Add MSM and Guard structs in polycommit scheme

src/arithmetic.rs

@@ -70,6 +70,10 @@ where
 }
 
 /// This is a 128-bit verifier challenge.
+///
+/// The verifier samples its challenge here as u^2, i.e. the square of the
+/// actual challenge. This is an optimisation that is documented in Section 6.3
+/// of the [Halo](https://eprint.iacr.org/2019/1021) paper.
 #[derive(Copy, Clone, Debug)]
 pub struct Challenge(pub(crate) u128);
 

src/plonk/verifier.rs

@@ -1,6 +1,9 @@
 use super::{hash_point, Proof, SRS};
 use crate::arithmetic::{get_challenge_scalar, Challenge, Curve, CurveAffine, Field};
-use crate::poly::{commitment::Params, Rotation};
+use crate::poly::{
+    commitment::{Params, MSM},
+    Rotation,
+};
 use crate::transcript::Hasher;
 
 impl<C: CurveAffine> Proof<C> {
@@ -261,12 +264,20 @@ impl<C: CurveAffine> Proof<C> {
         }
 
         // Verify the opening proof
-        self.opening.verify(
+        let (challenges, mut guard) = self
-            params,
+            .opening
-            &mut transcript,
+            .verify(
-            x_6,
+                params,
-            &f_commitment.to_affine(),
+                &mut MSM::default(&params),
-            f_eval,
+                &mut transcript,
-        )
+                x_6,
+                &f_commitment.to_affine(),
+                f_eval,
+            )
+            .unwrap();
+
+        let msm: MSM<C> = guard.use_challenges(challenges).unwrap();
+
+        msm.is_zero()
     }
 }

src/poly.rs

@@ -13,6 +13,14 @@ mod domain;
 
 pub use domain::*;
 
+/// This is an error that could occur during proving or circuit synthesis.
+// TODO: these errors need to be cleaned up
+#[derive(Debug)]
+pub enum Error {
+    /// OpeningProof is not well-formed
+    OpeningError,
+}
+
 /// The basis over which a polynomial is described.
 pub trait Basis: Clone + Debug + Send + Sync {}
 

src/poly/commitment.rs

@@ -3,8 +3,11 @@
 //!
 //! [halo]: https://eprint.iacr.org/2019/1021
 
-use super::{Coeff, LagrangeCoeff, Polynomial};
+use super::{Coeff, Error, LagrangeCoeff, Polynomial};
-use crate::arithmetic::{best_fft, best_multiexp, parallelize, Curve, CurveAffine, Field};
+use crate::arithmetic::{
+    best_fft, best_multiexp, get_challenge_scalar, parallelize, Challenge, Curve, CurveAffine,
+    Field,
+};
 use crate::transcript::Hasher;
 use std::ops::{Add, AddAssign, Mul, MulAssign};
 
@@ -21,6 +24,49 @@ pub struct OpeningProof<C: CurveAffine> {
     z2: C::Scalar,
 }
 
+/// A multiscalar multiplication in the polynomial commitment scheme
+#[derive(Debug)]
+pub struct MSM<C: CurveAffine> {
+    ///  Scalars in the multiscalar multiplication
+    pub scalars: Vec<C::Scalar>,
+
+    /// Points in the multiscalar multiplication
+    pub bases: Vec<C>,
+}
+
+impl<'a, C: CurveAffine> MSM<C> {
+    /// Empty MSM
+    pub fn default(params: &'a Params<C>) -> Self {
+        let scalars: Vec<C::Scalar> =
+            Vec::with_capacity(params.k as usize * 2 + 4 + params.n as usize);
+        let bases: Vec<C> = Vec::with_capacity(params.k as usize * 2 + 4 + params.n as usize);
+
+        MSM { scalars, bases }
+    }
+
+    /// Add arbitrary term (the scalar and the point)
+    pub fn add_term(&mut self, scalar: C::Scalar, point: C) {
+        &self.scalars.push(scalar);
+        &self.bases.push(point);
+    }
+    /// Add term to g
+    pub fn mutate_g(&mut self, scalar: C::Scalar, point: C) -> Self {
+        unimplemented!()
+    }
+    /// Add term to h
+    pub fn mutate_h(&mut self, scalar: C::Scalar, point: C) -> Self {
+        unimplemented!()
+    }
+    /// Scale by a random blinding factor
+    pub fn scale(&self, scalar: C::Scalar) -> Self {
+        unimplemented!()
+    }
+    /// Perform multiexp and check that it results in zero
+    pub fn is_zero(&self) -> bool {
+        bool::from(best_multiexp(&self.scalars, &self.bases).is_zero())
+    }
+}
+
 /// These are the public parameters for the polynomial commitment scheme.
 #[derive(Debug)]
 pub struct Params<C: CurveAffine> {
@@ -154,6 +200,85 @@ impl<C: CurveAffine> Params<C> {
     }
 }
 
+/// A guard returned by the verifier
+#[derive(Debug)]
+pub struct Guard<'a, C: CurveAffine> {
+    /// Negation of z1 value in the OpeningProof
+    pub neg_z1: C::Scalar,
+
+    /// Params that were used by the verifier
+    pub params: &'a Params<C>,
+
+    /// Scalars produced by the verifier for multiscalar multiplication
+    pub scalars: Vec<C::Scalar>,
+
+    /// Points produced by the verifier for multiscalar multiplication
+    pub bases: Vec<C>,
+}
+
+impl<'a, C: CurveAffine> Guard<'a, C> {
+    /// Lets caller supply the challenges and obtain an MSM with updated
+    /// scalars and points.
+    pub fn use_challenges(
+        &mut self,
+        challenges_sq_packed: Vec<Challenge>,
+    ) -> Result<MSM<C>, Error> {
+        // - [z1] G
+        let mut allinv = C::Scalar::one();
+        let mut challenges_sq = Vec::with_capacity(self.params.k as usize);
+
+        for challenge_sq_packed in challenges_sq_packed {
+            let challenge_sq: C::Scalar = get_challenge_scalar(challenge_sq_packed);
+            challenges_sq.push(challenge_sq);
+
+            let challenge = challenge_sq.deterministic_sqrt();
+            if challenge.is_none() {
+                // We didn't sample a square.
+                return Err(Error::OpeningError);
+            }
+            let challenge = challenge.unwrap();
+
+            let challenge_inv = challenge.invert();
+            if bool::from(challenge_inv.is_none()) {
+                // We sampled zero for some reason, unlikely to happen by
+                // chance.
+                return Err(Error::OpeningError);
+            }
+            let challenge_inv = challenge_inv.unwrap();
+            allinv *= &challenge_inv;
+        }
+
+        self.bases.extend(&self.params.g);
+        let mut s = compute_s(&challenges_sq, allinv);
+        // TODO: parallelize
+        for s in &mut s {
+            *s *= &self.neg_z1;
+        }
+        self.scalars.extend(s);
+
+        Ok(MSM {
+            scalars: self.scalars.clone(),
+            bases: self.bases.clone(),
+        })
+    }
+
+    /// Lets caller supply the purported G point and simply appends it to
+    /// return an updated MSM.
+    pub fn use_s(&mut self, mut s: Vec<C::Scalar>) -> Result<MSM<C>, Error> {
+        // - [z1] G
+        self.bases.extend(&self.params.g);
+        for s in &mut s {
+            *s *= &self.neg_z1;
+        }
+        self.scalars.extend(s);
+
+        Ok(MSM {
+            scalars: self.scalars.clone(),
+            bases: self.bases.clone(),
+        })
+    }
+}
+
 /// Wrapper type around a blinding factor.
 #[derive(Copy, Clone, Eq, PartialEq, Debug)]
 pub struct Blind<F>(pub F);
@@ -273,8 +398,39 @@ fn test_opening_proof() {
             transcript.absorb(Field::one());
         } else {
             let opening_proof = opening_proof.unwrap();
-            assert!(opening_proof.verify(&params, &mut transcript_dup, x, &p, v));
+            // Verify the opening proof
+            let (challenges, mut guard) = opening_proof
+                .verify(
+                    &params,
+                    &mut MSM::default(&params),
+                    &mut transcript_dup,
+                    x,
+                    &p,
+                    v,
+                )
+                .unwrap();
+
+            let msm = guard.use_challenges(challenges).unwrap();
+
+            assert!(msm.is_zero());
             break;
         }
     }
 }
+
+// TODO: parallelize
+fn compute_s<F: Field>(challenges_sq: &[F], allinv: F) -> Vec<F> {
+    let lg_n = challenges_sq.len();
+    let n = 1 << lg_n;
+
+    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;
+        let u_lg_i_sq = challenges_sq[(lg_n - 1) - lg_i];
+        s.push(s[i - k] * u_lg_i_sq);
+    }
+
+    s
+}

src/poly/commitment/verifier.rs

@@ -1,25 +1,25 @@
-use super::{OpeningProof, Params};
+use super::super::Error;
+use super::{Guard, OpeningProof, Params, MSM};
 use crate::transcript::Hasher;
 
-use crate::arithmetic::{
+use crate::arithmetic::{get_challenge_scalar, Challenge, CurveAffine, Field};
-    best_multiexp, get_challenge_scalar, Challenge, Curve, CurveAffine, Field,
-};
 
 impl<C: CurveAffine> OpeningProof<C> {
     /// Checks to see if an [`OpeningProof`] is valid given the current
     /// `transcript`, and a point `x` that the polynomial commitment `p` opens
     /// purportedly to the value `v`.
-    pub fn verify<H: Hasher<C::Base>>(
+    pub fn verify<'a, H: Hasher<C::Base>>(
         &self,
-        params: &Params<C>,
+        params: &'a Params<C>,
+        msm: &mut MSM<C>,
         transcript: &mut H,
         x: C::Scalar,
         p: &C,
         v: C::Scalar,
-    ) -> bool {
+    ) -> Result<(Vec<Challenge>, Guard<'a, C>), Error> {
         // Check for well-formedness
         if self.rounds.len() != params.k as usize {
-            return false;
+            return Err(Error::OpeningError);
         }
 
         transcript.absorb(C::Base::from_u64(self.fork as u64));
@@ -31,28 +31,25 @@ impl<C: CurveAffine> OpeningProof<C> {
             let u_y2 = u_x.square() * &u_x + &C::b();
             let u_y = u_y2.deterministic_sqrt();
             if u_y.is_none() {
-                return false;
+                return Err(Error::OpeningError);
             }
             let u_y = u_y.unwrap();
 
             C::from_xy(u_x, u_y).unwrap()
         };
 
-        let mut extra_scalars = Vec::with_capacity(self.rounds.len() * 2 + 4 + params.n as usize);
-        let mut extra_bases = Vec::with_capacity(self.rounds.len() * 2 + 4 + params.n as usize);
-
         // Data about the challenges from each of the rounds.
         let mut challenges = Vec::with_capacity(self.rounds.len());
         let mut challenges_inv = Vec::with_capacity(self.rounds.len());
         let mut challenges_sq = Vec::with_capacity(self.rounds.len());
-        let mut allinv = Field::one();
+        let mut challenges_sq_packed: Vec<Challenge> = Vec::with_capacity(self.rounds.len());
 
         for round in &self.rounds {
             // Feed L and R into the transcript.
             let l = round.0.get_xy();
             let r = round.1.get_xy();
             if bool::from(l.is_none() | r.is_none()) {
-                return false;
+                return Err(Error::OpeningError);
             }
             let l = l.unwrap();
             let r = r.unwrap();
@@ -66,7 +63,7 @@ impl<C: CurveAffine> OpeningProof<C> {
             let challenge = challenge_sq.deterministic_sqrt();
             if challenge.is_none() {
                 // We didn't sample a square.
-                return false;
+                return Err(Error::OpeningError);
             }
             let challenge = challenge.unwrap();
 
@@ -74,26 +71,26 @@ impl<C: CurveAffine> OpeningProof<C> {
             if bool::from(challenge_inv.is_none()) {
                 // We sampled zero for some reason, unlikely to happen by
                 // chance.
-                return false;
+                return Err(Error::OpeningError);
             }
             let challenge_inv = challenge_inv.unwrap();
-            allinv *= challenge_inv;
 
             let challenge_sq_inv = challenge_inv.square();
 
-            extra_scalars.push(challenge_sq);
+            msm.scalars.push(challenge_sq);
-            extra_bases.push(round.0);
+            msm.bases.push(round.0);
-            extra_scalars.push(challenge_sq_inv);
+            msm.scalars.push(challenge_sq_inv);
-            extra_bases.push(round.1);
+            msm.bases.push(round.1);
 
             challenges.push(challenge);
             challenges_inv.push(challenge_inv);
             challenges_sq.push(challenge_sq);
+            challenges_sq_packed.push(Challenge(challenge_sq_packed));
         }
 
         let delta = self.delta.get_xy();
         if bool::from(delta.is_none()) {
-            return false;
+            return Err(Error::OpeningError);
         }
         let delta = delta.unwrap();
 
@@ -109,7 +106,7 @@ impl<C: CurveAffine> OpeningProof<C> {
         // [c] P + [c * v] U + [c] sum(L_i * u_i^2) + [c] sum(R_i * u_i^-2) + delta - [z1] G - [z1 * b] U - [z2] H
         // = 0
 
-        for scalar in &mut extra_scalars {
+        for scalar in &mut msm.scalars {
             *scalar *= &c;
         }
 
@@ -118,31 +115,29 @@ impl<C: CurveAffine> OpeningProof<C> {
         let neg_z1 = -self.z1;
 
         // [c] P
-        extra_bases.push(*p);
+        msm.bases.push(*p);
-        extra_scalars.push(c);
+        msm.scalars.push(c);
 
         // [c * v] U - [z1 * b] U
-        extra_bases.push(u);
+        msm.bases.push(u);
-        extra_scalars.push((c * &v) + &(neg_z1 * &b));
+        msm.scalars.push((c * &v) + &(neg_z1 * &b));
 
         // delta
-        extra_bases.push(self.delta);
+        msm.bases.push(self.delta);
-        extra_scalars.push(Field::one());
+        msm.scalars.push(Field::one());
 
         // - [z2] H
-        extra_bases.push(params.h);
+        msm.bases.push(params.h);
-        extra_scalars.push(-self.z2);
+        msm.scalars.push(-self.z2);
 
-        // - [z1] G
+        let guard = Guard::<'a, _> {
-        extra_bases.extend(&params.g);
+            neg_z1,
-        let mut s = compute_s(&challenges_sq, allinv);
+            params,
-        // TODO: parallelize
+            scalars: msm.scalars.clone(),
-        for s in &mut s {
+            bases: msm.bases.clone(),
-            *s *= &neg_z1;
+        };
-        }
-        extra_scalars.extend(s);
 
-        bool::from(best_multiexp(&extra_scalars, &extra_bases).is_zero())
+        Ok((challenges_sq_packed, guard))
     }
 }
 
@@ -160,20 +155,3 @@ fn compute_b<F: Field>(x: F, challenges: &[F], challenges_inv: &[F]) -> F {
             )
     }
 }
-
-// TODO: parallelize
-fn compute_s<F: Field>(challenges_sq: &[F], allinv: F) -> Vec<F> {
-    let lg_n = challenges_sq.len();
-    let n = 1 << lg_n;
-
-    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;
-        let u_lg_i_sq = challenges_sq[(lg_n - 1) - lg_i];
-        s.push(s[i - k] * u_lg_i_sq);
-    }
-
-    s
-}