    )
+    .compress();
+
+    // verify the joint claim with a sum-check protocol
+    let (comm_claim_post_phase2, ry) = self.sc_proof_phase2.verify(
+      &comm_claim_phase2,
+      num_rounds_y,
+      2,
+      &gens.gens_sc.gens_1,
+      &gens.gens_sc.gens_3,
+      transcript,
+    )?;
+
+    // verify Z(ry) proof against the initial commitment
+    self.proof_eval_vars_at_ry.verify(
+      &gens.gens_pc,
+      transcript,
+      &ry[1..],
+      &self.comm_vars_at_ry,
+      &self.comm_vars,
+    )?;
+
+    let poly_input_eval = {
+      // constant term
+      let mut input_as_sparse_poly_entries = vec![SparsePolyEntry::new(0, Scalar::one())];
+      //remaining inputs
+      input_as_sparse_poly_entries.extend(
+        (0..input.len())
+          .map(|i| SparsePolyEntry::new(i + 1, input[i]))
+          .collect::<Vec<SparsePolyEntry>>(),
+      );
+      SparsePolynomial::new(n.log_2(), input_as_sparse_poly_entries).evaluate(&ry[1..])
+    };
+
+    // compute commitment to eval_Z_at_ry = (Scalar::one() - ry[0]) * self.eval_vars_at_ry + ry[0] * poly_input_eval
+    let comm_eval_Z_at_ry = GroupElement::vartime_multiscalar_mul(
+      iter::once(Scalar::one() - ry[0])
+        .chain(iter::once(ry[0]))
+        .map(|s| s)
+        .collect(),
+      iter::once(self.comm_vars_at_ry.decompress().unwrap())
+        .chain(iter::once(
+          poly_input_eval.commit(&Scalar::zero(), &gens.gens_pc.gens.gens_1),
+        ))
+        .collect(),
+    );
+
+    // perform the final check in the second sum-check protocol
+    let (eval_A_r, eval_B_r, eval_C_r) = evals;
+    let expected_claim_post_phase2 =
+      ((r_A * eval_A_r + r_B * eval_B_r + r_C * eval_C_r) * comm_eval_Z_at_ry).compress();
+    // verify proof that expected_claim_post_phase1 == claim_post_phase1
+    self.proof_eq_sc_phase2.verify(
+      &gens.gens_sc.gens_1,
+      transcript,
+      &expected_claim_post_phase2,
+      &comm_claim_post_phase2,
+    )?;
+
+    Ok((rx, ry))
+  }
+}
+
+#[cfg(test)]
+mod tests {
+  use super::*;
+  use rand_core::OsRng;
+
+  fn produce_tiny_r1cs() -> (R1CSInstance, Vec<Scalar>, Vec<Scalar>) {
+    // three constraints over five variables Z1, Z2, Z3, Z4, and Z5
+    // rounded to the nearest power of two
+    let num_cons = 128;
+    let num_vars = 256;
+    let num_inputs = 2;
+
+    // encode the above constraints into three matrices
+    let mut A: Vec<(usize, usize, Scalar)> = Vec::new();
+    let mut B: Vec<(usize, usize, Scalar)> = Vec::new();
+    let mut C: Vec<(usize, usize, Scalar)> = Vec::new();
+
+    let one = Scalar::one();
+    // constraint 0 entries
+    // (Z1 + Z2) * I0 - Z3 = 0;
+    A.push((0, 0, one));
+    A.push((0, 1, one));
+    B.push((0, num_vars + 1, one));
+    C.push((0, 2, one));
+
+    // constraint 1 entries
+    // (Z1 + I1) * (Z3) - Z4 = 0
+    A.push((1, 0, one));
+    A.push((1, num_vars + 2, one));
+    B.push((1, 2, one));
+    C.push((1, 3, one));
+    // constraint 3 entries
+    // Z5 * 1 - 0 = 0
+    A.push((2, 4, one));
+    B.push((2, num_vars, one));
+
+    let inst = R1CSInstance::new(num_cons, num_vars, num_inputs, &A, &B, &C);
+
+    // compute a satisfying assignment
+    let mut csprng: OsRng = OsRng;
+    let i0 = Scalar::random(&mut csprng);
+    let i1 = Scalar::random(&mut csprng);
+    let z1 = Scalar::random(&mut csprng);
+    let z2 = Scalar::random(&mut csprng);
+    let z3 = (z1 + z2) * i0; // constraint 1: (Z1 + Z2) * I0 - Z3 = 0;
+    let z4 = (z1 + i1) * z3; // constraint 2: (Z1 + I1) * (Z3) - Z4 = 0
+    let z5 = Scalar::zero(); //constraint 3
+
+    let mut vars = vec![Scalar::zero(); num_vars];
+    vars[0] = z1;
+    vars[1] = z2;
+    vars[2] = z3;
+    vars[3] = z4;
+    vars[4] = z5;
+
+    let mut input = vec![Scalar::zero(); num_inputs];
+    input[0] = i0;
+    input[1] = i1;
+
+    (inst, vars, input)
+  }
+
+  #[test]
+  fn test_tiny_r1cs() {
+    let (inst, vars, input) = tests::produce_tiny_r1cs();
+    let is_sat = inst.is_sat(&vars, &input);
+    assert!(is_sat);
+  }
+
+  #[test]
+  fn test_synthetic_r1cs() {
+    let (inst, vars, input) = R1CSInstance::produce_synthetic_r1cs(1024, 1024, 10);
+    let is_sat = inst.is_sat(&vars, &input);
+    assert!(is_sat);
+  }
+
+  #[test]
+  pub fn check_r1cs_proof() {
+    let num_vars = 1024;
+    let num_cons = num_vars;
+    let num_inputs = 10;
+    let (inst, vars, input) = R1CSInstance::produce_synthetic_r1cs(num_cons, num_vars, num_inputs);
+
+    let gens = R1CSGens::new(b"test-m", num_cons, num_vars);
+
+    let mut random_tape = RandomTape::new(b"proof");
+    let mut prover_transcript = Transcript::new(b"example");
+    let (proof, rx, ry) = R1CSProof::prove(
+      &inst,
+      vars,
+      &input,
+      &gens,
+      &mut prover_transcript,
+      &mut random_tape,
+    );
+
+    let inst_evals = inst.evaluate(&rx, &ry);
+
+    let mut verifier_transcript = Transcript::new(b"example");
+    assert!(proof
+      .verify(
+        inst.get_num_vars(),
+        inst.get_num_cons(),
+        &input,
+        &inst_evals,
+        &mut verifier_transcript,
+        &gens,
+      )
+      .is_ok());
+  }
+}

packages/Spartan-secq/src/random.rs

@@ -0,0 +1,27 @@
+use super::scalar::Scalar;
+use super::transcript::ProofTranscript;
+use merlin::Transcript;
+use rand_core::OsRng;
+pub struct RandomTape {
+  tape: Transcript,
+}
+
+impl RandomTape {
+  pub fn new(name: &'static [u8]) -> Self {
+    let tape = {
+      let mut rng = OsRng::default();
+      let mut tape = Transcript::new(name);
+      tape.append_scalar(b"init_randomness", &Scalar::random(&mut rng));
+      tape
+    };
+    Self { tape }
+  }
+
+  pub fn random_scalar(&mut self, label: &'static [u8]) -> Scalar {
+    self.tape.challenge_scalar(label)
+  }
+
+  pub fn random_vector(&mut self, label: &'static [u8], len: usize) -> Vec<Scalar> {
+    self.tape.challenge_vector(label, len)
+  }
+}

packages/Spartan-secq/src/scalar/mod.rs

@@ -0,0 +1,46 @@
+use secq256k1::elliptic_curve::ops::Reduce;
+use secq256k1::U256;
+
+mod scalar;
+
+pub type Scalar = scalar::Scalar;
+pub type ScalarBytes = secq256k1::Scalar;
+
+pub trait ScalarFromPrimitives {
+  fn to_scalar(self) -> Scalar;
+}
+
+impl ScalarFromPrimitives for usize {
+  #[inline]
+  fn to_scalar(self) -> Scalar {
+    (0..self).map(|_i| Scalar::one()).sum()
+  }
+}
+
+impl ScalarFromPrimitives for bool {
+  #[inline]
+  fn to_scalar(self) -> Scalar {
+    if self {
+      Scalar::one()
+    } else {
+      Scalar::zero()
+    }
+  }
+}
+
+pub trait ScalarBytesFromScalar {
+  fn decompress_scalar(s: &Scalar) -> ScalarBytes;
+  fn decompress_vector(s: &[Scalar]) -> Vec<ScalarBytes>;
+}
+
+impl ScalarBytesFromScalar for Scalar {
+  fn decompress_scalar(s: &Scalar) -> ScalarBytes {
+    ScalarBytes::from_uint_reduced(U256::from_le_slice(&s.to_bytes()))
+  }
+
+  fn decompress_vector(s: &[Scalar]) -> Vec<ScalarBytes> {
+    (0..s.len())
+      .map(|i| Scalar::decompress_scalar(&s[i]))
+      .collect::<Vec<ScalarBytes>>()
+  }
+}

packages/Spartan-secq/src/sumcheck.rs

@@ -0,0 +1,778 @@
+#![allow(clippy::too_many_arguments)]
+#![allow(clippy::type_complexity)]
+use super::commitments::{Commitments, MultiCommitGens};
+use super::dense_mlpoly::DensePolynomial;
+use super::errors::ProofVerifyError;
+use super::group::{CompressedGroup, GroupElement, VartimeMultiscalarMul};
+use super::nizk::DotProductProof;
+use super::random::RandomTape;
+use super::scalar::Scalar;
+use super::transcript::{AppendToTranscript, ProofTranscript};
+use super::unipoly::{CompressedUniPoly, UniPoly};
+use crate::group::DecompressEncodedPoint;
+use core::iter;
+use itertools::izip;
+use merlin::Transcript;
+use serde::{Deserialize, Serialize};
+
+#[derive(Serialize, Deserialize, Debug)]
+pub struct SumcheckInstanceProof {
+  compressed_polys: Vec<CompressedUniPoly>,
+}
+
+impl SumcheckInstanceProof {
+  pub fn new(compressed_polys: Vec<CompressedUniPoly>) -> SumcheckInstanceProof {
+    SumcheckInstanceProof { compressed_polys }
+  }
+
+  pub fn verify(
+    &self,
+    claim: Scalar,
+    num_rounds: usize,
+    degree_bound: usize,
+    transcript: &mut Transcript,
+  ) -> Result<(Scalar, Vec<Scalar>), ProofVerifyError> {
+    let mut e = claim;
+    let mut r: Vec<Scalar> = Vec::new();
+
+    // verify that there is a univariate polynomial for each round
+    assert_eq!(self.compressed_polys.len(), num_rounds);
+    for i in 0..self.compressed_polys.len() {
+      let poly = self.compressed_polys[i].decompress(&e);
+
+      // verify degree bound
+      assert_eq!(poly.degree(), degree_bound);
+
+      // check if G_k(0) + G_k(1) = e
+      assert_eq!(poly.eval_at_zero() + poly.eval_at_one(), e);
+
+      // append the prover's message to the transcript
+      poly.append_to_transcript(b"poly", transcript);
+
+      //derive the verifier's challenge for the next round
+      let r_i = transcript.challenge_scalar(b"challenge_nextround");
+
+      r.push(r_i);
+
+      // evaluate the claimed degree-ell polynomial at r_i
+      e = poly.evaluate(&r_i);
+    }
+
+    Ok((e, r))
+  }
+}
+
+#[derive(Serialize, Deserialize, Debug)]
+pub struct ZKSumcheckInstanceProof {
+  comm_polys: Vec<CompressedGroup>,
+  comm_evals: Vec<CompressedGroup>,
+  proofs: Vec<DotProductProof>,
+}
+
+impl ZKSumcheckInstanceProof {
+  pub fn new(
+    comm_polys: Vec<CompressedGroup>,
+    comm_evals: Vec<CompressedGroup>,
+    proofs: Vec<DotProductProof>,
+  ) -> Self {
+    ZKSumcheckInstanceProof {
+      comm_polys,
+      comm_evals,
+      proofs,
+    }
+  }
+
+  pub fn verify(
+    &self,
+    comm_claim: &CompressedGroup,
+    num_rounds: usize,
+    degree_bound: usize,
+    gens_1: &MultiCommitGens,
+    gens_n: &MultiCommitGens,
+    transcript: &mut Transcript,
+  ) -> Result<(CompressedGroup, Vec<Scalar>), ProofVerifyError> {
+    // verify degree bound
+    assert_eq!(gens_n.n, degree_bound + 1);
+
+    // verify that there is a univariate polynomial for each round
+    assert_eq!(self.comm_polys.len(), num_rounds);
+    assert_eq!(self.comm_evals.len(), num_rounds);
+
+    let mut r: Vec<Scalar> = Vec::new();
+    for i in 0..self.comm_polys.len() {
+      let comm_poly = &self.comm_polys[i];
+
+      // append the prover's polynomial to the transcript
+      comm_poly.append_to_transcript(b"comm_poly", transcript);
+
+      //derive the verifier's challenge for the next round
+      let r_i = transcript.challenge_scalar(b"challenge_nextround");
+
+      // verify the proof of sum-check and evals
+      let res = {
+        let comm_claim_per_round = if i == 0 {
+          comm_claim
+        } else {
+          &self.comm_evals[i - 1]
+        };
+        let comm_eval = &self.comm_evals[i];
+
+        // add two claims to transcript
+        comm_claim_per_round.append_to_transcript(b"comm_claim_per_round", transcript);
+        comm_eval.append_to_transcript(b"comm_eval", transcript);
+
+        // produce two weights
+        let w = transcript.challenge_vector(b"combine_two_claims_to_one", 2);
+
+        // compute a weighted sum of the RHS
+        let comm_target = GroupElement::vartime_multiscalar_mul(
+          w.clone(),
+          iter::once(&comm_claim_per_round)
+            .chain(iter::once(&comm_eval))
+            .map(|pt| pt.decompress().unwrap())
+            .collect(),
+        )
+        .compress();
+
+        let a = {
+          // the vector to use to decommit for sum-check test
+          let a_sc = {
+            let mut a = vec![Scalar::one(); degree_bound + 1];
+            a[0] += Scalar::one();
+            a
+          };
+
+          // the vector to use to decommit for evaluation
+          let a_eval = {
+            let mut a = vec![Scalar::one(); degree_bound + 1];
+            for j in 1..a.len() {
+              a[j] = a[j - 1] * r_i;
+            }
+            a
+          };
+
+          // take weighted sum of the two vectors using w
+          assert_eq!(a_sc.len(), a_eval.len());
+          (0..a_sc.len())
+            .map(|i| w[0] * a_sc[i] + w[1] * a_eval[i])
+            .collect::<Vec<Scalar>>()
+        };
+
+        self.proofs[i]
+          .verify(
+            gens_1,
+            gens_n,
+            transcript,
+            &a,
+            &self.comm_polys[i],
+            &comm_target,
+          )
+          .is_ok()
+      };
+      if !res {
+        return Err(ProofVerifyError::InternalError);
+      }
+
+      r.push(r_i);
+    }
+
+    Ok((self.comm_evals[self.comm_evals.len() - 1], r))
+  }
+}
+
+impl SumcheckInstanceProof {
+  pub fn prove_cubic<F>(
+    claim: &Scalar,
+    num_rounds: usize,
+    poly_A: &mut DensePolynomial,
+    poly_B: &mut DensePolynomial,
+    poly_C: &mut DensePolynomial,
+    comb_func: F,
+    transcript: &mut Transcript,
+  ) -> (Self, Vec<Scalar>, Vec<Scalar>)
+  where
+    F: Fn(&Scalar, &Scalar, &Scalar) -> Scalar,
+  {
+    let mut e = *claim;
+    let mut r: Vec<Scalar> = Vec::new();
+    let mut cubic_polys: Vec<CompressedUniPoly> = Vec::new();
+    for _j in 0..num_rounds {
+      let mut eval_point_0 = Scalar::zero();
+      let mut eval_point_2 = Scalar::zero();
+      let mut eval_point_3 = Scalar::zero();
+
+      let len = poly_A.len() / 2;
+      for i in 0..len {
+        // eval 0: bound_func is A(low)
+        eval_point_0 += comb_func(&poly_A[i], &poly_B[i], &poly_C[i]);
+
+        // eval 2: bound_func is -A(low) + 2*A(high)
+        let poly_A_bound_point = poly_A[len + i] + poly_A[len + i] - poly_A[i];
+        let poly_B_bound_point = poly_B[len + i] + poly_B[len + i] - poly_B[i];
+        let poly_C_bound_point = poly_C[len + i] + poly_C[len + i] - poly_C[i];
+        eval_point_2 += comb_func(
+          &poly_A_bound_point,
+          &poly_B_bound_point,
+          &poly_C_bound_point,
+        );
+
+        // eval 3: bound_func is -2A(low) + 3A(high); computed incrementally with bound_func applied to eval(2)
+        let poly_A_bound_point = poly_A_bound_point + poly_A[len + i] - poly_A[i];
+        let poly_B_bound_point = poly_B_bound_point + poly_B[len + i] - poly_B[i];
+        let poly_C_bound_point = poly_C_bound_point + poly_C[len + i] - poly_C[i];
+
+        eval_point_3 += comb_func(
+          &poly_A_bound_point,
+          &poly_B_bound_point,
+          &poly_C_bound_point,
+        );
+      }
+
+      let evals = vec![eval_point_0, e - eval_point_0, eval_point_2, eval_point_3];
+      let poly = UniPoly::from_evals(&evals);
+
+      // append the prover's message to the transcript
+      poly.append_to_transcript(b"poly", transcript);
+
+      //derive the verifier's challenge for the next round
+      let r_j = transcript.challenge_scalar(b"challenge_nextround");
+      r.push(r_j);
+      // bound all tables to the verifier's challenege
+      poly_A.bound_poly_var_top(&r_j);
+      poly_B.bound_poly_var_top(&r_j);
+      poly_C.bound_poly_var_top(&r_j);
+      e = poly.evaluate(&r_j);
+      cubic_polys.push(poly.compress());
+    }
+
+    (
+      SumcheckInstanceProof::new(cubic_polys),
+      r,
+      vec![poly_A[0], poly_B[0], poly_C[0]],
+    )
+  }
+
+  pub fn prove_cubic_batched<F>(
+    claim: &Scalar,
+    num_rounds: usize,
+    poly_vec_par: (
+      &mut Vec<&mut DensePolynomial>,
+      &mut Vec<&mut DensePolynomial>,
+      &mut DensePolynomial,
+    ),
+    poly_vec_seq: (
+      &mut Vec<&mut DensePolynomial>,
+      &mut Vec<&mut DensePolynomial>,
+      &mut Vec<&mut DensePolynomial>,
+    ),
+    coeffs: &[Scalar],
+    comb_func: F,
+    transcript: &mut Transcript,
+  ) -> (
+    Self,
+    Vec<Scalar>,
+    (Vec<Scalar>, Vec<Scalar>, Scalar),
+    (Vec<Scalar>, Vec<Scalar>, Vec<Scalar>),
+  )
+  where
+    F: Fn(&Scalar, &Scalar, &Scalar) -> Scalar,
+  {
+    let (poly_A_vec_par, poly_B_vec_par, poly_C_par) = poly_vec_par;
+    let (poly_A_vec_seq, poly_B_vec_seq, poly_C_vec_seq) = poly_vec_seq;
+
+    //let (poly_A_vec_seq, poly_B_vec_seq, poly_C_vec_seq) = poly_vec_seq;
+    let mut e = *claim;
+    let mut r: Vec<Scalar> = Vec::new();
+    let mut cubic_polys: Vec<CompressedUniPoly> = Vec::new();
+
+    for _j in 0..num_rounds {
+      let mut evals: Vec<(Scalar, Scalar, Scalar)> = Vec::new();
+
+      for (poly_A, poly_B) in poly_A_vec_par.iter().zip(poly_B_vec_par.iter()) {
+        let mut eval_point_0 = Scalar::zero();
+        let mut eval_point_2 = Scalar::zero();
+        let mut eval_point_3 = Scalar::zero();
+
+        let len = poly_A.len() / 2;
+        for i in 0..len {
+          // eval 0: bound_func is A(low)
+          eval_point_0 += comb_func(&poly_A[i], &poly_B[i], &poly_C_par[i]);
+
+          // eval 2: bound_func is -A(low) + 2*A(high)
+          let poly_A_bound_point = poly_A[len + i] + poly_A[len + i] - poly_A[i];
+          let poly_B_bound_point = poly_B[len + i] + poly_B[len + i] - poly_B[i];
+          let poly_C_bound_point = poly_C_par[len + i] + poly_C_par[len + i] - poly_C_par[i];
+          eval_point_2 += comb_func(
+            &poly_A_bound_point,
+            &poly_B_bound_point,
+            &poly_C_bound_point,
+          );
+
+          // eval 3: bound_func is -2A(low) + 3A(high); computed incrementally with bound_func applied to eval(2)
+          let poly_A_bound_point = poly_A_bound_point + poly_A[len + i] - poly_A[i];
+          let poly_B_bound_point = poly_B_bound_point + poly_B[len + i] - poly_B[i];
+          let poly_C_bound_point = poly_C_bound_point + poly_C_par[len + i] - poly_C_par[i];
+
+          eval_point_3 += comb_func(
+            &poly_A_bound_point,
+            &poly_B_bound_point,
+            &poly_C_bound_point,
+          );
+        }
+
+        evals.push((eval_point_0, eval_point_2, eval_point_3));
+      }
+
+      for (poly_A, poly_B, poly_C) in izip!(
+        poly_A_vec_seq.iter(),
+        poly_B_vec_seq.iter(),
+        poly_C_vec_seq.iter()
+      ) {
+        let mut eval_point_0 = Scalar::zero();
+        let mut eval_point_2 = Scalar::zero();
+        let mut eval_point_3 = Scalar::zero();
+        let len = poly_A.len() / 2;
+        for i in 0..len {
+          // eval 0: bound_func is A(low)
+          eval_point_0 += comb_func(&poly_A[i], &poly_B[i], &poly_C[i]);
+          // eval 2: bound_func is -A(low) + 2*A(high)
+          let poly_A_bound_point = poly_A[len + i] + poly_A[len + i] - poly_A[i];
+          let poly_B_bound_point = poly_B[len + i] + poly_B[len + i] - poly_B[i];
+          let poly_C_bound_point = poly_C[len + i] + poly_C[len + i] - poly_C[i];
+          eval_point_2 += comb_func(
+            &poly_A_bound_point,
+            &poly_B_bound_point,
+            &poly_C_bound_point,
+          );
+          // eval 3: bound_func is -2A(low) + 3A(high); computed incrementally with bound_func applied to eval(2)
+          let poly_A_bound_point = poly_A_bound_point + poly_A[len + i] - poly_A[i];
+          let poly_B_bound_point = poly_B_bound_point + poly_B[len + i] - poly_B[i];
+          let poly_C_bound_point = poly_C_bound_point + poly_C[len + i] - poly_C[i];
+          eval_point_3 += comb_func(
+            &poly_A_bound_point,
+            &poly_B_bound_point,
+            &poly_C_bound_point,
+          );
+        }
+        evals.push((eval_point_0, eval_point_2, eval_point_3));
+      }
+
+      let evals_combined_0 = (0..evals.len()).map(|i| evals[i].0 * coeffs[i]).sum();
+      let evals_combined_2 = (0..evals.len()).map(|i| evals[i].1 * coeffs[i]).sum();
+      let evals_combined_3 = (0..evals.len()).map(|i| evals[i].2 * coeffs[i]).sum();
+
+      let evals = vec![
+        evals_combined_0,
+        e - evals_combined_0,
+        evals_combined_2,
+        evals_combined_3,
+      ];
+      let poly = UniPoly::from_evals(&evals);
+
+      // append the prover's message to the transcript
+      poly.append_to_transcript(b"poly", transcript);
+
+      //derive the verifier's challenge for the next round
+      let r_j = transcript.challenge_scalar(b"challenge_nextround");
+      r.push(r_j);
+
+      // bound all tables to the verifier's challenege
+      for (poly_A, poly_B) in poly_A_vec_par.iter_mut().zip(poly_B_vec_par.iter_mut()) {
+        poly_A.bound_poly_var_top(&r_j);
+        poly_B.bound_poly_var_top(&r_j);
+      }
+      poly_C_par.bound_poly_var_top(&r_j);
+
+      for (poly_A, poly_B, poly_C) in izip!(
+        poly_A_vec_seq.iter_mut(),
+        poly_B_vec_seq.iter_mut(),
+        poly_C_vec_seq.iter_mut()
+      ) {
+        poly_A.bound_poly_var_top(&r_j);
+        poly_B.bound_poly_var_top(&r_j);
+        poly_C.bound_poly_var_top(&r_j);
+      }
+
+      e = poly.evaluate(&r_j);
+      cubic_polys.push(poly.compress());
+    }
+
+    let poly_A_par_final = (0..poly_A_vec_par.len())
+      .map(|i| poly_A_vec_par[i][0])
+      .collect();
+    let poly_B_par_final = (0..poly_B_vec_par.len())
+      .map(|i| poly_B_vec_par[i][0])
+      .collect();
+    let claims_prod = (poly_A_par_final, poly_B_par_final, poly_C_par[0]);
+
+    let poly_A_seq_final = (0..poly_A_vec_seq.len())
+      .map(|i| poly_A_vec_seq[i][0])
+      .collect();
+    let poly_B_seq_final = (0..poly_B_vec_seq.len())
+      .map(|i| poly_B_vec_seq[i][0])
+      .collect();
+    let poly_C_seq_final = (0..poly_C_vec_seq.len())
+      .map(|i| poly_C_vec_seq[i][0])
+      .collect();
+    let claims_dotp = (poly_A_seq_final, poly_B_seq_final, poly_C_seq_final);
+
+    (
+      SumcheckInstanceProof::new(cubic_polys),
+      r,
+      claims_prod,
+      claims_dotp,
+    )
+  }
+}
+
+impl ZKSumcheckInstanceProof {
+  pub fn prove_quad<F>(
+    claim: &Scalar,
+    blind_claim: &Scalar,
+    num_rounds: usize,
+    poly_A: &mut DensePolynomial,
+    poly_B: &mut DensePolynomial,
+    comb_func: F,
+    gens_1: &MultiCommitGens,
+    gens_n: &MultiCommitGens,
+    transcript: &mut Transcript,
+    random_tape: &mut RandomTape,
+  ) -> (Self, Vec<Scalar>, Vec<Scalar>, Scalar)
+  where
+    F: Fn(&Scalar, &Scalar) -> Scalar,
+  {
+    let (blinds_poly, blinds_evals) = (
+      random_tape.random_vector(b"blinds_poly", num_rounds),
+      random_tape.random_vector(b"blinds_evals", num_rounds),
+    );
+    let mut claim_per_round = *claim;
+    let mut comm_claim_per_round = claim_per_round.commit(blind_claim, gens_1).compress();
+
+    let mut r: Vec<Scalar> = Vec::new();
+    let mut comm_polys: Vec<CompressedGroup> = Vec::new();
+    let mut comm_evals: Vec<CompressedGroup> = Vec::new();
+    let mut proofs: Vec<DotProductProof> = Vec::new();
+
+    for j in 0..num_rounds {
+      let (poly, comm_poly) = {
+        let mut eval_point_0 = Scalar::zero();
+        let mut eval_point_2 = Scalar::zero();
+
+        let len = poly_A.len() / 2;
+        for i in 0..len {
+          // eval 0: bound_func is A(low)
+          eval_point_0 += comb_func(&poly_A[i], &poly_B[i]);
+
+          // eval 2: bound_func is -A(low) + 2*A(high)
+          let poly_A_bound_point = poly_A[len + i] + poly_A[len + i] - poly_A[i];
+          let poly_B_bound_point = poly_B[len + i] + poly_B[len + i] - poly_B[i];
+          eval_point_2 += comb_func(&poly_A_bound_point, &poly_B_bound_point);
+        }
+
+        let evals = vec![eval_point_0, claim_per_round - eval_point_0, eval_point_2];
+        let poly = UniPoly::from_evals(&evals);
+        let comm_poly = poly.commit(gens_n, &blinds_poly[j]).compress();
+        (poly, comm_poly)
+      };
+
+      // append the prover's message to the transcript
+      comm_poly.append_to_transcript(b"comm_poly", transcript);
+      comm_polys.push(comm_poly);
+
+      //derive the verifier's challenge for the next round
+      let r_j = transcript.challenge_scalar(b"challenge_nextround");
+
+      // bound all tables to the verifier's challenege
+      poly_A.bound_poly_var_top(&r_j);
+      poly_B.bound_poly_var_top(&r_j);
+
+      // produce a proof of sum-check and of evaluation
+      let (proof, claim_next_round, comm_claim_next_round) = {
+        let eval = poly.evaluate(&r_j);
+        let comm_eval = eval.commit(&blinds_evals[j], gens_1).compress();
+
+        // we need to prove the following under homomorphic commitments:
+        // (1) poly(0) + poly(1) = claim_per_round
+        // (2) poly(r_j) = eval
+
+        // Our technique is to leverage dot product proofs:
+        // (1) we can prove: <poly_in_coeffs_form, (2, 1, 1, 1)> = claim_per_round
+        // (2) we can prove: <poly_in_coeffs_form, (1, r_j, r^2_j, ..) = eval
+        // for efficiency we batch them using random weights
+
+        // add two claims to transcript
+        comm_claim_per_round.append_to_transcript(b"comm_claim_per_round", transcript);
+        comm_eval.append_to_transcript(b"comm_eval", transcript);
+
+        // produce two weights
+        let w = transcript.challenge_vector(b"combine_two_claims_to_one", 2);
+
+        // compute a weighted sum of the RHS
+        let target = w[0] * claim_per_round + w[1] * eval;
+        let comm_target = GroupElement::vartime_multiscalar_mul(
+          w.clone(),
+          iter::once(&comm_claim_per_round)
+            .chain(iter::once(&comm_eval))
+            .map(|pt| pt.decompress().unwrap())
+            .collect(),
+        )
+        .compress();
+
+        let blind = {
+          let blind_sc = if j == 0 {
+            blind_claim
+          } else {
+            &blinds_evals[j - 1]
+          };
+
+          let blind_eval = &blinds_evals[j];
+
+          w[0] * blind_sc + w[1] * blind_eval
+        };
+        assert_eq!(target.commit(&blind, gens_1).compress(), comm_target);
+
+        let a = {
+          // the vector to use to decommit for sum-check test
+          let a_sc = {
+            let mut a = vec![Scalar::one(); poly.degree() + 1];
+            a[0] += Scalar::one();
+            a
+          };
+
+          // the vector to use to decommit for evaluation
+          let a_eval = {
+            let mut a = vec![Scalar::one(); poly.degree() + 1];
+            for j in 1..a.len() {
+              a[j] = a[j - 1] * r_j;
+            }
+            a
+          };
+
+          // take weighted sum of the two vectors using w
+          assert_eq!(a_sc.len(), a_eval.len());
+          (0..a_sc.len())
+            .map(|i| w[0] * a_sc[i] + w[1] * a_eval[i])
+            .collect::<Vec<Scalar>>()
+        };
+
+        let (proof, _comm_poly, _comm_sc_eval) = DotProductProof::prove(
+          gens_1,
+          gens_n,
+          transcript,
+          random_tape,
+          &poly.as_vec(),
+          &blinds_poly[j],
+          &a,
+          &target,
+          &blind,
+        );
+
+        (proof, eval, comm_eval)
+      };
+
+      claim_per_round = claim_next_round;
+      comm_claim_per_round = comm_claim_next_round;
+
+      proofs.push(proof);
+      r.push(r_j);
+      comm_evals.push(comm_claim_per_round);
+    }
+
+    (
+      ZKSumcheckInstanceProof::new(comm_polys, comm_evals, proofs),
+      r,
+      vec![poly_A[0], poly_B[0]],
+      blinds_evals[num_rounds - 1],
+    )
+  }
+
+  pub fn prove_cubic_with_additive_term<F>(
+    claim: &Scalar,
+    blind_claim: &Scalar,
+    num_rounds: usize,
+    poly_A: &mut DensePolynomial,
+    poly_B: &mut DensePolynomial,
+    poly_C: &mut DensePolynomial,
+    poly_D: &mut DensePolynomial,
+    comb_func: F,
+    gens_1: &MultiCommitGens,
+    gens_n: &MultiCommitGens,
+    transcript: &mut Transcript,
+    random_tape: &mut RandomTape,
+  ) -> (Self, Vec<Scalar>, Vec<Scalar>, Scalar)
+  where
+    F: Fn(&Scalar, &Scalar, &Scalar, &Scalar) -> Scalar,
+  {
+    let (blinds_poly, blinds_evals) = (
+      random_tape.random_vector(b"blinds_poly", num_rounds),
+      random_tape.random_vector(b"blinds_evals", num_rounds),
+    );
+
+    let mut claim_per_round = *claim;
+    let mut comm_claim_per_round = claim_per_round.commit(blind_claim, gens_1).compress();
+
+    let mut r: Vec<Scalar> = Vec::new();
+    let mut comm_polys: Vec<CompressedGroup> = Vec::new();
+    let mut comm_evals: Vec<CompressedGroup> = Vec::new();
+    let mut proofs: Vec<DotProductProof> = Vec::new();
+
+    for j in 0..num_rounds {
+      let (poly, comm_poly) = {
+        let mut eval_point_0 = Scalar::zero();
+        let mut eval_point_2 = Scalar::zero();
+        let mut eval_point_3 = Scalar::zero();
+
+        let len = poly_A.len() / 2;
+        for i in 0..len {
+          // eval 0: bound_func is A(low)
+          eval_point_0 += comb_func(&poly_A[i], &poly_B[i], &poly_C[i], &poly_D[i]);
+
+          // eval 2: bound_func is -A(low) + 2*A(high)
+          let poly_A_bound_point = poly_A[len + i] + poly_A[len + i] - poly_A[i];
+          let poly_B_bound_point = poly_B[len + i] + poly_B[len + i] - poly_B[i];
+          let poly_C_bound_point = poly_C[len + i] + poly_C[len + i] - poly_C[i];
+          let poly_D_bound_point = poly_D[len + i] + poly_D[len + i] - poly_D[i];
+          eval_point_2 += comb_func(
+            &poly_A_bound_point,
+            &poly_B_bound_point,
+            &poly_C_bound_point,
+            &poly_D_bound_point,
+          );
+
+          // eval 3: bound_func is -2A(low) + 3A(high); computed incrementally with bound_func applied to eval(2)
+          let poly_A_bound_point = poly_A_bound_point + poly_A[len + i] - poly_A[i];
+          let poly_B_bound_point = poly_B_bound_point + poly_B[len + i] - poly_B[i];
+          let poly_C_bound_point = poly_C_bound_point + poly_C[len + i] - poly_C[i];
+          let poly_D_bound_point = poly_D_bound_point + poly_D[len + i] - poly_D[i];
+          eval_point_3 += comb_func(
+            &poly_A_bound_point,
+            &poly_B_bound_point,
+            &poly_C_bound_point,
+            &poly_D_bound_point,
+          );
+        }
+
+        let evals = vec![
+          eval_point_0,
+          claim_per_round - eval_point_0,
+          eval_point_2,
+          eval_point_3,
+        ];
+        let poly = UniPoly::from_evals(&evals);
+        let comm_poly = poly.commit(gens_n, &blinds_poly[j]).compress();
+        (poly, comm_poly)
+      };
+
+      // append the prover's message to the transcript
+      comm_poly.append_to_transcript(b"comm_poly", transcript);
+      comm_polys.push(comm_poly);
+
+      //derive the verifier's challenge for the next round
+      let r_j = transcript.challenge_scalar(b"challenge_nextround");
+
+      // bound all tables to the verifier's challenege
+      poly_A.bound_poly_var_top(&r_j);
+      poly_B.bound_poly_var_top(&r_j);
+      poly_C.bound_poly_var_top(&r_j);
+      poly_D.bound_poly_var_top(&r_j);
+
+      // produce a proof of sum-check and of evaluation
+      let (proof, claim_next_round, comm_claim_next_round) = {
+        let eval = poly.evaluate(&r_j);
+        let comm_eval = eval.commit(&blinds_evals[j], gens_1).compress();
+
+        // we need to prove the following under homomorphic commitments:
+        // (1) poly(0) + poly(1) = claim_per_round
+        // (2) poly(r_j) = eval
+
+        // Our technique is to leverage dot product proofs:
+        // (1) we can prove: <poly_in_coeffs_form, (2, 1, 1, 1)> = claim_per_round
+        // (2) we can prove: <poly_in_coeffs_form, (1, r_j, r^2_j, ..) = eval
+        // for efficiency we batch them using random weights
+
+        // add two claims to transcript
+        comm_claim_per_round.append_to_transcript(b"comm_claim_per_round", transcript);
+        comm_eval.append_to_transcript(b"comm_eval", transcript);
+
+        // produce two weights
+        let w = transcript.challenge_vector(b"combine_two_claims_to_one", 2);
+
+        // compute a weighted sum of the RHS
+        let target = w[0] * claim_per_round + w[1] * eval;
+        let comm_target = GroupElement::vartime_multiscalar_mul(
+          w.clone(),
+          iter::once(&comm_claim_per_round)
+            .chain(iter::once(&comm_eval))
+            .map(|pt| pt.decompress().unwrap())
+            .collect::<Vec<GroupElement>>(),
+        )
+        .compress();
+
+        let blind = {
+          let blind_sc = if j == 0 {
+            blind_claim
+          } else {
+            &blinds_evals[j - 1]
+          };
+
+          let blind_eval = &blinds_evals[j];
+
+          w[0] * blind_sc + w[1] * blind_eval
+        };
+
+        assert_eq!(target.commit(&blind, gens_1).compress(), comm_target);
+
+        let a = {
+          // the vector to use to decommit for sum-check test
+          let a_sc = {
+            let mut a = vec![Scalar::one(); poly.degree() + 1];
+            a[0] += Scalar::one();
+            a
+          };
+
+          // the vector to use to decommit for evaluation
+          let a_eval = {
+            let mut a = vec![Scalar::one(); poly.degree() + 1];
+            for j in 1..a.len() {
+              a[j] = a[j - 1] * r_j;
+            }
+            a
+          };
+
+          // take weighted sum of the two vectors using w
+          assert_eq!(a_sc.len(), a_eval.len());
+          (0..a_sc.len())
+            .map(|i| w[0] * a_sc[i] + w[1] * a_eval[i])
+            .collect::<Vec<Scalar>>()
+        };
+
+        let (proof, _comm_poly, _comm_sc_eval) = DotProductProof::prove(
+          gens_1,
+          gens_n,
+          transcript,
+          random_tape,
+          &poly.as_vec(),
+          &blinds_poly[j],
+          &a,
+          &target,
+          &blind,
+        );
+
+        (proof, eval, comm_eval)
+      };
+
+      proofs.push(proof);
+      claim_per_round = claim_next_round;
+      comm_claim_per_round = comm_claim_next_round;
+      r.push(r_j);
+      comm_evals.push(comm_claim_per_round);
+    }
+
+    (
+      ZKSumcheckInstanceProof::new(comm_polys, comm_evals, proofs),
+      r,
+      vec![poly_A[0], poly_B[0], poly_C[0], poly_D[0]],
+      blinds_evals[num_rounds - 1],
+    )
+  }
+}

packages/Spartan-secq/src/timer.rs

@@ -0,0 +1,88 @@
+#[cfg(feature = "profile")]
+use colored::Colorize;
+#[cfg(feature = "profile")]
+use core::sync::atomic::AtomicUsize;
+#[cfg(feature = "profile")]
+use core::sync::atomic::Ordering;
+#[cfg(feature = "profile")]
+use std::time::Instant;
+
+#[cfg(feature = "profile")]
+pub static CALL_DEPTH: AtomicUsize = AtomicUsize::new(0);
+
+#[cfg(feature = "profile")]
+pub struct Timer {
+  label: String,
+  timer: Instant,
+}
+
+#[cfg(feature = "profile")]
+impl Timer {
+  #[inline(always)]
+  pub fn new(label: &str) -> Self {
+    let timer = Instant::now();
+    CALL_DEPTH.fetch_add(1, Ordering::Relaxed);
+    let star = "* ";
+    println!(
+      "{:indent$}{}{}",
+      "",
+      star,
+      label.yellow().bold(),
+      indent = 2 * CALL_DEPTH.fetch_add(0, Ordering::Relaxed)
+    );
+    Self {
+      label: label.to_string(),
+      timer,
+    }
+  }
+
+  #[inline(always)]
+  pub fn stop(&self) {
+    let duration = self.timer.elapsed();
+    let star = "* ";
+    println!(
+      "{:indent$}{}{} {:?}",
+      "",
+      star,
+      self.label.blue().bold(),
+      duration,
+      indent = 2 * CALL_DEPTH.fetch_add(0, Ordering::Relaxed)
+    );
+    CALL_DEPTH.fetch_sub(1, Ordering::Relaxed);
+  }
+
+  #[inline(always)]
+  pub fn print(msg: &str) {
+    CALL_DEPTH.fetch_add(1, Ordering::Relaxed);
+    let star = "* ";
+    println!(
+      "{:indent$}{}{}",
+      "",
+      star,
+      msg.to_string().green().bold(),
+      indent = 2 * CALL_DEPTH.fetch_add(0, Ordering::Relaxed)
+    );
+    CALL_DEPTH.fetch_sub(1, Ordering::Relaxed);
+  }
+}
+
+#[cfg(not(feature = "profile"))]
+pub struct Timer {
+  _label: String,
+}
+
+#[cfg(not(feature = "profile"))]
+impl Timer {
+  #[inline(always)]
+  pub fn new(label: &str) -> Self {
+    Self {
+      _label: label.to_string(),
+    }
+  }
+
+  #[inline(always)]
+  pub fn stop(&self) {}
+
+  #[inline(always)]
+  pub fn print(_msg: &str) {}
+}

packages/Spartan-secq/src/transcript.rs

@@ -0,0 +1,63 @@
+use super::group::CompressedGroup;
+use super::scalar::Scalar;
+use merlin::Transcript;
+
+pub trait ProofTranscript {
+  fn append_protocol_name(&mut self, protocol_name: &'static [u8]);
+  fn append_scalar(&mut self, label: &'static [u8], scalar: &Scalar);
+  fn append_point(&mut self, label: &'static [u8], point: &CompressedGroup);
+  fn challenge_scalar(&mut self, label: &'static [u8]) -> Scalar;
+  fn challenge_vector(&mut self, label: &'static [u8], len: usize) -> Vec<Scalar>;
+}
+
+impl ProofTranscript for Transcript {
+  fn append_protocol_name(&mut self, protocol_name: &'static [u8]) {
+    self.append_message(b"protocol-name", protocol_name);
+  }
+
+  fn append_scalar(&mut self, label: &'static [u8], scalar: &Scalar) {
+    self.append_message(label, &scalar.to_bytes());
+  }
+
+  fn append_point(&mut self, label: &'static [u8], point: &CompressedGroup) {
+    self.append_message(label, point.as_bytes());
+  }
+
+  fn challenge_scalar(&mut self, label: &'static [u8]) -> Scalar {
+    let mut buf = [0u8; 64];
+    self.challenge_bytes(label, &mut buf);
+    Scalar::from_bytes_wide(&buf)
+  }
+
+  fn challenge_vector(&mut self, label: &'static [u8], len: usize) -> Vec<Scalar> {
+    (0..len)
+      .map(|_i| self.challenge_scalar(label))
+      .collect::<Vec<Scalar>>()
+  }
+}
+
+pub trait AppendToTranscript {
+  fn append_to_transcript(&self, label: &'static [u8], transcript: &mut Transcript);
+}
+
+impl AppendToTranscript for Scalar {
+  fn append_to_transcript(&self, label: &'static [u8], transcript: &mut Transcript) {
+    transcript.append_scalar(label, self);
+  }
+}
+
+impl AppendToTranscript for [Scalar] {
+  fn append_to_transcript(&self, label: &'static [u8], transcript: &mut Transcript) {
+    transcript.append_message(label, b"begin_append_vector");
+    for item in self {
+      transcript.append_scalar(label, item);
+    }
+    transcript.append_message(label, b"end_append_vector");
+  }
+}
+
+impl AppendToTranscript for CompressedGroup {
+  fn append_to_transcript(&self, label: &'static [u8], transcript: &mut Transcript) {
+    transcript.append_point(label, self);
+  }
+}

packages/Spartan-secq/src/unipoly.rs

@@ -0,0 +1,182 @@
+use super::commitments::{Commitments, MultiCommitGens};
+use super::group::GroupElement;
+use super::scalar::{Scalar, ScalarFromPrimitives};
+use super::transcript::{AppendToTranscript, ProofTranscript};
+use merlin::Transcript;
+use serde::{Deserialize, Serialize};
+
+// ax^2 + bx + c stored as vec![c,b,a]
+// ax^3 + bx^2 + cx + d stored as vec![d,c,b,a]
+#[derive(Debug)]
+pub struct UniPoly {
+  coeffs: Vec<Scalar>,
+}
+
+// ax^2 + bx + c stored as vec![c,a]
+// ax^3 + bx^2 + cx + d stored as vec![d,b,a]
+#[derive(Serialize, Deserialize, Debug)]
+pub struct CompressedUniPoly {
+  coeffs_except_linear_term: Vec<Scalar>,
+}
+
+impl UniPoly {
+  pub fn from_evals(evals: &[Scalar]) -> Self {
+    // we only support degree-2 or degree-3 univariate polynomials
+    assert!(evals.len() == 3 || evals.len() == 4);
+    let coeffs = if evals.len() == 3 {
+      // ax^2 + bx + c
+      let two_inv = (2_usize).to_scalar().invert().unwrap();
+
+      let c = evals[0];
+      let a = two_inv * (evals[2] - evals[1] - evals[1] + c);
+      let b = evals[1] - c - a;
+      vec![c, b, a]
+    } else {
+      // ax^3 + bx^2 + cx + d
+      let two_inv = (2_usize).to_scalar().invert().unwrap();
+      let six_inv = (6_usize).to_scalar().invert().unwrap();
+
+      let d = evals[0];
+      let a = six_inv
+        * (evals[3] - evals[2] - evals[2] - evals[2] + evals[1] + evals[1] + evals[1] - evals[0]);
+      let b = two_inv
+        * (evals[0] + evals[0] - evals[1] - evals[1] - evals[1] - evals[1] - evals[1]
+          + evals[2]
+          + evals[2]
+          + evals[2]
+          + evals[2]
+          - evals[3]);
+      let c = evals[1] - d - a - b;
+      vec![d, c, b, a]
+    };
+
+    UniPoly { coeffs }
+  }
+
+  pub fn degree(&self) -> usize {
+    self.coeffs.len() - 1
+  }
+
+  pub fn as_vec(&self) -> Vec<Scalar> {
+    self.coeffs.clone()
+  }
+
+  pub fn eval_at_zero(&self) -> Scalar {
+    self.coeffs[0]
+  }
+
+  pub fn eval_at_one(&self) -> Scalar {
+    (0..self.coeffs.len()).map(|i| self.coeffs[i]).sum()
+  }
+
+  pub fn evaluate(&self, r: &Scalar) -> Scalar {
+    let mut eval = self.coeffs[0];
+    let mut power = *r;
+    for i in 1..self.coeffs.len() {
+      eval += power * self.coeffs[i];
+      power *= r;
+    }
+    eval
+  }
+
+  pub fn compress(&self) -> CompressedUniPoly {
+    let coeffs_except_linear_term = [&self.coeffs[..1], &self.coeffs[2..]].concat();
+    assert_eq!(coeffs_except_linear_term.len() + 1, self.coeffs.len());
+    CompressedUniPoly {
+      coeffs_except_linear_term,
+    }
+  }
+
+  pub fn commit(&self, gens: &MultiCommitGens, blind: &Scalar) -> GroupElement {
+    self.coeffs.commit(blind, gens)
+  }
+}
+
+impl CompressedUniPoly {
+  // we require eval(0) + eval(1) = hint, so we can solve for the linear term as:
+  // linear_term = hint - 2 * constant_term - deg2 term - deg3 term
+  pub fn decompress(&self, hint: &Scalar) -> UniPoly {
+    let mut linear_term =
+      hint - self.coeffs_except_linear_term[0] - self.coeffs_except_linear_term[0];
+    for i in 1..self.coeffs_except_linear_term.len() {
+      linear_term -= self.coeffs_except_linear_term[i];
+    }
+
+    let mut coeffs = vec![self.coeffs_except_linear_term[0], linear_term];
+    coeffs.extend(&self.coeffs_except_linear_term[1..]);
+    assert_eq!(self.coeffs_except_linear_term.len() + 1, coeffs.len());
+    UniPoly { coeffs }
+  }
+}
+
+impl AppendToTranscript for UniPoly {
+  fn append_to_transcript(&self, label: &'static [u8], transcript: &mut Transcript) {
+    transcript.append_message(label, b"UniPoly_begin");
+    for i in 0..self.coeffs.len() {
+      transcript.append_scalar(b"coeff", &self.coeffs[i]);
+    }
+    transcript.append_message(label, b"UniPoly_end");
+  }
+}
+
+#[cfg(test)]
+mod tests {
+
+  use super::*;
+
+  #[test]
+  fn test_from_evals_quad() {
+    // polynomial is 2x^2 + 3x + 1
+    let e0 = Scalar::one();
+    let e1 = (6_usize).to_scalar();
+    let e2 = (15_usize).to_scalar();
+    let evals = vec![e0, e1, e2];
+    let poly = UniPoly::from_evals(&evals);
+
+    assert_eq!(poly.eval_at_zero(), e0);
+    assert_eq!(poly.eval_at_one(), e1);
+    assert_eq!(poly.coeffs.len(), 3);
+    assert_eq!(poly.coeffs[0], Scalar::one());
+    assert_eq!(poly.coeffs[1], (3_usize).to_scalar());
+    assert_eq!(poly.coeffs[2], (2_usize).to_scalar());
+
+    let hint = e0 + e1;
+    let compressed_poly = poly.compress();
+    let decompressed_poly = compressed_poly.decompress(&hint);
+    for i in 0..decompressed_poly.coeffs.len() {
+      assert_eq!(decompressed_poly.coeffs[i], poly.coeffs[i]);
+    }
+
+    let e3 = (28_usize).to_scalar();
+    assert_eq!(poly.evaluate(&(3_usize).to_scalar()), e3);
+  }
+
+  #[test]
+  fn test_from_evals_cubic() {
+    // polynomial is x^3 + 2x^2 + 3x + 1
+    let e0 = Scalar::one();
+    let e1 = (7_usize).to_scalar();
+    let e2 = (23_usize).to_scalar();
+    let e3 = (55_usize).to_scalar();
+    let evals = vec![e0, e1, e2, e3];
+    let poly = UniPoly::from_evals(&evals);
+
+    assert_eq!(poly.eval_at_zero(), e0);
+    assert_eq!(poly.eval_at_one(), e1);
+    assert_eq!(poly.coeffs.len(), 4);
+    assert_eq!(poly.coeffs[0], Scalar::one());
+    assert_eq!(poly.coeffs[1], (3_usize).to_scalar());
+    assert_eq!(poly.coeffs[2], (2_usize).to_scalar());
+    assert_eq!(poly.coeffs[3], (1_usize).to_scalar());
+
+    let hint = e0 + e1;
+    let compressed_poly = poly.compress();
+    let decompressed_poly = compressed_poly.decompress(&hint);
+    for i in 0..decompressed_poly.coeffs.len() {
+      assert_eq!(decompressed_poly.coeffs[i], poly.coeffs[i]);
+    }
+
+    let e4 = (109_usize).to_scalar();
+    assert_eq!(poly.evaluate(&(4_usize).to_scalar()), e4);
+  }
+}