Use a ring per 2 bits instead of per bit

Reduces proof size by 21.5% without notable computational complexity 
changes. I wouldn't be surprised if it has minor ones, yet I can't 
comment in which way they go without further review.

Bit now verifies it can successfully complete the ring under debug, 
slightly increasing debug times.

crypto/dleq/src/cross_group/mod.rs

@@ -3,7 +3,7 @@ use rand_core::{RngCore, CryptoRng};
 
 use digest::Digest;
 
-use subtle::{Choice, ConditionallySelectable};
+use subtle::{ConstantTimeEq, ConditionallySelectable};
 
 use transcript::Transcript;
 
@@ -34,24 +34,168 @@ pub(crate) fn read_point<R: Read, G: PrimeGroup>(r: &mut R) -> std::io::Result<G
 }
 
 #[derive(Clone, PartialEq, Eq, Debug)]
-pub struct Bit<G0: PrimeGroup, G1: PrimeGroup> {
+pub struct Bits<G0: PrimeGroup, G1: PrimeGroup, const POSSIBLE_VALUES: usize> {
   commitments: (G0, G1),
   // Merged challenges have a slight security reduction, yet one already applied to the scalar
   // being proven for, and this saves ~8kb. Alternatively, challenges could be redefined as a seed,
   // present here, which is then hashed for each of the two challenges, remaining unbiased/unique
   // while maintaining the bandwidth savings, yet also while adding 252 hashes for
   // Secp256k1/Ed25519
-  e: G0::Scalar,
+  e_0: G0::Scalar,
-  s: [(G0::Scalar, G1::Scalar); 2]
+  s: [(G0::Scalar, G1::Scalar); POSSIBLE_VALUES]
 }
 
-impl<G0: PrimeGroup, G1: PrimeGroup> Bit<G0, G1> {
+impl<G0: PrimeGroup, G1: PrimeGroup, const POSSIBLE_VALUES: usize> Bits<G0, G1, POSSIBLE_VALUES>
+  where G0::Scalar: PrimeFieldBits, G1::Scalar: PrimeFieldBits {
+  pub fn transcript<T: Transcript>(transcript: &mut T, i: usize, commitments: (G0, G1)) {
+    if i == 0 {
+      transcript.domain_separate(b"cross_group_dleq");
+    }
+    transcript.append_message(b"bit_group", &u16::try_from(i).unwrap().to_le_bytes());
+    transcript.append_message(b"commitment_0", commitments.0.to_bytes().as_ref());
+    transcript.append_message(b"commitment_1", commitments.1.to_bytes().as_ref());
+  }
+
+  #[allow(non_snake_case)]
+  fn nonces<T: Transcript>(mut transcript: T, nonces: (G0, G1)) -> (G0::Scalar, G1::Scalar) {
+    transcript.append_message(b"nonce_0", nonces.0.to_bytes().as_ref());
+    transcript.append_message(b"nonce_1", nonces.1.to_bytes().as_ref());
+    mutual_scalar_from_bytes(transcript.challenge(b"challenge").as_ref())
+  }
+
+  #[allow(non_snake_case)]
+  fn R(
+    generators: (Generators<G0>, Generators<G1>),
+    s: (G0::Scalar, G1::Scalar),
+    A: (G0, G1),
+    e: (G0::Scalar, G1::Scalar)
+  ) -> (G0, G1) {
+    (((generators.0.alt * s.0) - (A.0 * e.0)), ((generators.1.alt * s.1) - (A.1 * e.1)))
+  }
+
+  #[allow(non_snake_case)]
+  fn R_nonces<T: Transcript>(
+    transcript: T,
+    generators: (Generators<G0>, Generators<G1>),
+    s: (G0::Scalar, G1::Scalar),
+    A: (G0, G1),
+    e: (G0::Scalar, G1::Scalar)
+  ) -> (G0::Scalar, G1::Scalar) {
+    Self::nonces(transcript, Self::R(generators, s, A, e))
+  }
+
+  fn ring(pow_2: (G0, G1), commitments: (G0, G1)) -> [(G0, G1); POSSIBLE_VALUES] {
+    let mut res = [(G0::identity(), G1::identity()); POSSIBLE_VALUES];
+    res[POSSIBLE_VALUES - 1] = commitments;
+    for i in (0 .. (POSSIBLE_VALUES - 1)).rev() {
+      res[i] = (res[i + 1].0 - pow_2.0, res[i + 1].1 - pow_2.1);
+    }
+    res
+  }
+
+  pub fn prove<R: RngCore + CryptoRng, T: Clone + Transcript>(
+    rng: &mut R,
+    transcript: &mut T,
+    generators: (Generators<G0>, Generators<G1>),
+    i: usize,
+    pow_2: &mut (G0, G1),
+    bits: u8,
+    blinding_key: (G0::Scalar, G1::Scalar)
+  ) -> Bits<G0, G1, POSSIBLE_VALUES> {
+    // While it is possible to use larger values, it's not efficient to do so
+    // 2 + 2 == 2^2, yet 2 + 2 + 2 < 2^3
+    debug_assert!((POSSIBLE_VALUES == 2) || (POSSIBLE_VALUES == 4));
+
+    let mut commitments = (
+      (generators.0.alt * blinding_key.0),
+      (generators.1.alt * blinding_key.1)
+    );
+    commitments.0 += pow_2.0 * G0::Scalar::from(bits.into());
+    commitments.1 += pow_2.1 * G1::Scalar::from(bits.into());
+    Self::transcript(transcript, i, commitments);
+
+    let ring = Self::ring(*pow_2, commitments);
+    // Invert the index to get the raw blinding key's position in the ring
+    let actual = POSSIBLE_VALUES - 1 - usize::from(bits);
+
+    let mut e_0 = G0::Scalar::zero();
+    let mut s = [(G0::Scalar::zero(), G1::Scalar::zero()); POSSIBLE_VALUES];
+
+    let r = (G0::Scalar::random(&mut *rng), G1::Scalar::random(&mut *rng));
+    #[allow(non_snake_case)]
+    let original_R = (generators.0.alt * r.0, generators.1.alt * r.1);
+    #[allow(non_snake_case)]
+    let mut R = original_R;
+
+    for i in ((actual + 1) .. (actual + POSSIBLE_VALUES + 1)).map(|i| i % POSSIBLE_VALUES) {
+      let e = Self::nonces(transcript.clone(), R);
+      e_0 = G0::Scalar::conditional_select(&e_0, &e.0, usize::ct_eq(&i, &1));
+
+      // Solve for the real index
+      if i == actual {
+        s[i] = (r.0 + (e.0 * blinding_key.0), r.1 + (e.1 * blinding_key.1));
+        debug_assert_eq!(Self::R(generators, s[i], ring[actual], e), original_R);
+        break;
+      // Generate a decoy response
+      } else {
+        s[i] = (G0::Scalar::random(&mut *rng), G1::Scalar::random(&mut *rng));
+      }
+
+      R = Self::R(generators, s[i], ring[i], e);
+    }
+
+    pow_2.0 = pow_2.0.double();
+    pow_2.1 = pow_2.1.double();
+    if POSSIBLE_VALUES == 4 {
+      pow_2.0 = pow_2.0.double();
+      pow_2.1 = pow_2.1.double();
+    }
+
+    Bits { commitments, e_0, s }
+  }
+
+  pub fn verify<T: Clone + Transcript>(
+    &self,
+    transcript: &mut T,
+    generators: (Generators<G0>, Generators<G1>),
+    i: usize,
+    pow_2: &mut (G0, G1)
+  ) -> Result<(), DLEqError> {
+    debug_assert!((POSSIBLE_VALUES == 2) || (POSSIBLE_VALUES == 4));
+
+    Self::transcript(transcript, i, self.commitments);
+
+    let ring = Self::ring(*pow_2, self.commitments);
+    let e_0 = (self.e_0, scalar_convert(self.e_0).ok_or(DLEqError::InvalidChallenge)?);
+    let mut e = None;
+    for i in (1 .. (POSSIBLE_VALUES + 1)).map(|i| i % POSSIBLE_VALUES) {
+      e = Some(
+        Self::R_nonces(transcript.clone(), generators, self.s[i], ring[i], e.unwrap_or(e_0))
+      );
+    }
+
+    // Will panic if the above loop is never run somehow
+    // If e wasn't an Option, and instead initially set to e_0, it'd always pass
+    if e_0 != e.unwrap() {
+      return Err(DLEqError::InvalidProof);
+    }
+
+    pow_2.0 = pow_2.0.double();
+    pow_2.1 = pow_2.1.double();
+    if POSSIBLE_VALUES == 4 {
+      pow_2.0 = pow_2.0.double();
+      pow_2.1 = pow_2.1.double();
+    }
+
+    Ok(())
+  }
+
   #[cfg(feature = "serialize")]
   pub fn serialize<W: Write>(&self, w: &mut W) -> std::io::Result<()> {
     w.write_all(self.commitments.0.to_bytes().as_ref())?;
     w.write_all(self.commitments.1.to_bytes().as_ref())?;
-    w.write_all(self.e.to_repr().as_ref())?;
+    w.write_all(self.e_0.to_repr().as_ref())?;
-    for i in 0 .. 2 {
+    for i in 0 .. POSSIBLE_VALUES {
       w.write_all(self.s[i].0.to_repr().as_ref())?;
       w.write_all(self.s[i].1.to_repr().as_ref())?;
     }
@@ -59,17 +203,14 @@ impl<G0: PrimeGroup, G1: PrimeGroup> Bit<G0, G1> {
   }
 
   #[cfg(feature = "serialize")]
-  pub fn deserialize<R: Read>(r: &mut R) -> std::io::Result<Bit<G0, G1>> {
+  pub fn deserialize<R: Read>(r: &mut R) -> std::io::Result<Bits<G0, G1, POSSIBLE_VALUES>> {
-    Ok(
+    let commitments = (read_point(r)?, read_point(r)?);
-      Bit {
+    let e_0 = read_scalar(r)?;
-        commitments: (read_point(r)?, read_point(r)?),
+    let mut s = [(G0::Scalar::zero(), G1::Scalar::zero()); POSSIBLE_VALUES];
-        e: read_scalar(r)?,
+    for i in 0 .. POSSIBLE_VALUES {
-        s: [
+      s[i] = (read_scalar(r)?, read_scalar(r)?);
-          (read_scalar(r)?, read_scalar(r)?),
+    }
-          (read_scalar(r)?, read_scalar(r)?)
+    Ok(Bits { commitments, e_0, s })
-        ]
-      }
-    )
   }
 }
 
@@ -89,7 +230,8 @@ pub enum DLEqError {
 // anyone who wants it
 #[derive(Clone, PartialEq, Eq, Debug)]
 pub struct DLEqProof<G0: PrimeGroup, G1: PrimeGroup> {
-  bits: Vec<Bit<G0, G1>>,
+  bits: Vec<Bits<G0, G1, 4>>,
+  remainder: Option<Bits<G0, G1, 2>>,
   poks: (SchnorrPoK<G0>, SchnorrPoK<G1>)
 }
 
@@ -121,43 +263,17 @@ impl<G0: PrimeGroup, G1: PrimeGroup> DLEqProof<G0, G1>
     blinding_key
   }
 
-  #[allow(non_snake_case)]
-  fn nonces<T: Transcript>(mut transcript: T, nonces: (G0, G1)) -> (G0::Scalar, G1::Scalar) {
-    transcript.append_message(b"nonce_0", nonces.0.to_bytes().as_ref());
-    transcript.append_message(b"nonce_1", nonces.1.to_bytes().as_ref());
-    mutual_scalar_from_bytes(transcript.challenge(b"challenge").as_ref())
-  }
-
-  #[allow(non_snake_case)]
-  fn R_nonces<T: Transcript>(
-    transcript: T,
-    generators: (Generators<G0>, Generators<G1>),
-    s: (G0::Scalar, G1::Scalar),
-    A: (G0, G1),
-    e: (G0::Scalar, G1::Scalar)
-  ) -> (G0::Scalar, G1::Scalar) {
-    Self::nonces(
-      transcript,
-      (((generators.0.alt * s.0) - (A.0 * e.0)), ((generators.1.alt * s.1) - (A.1 * e.1)))
-    )
-  }
-
   fn reconstruct_keys(&self) -> (G0, G1) {
+    let remainder = self.remainder
+      .as_ref()
+      .map(|bit| bit.commitments)
+      .unwrap_or((G0::identity(), G1::identity()));
     (
-      self.bits.iter().map(|bit| bit.commitments.0).sum(),
+      self.bits.iter().map(|bit| bit.commitments.0).sum::<G0>() + remainder.0,
-      self.bits.iter().map(|bit| bit.commitments.1).sum()
+      self.bits.iter().map(|bit| bit.commitments.1).sum::<G1>() + remainder.1
     )
   }
 
-  fn transcript_bit<T: Transcript>(transcript: &mut T, i: usize, commitments: (G0, G1)) {
-    if i == 0 {
-      transcript.domain_separate(b"cross_group_dleq");
-    }
-    transcript.append_message(b"bit", &u16::try_from(i).unwrap().to_le_bytes());
-    transcript.append_message(b"commitment_0", commitments.0.to_bytes().as_ref());
-    transcript.append_message(b"commitment_1", commitments.1.to_bytes().as_ref());
-  }
-
   fn prove_internal<R: RngCore + CryptoRng, T: Clone + Transcript>(
     rng: &mut R,
     transcript: &mut T,
@@ -176,16 +292,7 @@ impl<G0: PrimeGroup, G1: PrimeGroup> DLEqProof<G0, G1>
     );
 
     let mut blinding_key_total = (G0::Scalar::zero(), G1::Scalar::zero());
-    let mut pow_2 = (generators.0.primary, generators.1.primary);
+    let mut blinding_key = |rng: &mut R, last| {
-
-    let raw_bits = f.0.to_le_bits();
-    let capacity = usize::try_from(G0::Scalar::CAPACITY.min(G1::Scalar::CAPACITY)).unwrap();
-    let mut bits = Vec::with_capacity(capacity);
-    for (i, bit) in raw_bits.iter().enumerate() {
-      let bit = *bit as u8;
-      debug_assert_eq!(bit | 1, 1);
-
-      let last = i == (capacity - 1);
       let blinding_key = (
         Self::blinding_key(&mut *rng, &mut blinding_key_total.0, last),
         Self::blinding_key(&mut *rng, &mut blinding_key_total.1, last)
@@ -194,45 +301,54 @@ impl<G0: PrimeGroup, G1: PrimeGroup> DLEqProof<G0, G1>
         debug_assert_eq!(blinding_key_total.0, G0::Scalar::zero());
         debug_assert_eq!(blinding_key_total.1, G1::Scalar::zero());
       }
+      blinding_key
+    };
 
-      let mut commitments = (
+    let mut pow_2 = (generators.0.primary, generators.1.primary);
-        (generators.0.alt * blinding_key.0),
-        (generators.1.alt * blinding_key.1)
-      );
-      commitments.0 += pow_2.0 * G0::Scalar::from(bit.into());
-      commitments.1 += pow_2.1 * G1::Scalar::from(bit.into());
-      Self::transcript_bit(transcript, i, commitments);
 
-      let nonces = (G0::Scalar::random(&mut *rng), G1::Scalar::random(&mut *rng));
+    let raw_bits = f.0.to_le_bits();
-      let e_0 = Self::nonces(
+    let capacity = usize::try_from(G0::Scalar::CAPACITY.min(G1::Scalar::CAPACITY)).unwrap();
-        transcript.clone(),
+    let mut bits = Vec::with_capacity(capacity);
-        ((generators.0.alt * nonces.0), (generators.1.alt * nonces.1))
+    let mut these_bits: u8 = 0;
-      );
+    for (i, bit) in raw_bits.iter().enumerate() {
-      let mut s_0 = (G0::Scalar::random(&mut *rng), G1::Scalar::random(&mut *rng));
+      if i > ((capacity / 2) * 2) {
-
-      let mut to_sign = commitments;
-      let bit = Choice::from(bit);
-      let inv_bit = (!bit).unwrap_u8();
-      to_sign.0 -= pow_2.0 * G0::Scalar::from(inv_bit.into());
-      to_sign.1 -= pow_2.1 * G1::Scalar::from(inv_bit.into());
-      let e_1 = Self::R_nonces(transcript.clone(), generators, (s_0.0, s_0.1), to_sign, e_0);
-      let mut s_1 = (nonces.0 + (e_1.0 * blinding_key.0), nonces.1 + (e_1.1 * blinding_key.1));
-
-      let e = G0::Scalar::conditional_select(&e_1.0, &e_0.0, bit);
-      G0::Scalar::conditional_swap(&mut s_1.0, &mut s_0.0, bit);
-      G1::Scalar::conditional_swap(&mut s_1.1, &mut s_0.1, bit);
-      bits.push(Bit { commitments, e, s: [s_0, s_1] });
-
-      // Break in order to not generate commitments for unused bits
-      if last {
         break;
       }
 
-      pow_2.0 = pow_2.0.double();
+      let bit = *bit as u8;
-      pow_2.1 = pow_2.1.double();
+      debug_assert_eq!(bit | 1, 1);
+
+      if (i % 2) == 0 {
+        these_bits = bit;
+        continue;
+      } else {
+        these_bits += bit << 1;
+      }
+
+      let last = i == (capacity - 1);
+      let blinding_key = blinding_key(&mut *rng, last);
+      bits.push(
+        Bits::prove(&mut *rng, transcript, generators, i / 2, &mut pow_2, these_bits, blinding_key)
+      );
     }
 
-    let proof = DLEqProof { bits, poks };
+    let mut remainder = None;
+    if (capacity % 2) == 1 {
+      let blinding_key = blinding_key(&mut *rng, true);
+      remainder = Some(
+        Bits::prove(
+          &mut *rng,
+          transcript,
+          generators,
+          capacity / 2,
+          &mut pow_2,
+          these_bits,
+          blinding_key
+        )
+      );
+    }
+
+    let proof = DLEqProof { bits, remainder, poks };
     debug_assert_eq!(
       proof.reconstruct_keys(),
       (generators.0.primary * f.0, generators.1.primary * f.1)
@@ -280,7 +396,11 @@ impl<G0: PrimeGroup, G1: PrimeGroup> DLEqProof<G0, G1>
     generators: (Generators<G0>, Generators<G1>)
   ) -> Result<(G0, G1), DLEqError> {
     let capacity = G0::Scalar::CAPACITY.min(G1::Scalar::CAPACITY);
-    if self.bits.len() != capacity.try_into().unwrap() {
+    if (self.bits.len() != (capacity / 2).try_into().unwrap()) || (
+      // This shouldn't be possible, as deserialize ensures this is present for fields with this
+      // characteristic, and proofs locally generated will have it. Regardless, best to ensure
+      self.remainder.is_none() && ((capacity % 2) == 1)
+    ) {
       return Err(DLEqError::InvalidProofLength);
     }
 
@@ -294,31 +414,11 @@ impl<G0: PrimeGroup, G1: PrimeGroup> DLEqProof<G0, G1>
     }
 
     let mut pow_2 = (generators.0.primary, generators.1.primary);
-    for (i, bit) in self.bits.iter().enumerate() {
+    for (i, bits) in self.bits.iter().enumerate() {
-      Self::transcript_bit(transcript, i, bit.commitments);
+      bits.verify(transcript, generators, i, &mut pow_2)?;
-
+    }
-      let bit_e = (bit.e, scalar_convert(bit.e).ok_or(DLEqError::InvalidChallenge)?);
+    if let Some(bit) = &self.remainder {
-      if bit_e != Self::R_nonces(
+      bit.verify(transcript, generators, self.bits.len(), &mut pow_2)?;
-        transcript.clone(),
-        generators,
-        bit.s[0],
-        (
-          bit.commitments.0 - pow_2.0,
-          bit.commitments.1 - pow_2.1
-        ),
-        Self::R_nonces(
-          transcript.clone(),
-          generators,
-          bit.s[1],
-          bit.commitments,
-          bit_e
-        )
-      ) {
-        return Err(DLEqError::InvalidProof);
-      }
-
-      pow_2.0 = pow_2.0.double();
-      pow_2.1 = pow_2.1.double();
     }
 
     Ok(keys)
@@ -329,6 +429,9 @@ impl<G0: PrimeGroup, G1: PrimeGroup> DLEqProof<G0, G1>
     for bit in &self.bits {
       bit.serialize(w)?;
     }
+    if let Some(bit) = &self.remainder {
+      bit.serialize(w)?;
+    }
     self.poks.0.serialize(w)?;
     self.poks.1.serialize(w)
   }
@@ -337,9 +440,19 @@ impl<G0: PrimeGroup, G1: PrimeGroup> DLEqProof<G0, G1>
   pub fn deserialize<R: Read>(r: &mut R) -> std::io::Result<DLEqProof<G0, G1>> {
     let capacity = G0::Scalar::CAPACITY.min(G1::Scalar::CAPACITY);
     let mut bits = Vec::with_capacity(capacity.try_into().unwrap());
-    for _ in 0 .. capacity {
+    for _ in 0 .. (capacity / 2) {
-      bits.push(Bit::deserialize(r)?);
+      bits.push(Bits::deserialize(r)?);
     }
-    Ok(DLEqProof { bits, poks: (SchnorrPoK::deserialize(r)?, SchnorrPoK::deserialize(r)?) })
+    let mut remainder = None;
+    if (capacity % 2) == 1 {
+      remainder = Some(Bits::deserialize(r)?);
+    }
+    Ok(
+      DLEqProof {
+        bits,
+        remainder,
+        poks: (SchnorrPoK::deserialize(r)?, SchnorrPoK::deserialize(r)?)
+      }
+    )
   }
 }

crypto/dleq/src/tests/cross_group/mod.rs

@@ -14,7 +14,7 @@ use blake2::{Digest, Blake2b512};
 
 use transcript::RecommendedTranscript;
 
-use crate::{Generators, cross_group::DLEqProof};
+use crate::{Generators, cross_group::{DLEqProof, scalar::mutual_scalar_from_bytes}};
 
 fn transcript() -> RecommendedTranscript {
   RecommendedTranscript::new(b"Cross-Group DLEq Proof Test")
@@ -104,3 +104,35 @@ fn test_cross_group_dleq() {
     }
   }
 }
+
+#[test]
+fn test_remainder() {
+  // Uses Secp256k1 for both to achieve an odd capacity of 255
+  assert_eq!(Scalar::CAPACITY, 255);
+  let generators = (generators().0, generators().0);
+  let keys = mutual_scalar_from_bytes(&[0xFF; 32]);
+  assert_eq!(keys.0, keys.1);
+
+  let (proof, res) = DLEqProof::prove_without_bias(
+    &mut OsRng,
+    &mut transcript(),
+    generators,
+    keys.0
+  ).unwrap();
+  assert_eq!(keys, res);
+
+  let public_keys = proof.verify(&mut transcript(), generators).unwrap();
+  assert_eq!(generators.0.primary * keys.0, public_keys.0);
+  assert_eq!(generators.1.primary * keys.1, public_keys.1);
+
+  #[cfg(feature = "serialize")]
+  {
+    let mut buf = vec![];
+    proof.serialize(&mut buf).unwrap();
+    let deserialized = DLEqProof::<ProjectivePoint, ProjectivePoint>::deserialize(
+      &mut std::io::Cursor::new(&buf)
+    ).unwrap();
+    assert_eq!(proof, deserialized);
+    deserialized.verify(&mut transcript(), generators).unwrap();
+  }
+}