Implement Schnorr half-aggregation from https://eprint.iacr.org/2021/350.pdf

Relevant to https://github.com/serai-dex/serai/issues/99.

crypto/schnorr/Cargo.toml

@@ -17,11 +17,14 @@ rand_core = "0.6"
 
 zeroize = { version = "1.5", features = ["zeroize_derive"] }
 
+digest = "0.10"
+
 group = "0.12"
 ciphersuite = { path = "../ciphersuite", version = "0.1" }
 
 multiexp = { path = "../multiexp", version = "0.2", features = ["batch"] }
 
 [dev-dependencies]
+blake2 = "0.10"
 dalek-ff-group = { path =  "../dalek-ff-group", version = "^0.1.2" }
 ciphersuite = { path = "../ciphersuite", version = "0.1", features = ["ristretto"] }

crypto/schnorr/src/aggregate.rs

@@ -0,0 +1,168 @@
+use std::io::{self, Read, Write};
+
+use zeroize::Zeroize;
+
+use digest::Digest;
+
+use group::{
+  ff::{Field, PrimeField},
+  Group, GroupEncoding,
+  prime::PrimeGroup,
+};
+
+use multiexp::multiexp_vartime;
+
+use ciphersuite::Ciphersuite;
+
+use crate::SchnorrSignature;
+
+fn digest<D: Digest>() -> D {
+  D::new_with_prefix(b"Schnorr Aggregate")
+}
+
+// A secure challenge will include the nonce and whatever message
+// Depending on the environment, a secure challenge *may* not include the public key, even if
+// the modern consensus is it should
+// Accordingly, transcript both here, even if ideally only the latter would need to be
+fn digest_accumulate<D: Digest, G: PrimeGroup>(digest: &mut D, key: G, challenge: G::Scalar) {
+  digest.update(key.to_bytes().as_ref());
+  digest.update(challenge.to_repr().as_ref());
+}
+
+// Performs a big-endian modular reduction of the hash value
+// This is used by the below aggregator to prevent mutability
+// Only an 128-bit scalar is needed to offer 128-bits of security against malleability per
+// https://cr.yp.to/badbatch/badbatch-20120919.pdf
+// Accordingly, while a 256-bit hash used here with a 256-bit ECC will have bias, it shouldn't be
+// an issue
+fn scalar_from_digest<D: Digest, F: PrimeField>(digest: D) -> F {
+  let bytes = digest.finalize();
+  debug_assert_eq!(bytes.len() % 8, 0);
+
+  let mut res = F::zero();
+  let mut i = 0;
+  while i < bytes.len() {
+    if i != 0 {
+      for _ in 0 .. 8 {
+        res += res;
+      }
+    }
+    res += F::from(u64::from_be_bytes(bytes[i .. (i + 8)].try_into().unwrap()));
+    i += 8;
+  }
+  res
+}
+
+fn digest_yield<D: Digest, F: PrimeField>(digest: D, i: usize) -> F {
+  scalar_from_digest(digest.chain_update(
+    u32::try_from(i).expect("more than 4 billion signatures in aggregate").to_le_bytes(),
+  ))
+}
+
+/// Aggregate Schnorr signature as defined in https://eprint.iacr.org/2021/350.pdf.
+#[allow(non_snake_case)]
+#[derive(Clone, PartialEq, Eq, Debug, Zeroize)]
+pub struct SchnorrAggregate<C: Ciphersuite> {
+  pub Rs: Vec<C::G>,
+  pub s: C::F,
+}
+
+impl<C: Ciphersuite> SchnorrAggregate<C> {
+  /// Read a SchnorrAggregate from something implementing Read.
+  pub fn read<R: Read>(reader: &mut R) -> io::Result<Self> {
+    let mut len = [0; 4];
+    reader.read_exact(&mut len)?;
+
+    #[allow(non_snake_case)]
+    let mut Rs = vec![];
+    for _ in 0 .. u32::from_le_bytes(len) {
+      Rs.push(C::read_G(reader)?);
+    }
+
+    Ok(SchnorrAggregate { Rs, s: C::read_F(reader)? })
+  }
+
+  /// Write a SchnorrAggregate to something implementing Read.
+  pub fn write<W: Write>(&self, writer: &mut W) -> io::Result<()> {
+    writer.write_all(
+      &u32::try_from(self.Rs.len())
+        .expect("more than 4 billion signatures in aggregate")
+        .to_le_bytes(),
+    )?;
+    #[allow(non_snake_case)]
+    for R in &self.Rs {
+      writer.write_all(R.to_bytes().as_ref())?;
+    }
+    writer.write_all(self.s.to_repr().as_ref())
+  }
+
+  /// Serialize a SchnorrAggregate, returning a Vec<u8>.
+  pub fn serialize(&self) -> Vec<u8> {
+    let mut buf = vec![];
+    self.write(&mut buf).unwrap();
+    buf
+  }
+
+  /// Perform signature verification.
+  #[must_use]
+  pub fn verify<D: Clone + Digest>(&self, keys_and_challenges: &[(C::G, C::F)]) -> bool {
+    if self.Rs.len() != keys_and_challenges.len() {
+      return false;
+    }
+
+    let mut digest = digest::<D>();
+    for (key, challenge) in keys_and_challenges {
+      digest_accumulate(&mut digest, *key, *challenge);
+    }
+
+    let mut pairs = Vec::with_capacity((2 * keys_and_challenges.len()) + 1);
+    for (i, (key, challenge)) in keys_and_challenges.iter().enumerate() {
+      let z = digest_yield(digest.clone(), i);
+      pairs.push((z, self.Rs[i]));
+      pairs.push((z * challenge, *key));
+    }
+    pairs.push((-self.s, C::generator()));
+    multiexp_vartime(&pairs).is_identity().into()
+  }
+}
+
+#[allow(non_snake_case)]
+#[derive(Clone, Debug, Zeroize)]
+pub struct SchnorrAggregator<D: Clone + Digest, C: Ciphersuite> {
+  digest: D,
+  sigs: Vec<SchnorrSignature<C>>,
+}
+
+impl<D: Clone + Digest, C: Ciphersuite> Default for SchnorrAggregator<D, C> {
+  fn default() -> Self {
+    Self { digest: digest(), sigs: vec![] }
+  }
+}
+
+impl<D: Clone + Digest, C: Ciphersuite> SchnorrAggregator<D, C> {
+  /// Create a new aggregator.
+  pub fn new() -> Self {
+    Self::default()
+  }
+
+  /// Aggregate a signature.
+  pub fn aggregate(&mut self, public_key: C::G, challenge: C::F, sig: SchnorrSignature<C>) {
+    digest_accumulate(&mut self.digest, public_key, challenge);
+    self.sigs.push(sig);
+  }
+
+  /// Complete aggregation, returning None if none were aggregated.
+  pub fn complete(self) -> Option<SchnorrAggregate<C>> {
+    if self.sigs.is_empty() {
+      return None;
+    }
+
+    let mut aggregate =
+      SchnorrAggregate { Rs: Vec::with_capacity(self.sigs.len()), s: C::F::zero() };
+    for i in 0 .. self.sigs.len() {
+      aggregate.Rs.push(self.sigs[i].R);
+      aggregate.s += self.sigs[i].s * digest_yield::<_, C::F>(self.digest.clone(), i);
+    }
+    Some(aggregate)
+  }
+}

crypto/schnorr/src/lib.rs

@@ -13,6 +13,8 @@ use multiexp::BatchVerifier;
 
 use ciphersuite::Ciphersuite;
 
+pub mod aggregate;
+
 #[cfg(test)]
 mod tests;
 

crypto/schnorr/src/tests.rs

@@ -1,13 +1,19 @@
 use rand_core::OsRng;
 
+use blake2::{digest::typenum::U32, Blake2b};
+type Blake2b256 = Blake2b<U32>;
+
 use group::{ff::Field, Group};
 
 use multiexp::BatchVerifier;
 
 use ciphersuite::{Ciphersuite, Ristretto};
-use crate::SchnorrSignature;
+use crate::{
+  SchnorrSignature,
+  aggregate::{SchnorrAggregator, SchnorrAggregate},
+};
 
-pub(crate) fn core_sign<C: Ciphersuite>() {
+pub(crate) fn sign<C: Ciphersuite>() {
   let private_key = C::random_nonzero_F(&mut OsRng);
   let nonce = C::random_nonzero_F(&mut OsRng);
   let challenge = C::random_nonzero_F(&mut OsRng); // Doesn't bother to craft an HRAm
@@ -18,12 +24,12 @@ pub(crate) fn core_sign<C: Ciphersuite>() {
 // The above sign function verifies signing works
 // This verifies invalid signatures don't pass, using zero signatures, which should effectively be
 // random
-pub(crate) fn core_verify<C: Ciphersuite>() {
+pub(crate) fn verify<C: Ciphersuite>() {
   assert!(!SchnorrSignature::<C> { R: C::G::identity(), s: C::F::zero() }
     .verify(C::generator() * C::random_nonzero_F(&mut OsRng), C::random_nonzero_F(&mut OsRng)));
 }
 
-pub(crate) fn core_batch_verify<C: Ciphersuite>() {
+pub(crate) fn batch_verify<C: Ciphersuite>() {
   // Create 5 signatures
   let mut keys = vec![];
   let mut challenges = vec![];
@@ -64,9 +70,38 @@ pub(crate) fn core_batch_verify<C: Ciphersuite>() {
   }
 }
 
+pub(crate) fn aggregate<C: Ciphersuite>() {
+  // Create 5 signatures
+  let mut keys = vec![];
+  let mut challenges = vec![];
+  let mut aggregator = SchnorrAggregator::<Blake2b256, C>::new();
+  for i in 0 .. 5 {
+    keys.push(C::random_nonzero_F(&mut OsRng));
+    challenges.push(C::random_nonzero_F(&mut OsRng));
+    aggregator.aggregate(
+      C::generator() * keys[i],
+      challenges[i],
+      SchnorrSignature::<C>::sign(keys[i], C::random_nonzero_F(&mut OsRng), challenges[i]),
+    );
+  }
+
+  let aggregate = aggregator.complete().unwrap();
+  let aggregate =
+    SchnorrAggregate::<C>::read::<&[u8]>(&mut aggregate.serialize().as_ref()).unwrap();
+  assert!(aggregate.verify::<Blake2b256>(
+    keys
+      .iter()
+      .map(|key| C::generator() * key)
+      .zip(challenges.iter().cloned())
+      .collect::<Vec<_>>()
+      .as_ref()
+  ));
+}
+
 #[test]
 fn test() {
-  core_sign::<Ristretto>();
-  core_verify::<Ristretto>();
-  core_batch_verify::<Ristretto>();
+  sign::<Ristretto>();
+  verify::<Ristretto>();
+  batch_verify::<Ristretto>();
+  aggregate::<Ristretto>();
 }