Update the DLEq proof for any amount of generators

The two-generator limit wasn't required nor beneficial. This does 
theoretically optimize FROST, yet not for any current constructions. A 
follow up proof which would optimize current constructions has been 
noted in #38.

Adds explicit no_std support to the core DLEq proof.

Closes #34.

coins/monero/src/frost.rs

@@ -9,7 +9,7 @@ use group::{Group, GroupEncoding};
 
 use transcript::{Transcript, RecommendedTranscript};
 use dalek_ff_group as dfg;
-use dleq::{Generators, DLEqProof};
+use dleq::DLEqProof;
 
 #[derive(Clone, Error, Debug)]
 pub enum MultisigError {
@@ -40,7 +40,7 @@ pub(crate) fn write_dleq<R: RngCore + CryptoRng>(
     // It'd be a poor API to have CLSAG define a new transcript solely to pass here, just to try to
     // merge later in some form, when it should instead just merge xH (as it does)
     &mut transcript(),
-    Generators::new(dfg::EdwardsPoint::generator(), dfg::EdwardsPoint(H)),
+    &[dfg::EdwardsPoint::generator(), dfg::EdwardsPoint(H)],
     dfg::Scalar(x)
   ).serialize(&mut res).unwrap();
   res
@@ -68,8 +68,8 @@ pub(crate) fn read_dleq<Re: Read>(
     serialized
   ).map_err(|_| MultisigError::InvalidDLEqProof(l))?.verify(
     &mut transcript(),
-    Generators::new(dfg::EdwardsPoint::generator(), dfg::EdwardsPoint(H)),
+    &[dfg::EdwardsPoint::generator(), dfg::EdwardsPoint(H)],
-    (xG, xH)
+    &[xG, xH]
   ).map_err(|_| MultisigError::InvalidDLEqProof(l))?;
 
   Ok(xH)

crypto/dleq/Cargo.toml

@@ -30,9 +30,10 @@ dalek-ff-group = { path = "../dalek-ff-group" }
 transcript = { package = "flexible-transcript", path = "../transcript", features = ["recommended"] }
 
 [features]
-serialize = []
+std = []
-experimental = ["multiexp"]
+serialize = ["std"]
+experimental = ["std", "multiexp"]
 secure_capacity_difference = []
 
-# Only applies to cross_group, yet is default to ensure security
+# Only applies to experimental, yet is default to ensure security
 default = ["secure_capacity_difference"]

crypto/dleq/src/cross_group/aos.rs

@@ -6,9 +6,8 @@ use group::{ff::{Field, PrimeFieldBits}, prime::PrimeGroup};
 
 use multiexp::BatchVerifier;
 
-use crate::{
+use crate::cross_group::{
-  Generators,
+  Generators, DLEqError, scalar::{scalar_convert, mutual_scalar_from_bytes}
-  cross_group::{DLEqError, scalar::{scalar_convert, mutual_scalar_from_bytes}}
 };
 
 #[cfg(feature = "serialize")]

crypto/dleq/src/cross_group/bits.rs

@@ -5,7 +5,7 @@ use transcript::Transcript;
 use group::{ff::PrimeFieldBits, prime::PrimeGroup};
 use multiexp::BatchVerifier;
 
-use crate::{Generators, cross_group::{DLEqError, aos::{Re, Aos}}};
+use crate::cross_group::{Generators, DLEqError, aos::{Re, Aos}};
 
 #[cfg(feature = "serialize")]
 use std::io::{Read, Write};

crypto/dleq/src/cross_group/mod.rs

@@ -8,8 +8,6 @@ use transcript::Transcript;
 use group::{ff::{Field, PrimeField, PrimeFieldBits}, prime::PrimeGroup};
 use multiexp::BatchVerifier;
 
-use crate::Generators;
-
 pub mod scalar;
 use scalar::{scalar_convert, mutual_scalar_from_bytes};
 
@@ -35,6 +33,24 @@ pub(crate) fn read_point<R: Read, G: PrimeGroup>(r: &mut R) -> std::io::Result<G
   Ok(point.unwrap())
 }
 
+#[derive(Clone, Copy, PartialEq, Eq)]
+pub struct Generators<G: PrimeGroup> {
+  pub primary: G,
+  pub alt: G
+}
+
+impl<G: PrimeGroup> Generators<G> {
+  pub fn new(primary: G, alt: G) -> Generators<G> {
+    Generators { primary, alt }
+  }
+
+  fn transcript<T: Transcript>(&self, transcript: &mut T) {
+    transcript.domain_separate(b"generators");
+    transcript.append_message(b"primary", self.primary.to_bytes().as_ref());
+    transcript.append_message(b"alternate", self.alt.to_bytes().as_ref());
+  }
+}
+
 #[derive(Error, PartialEq, Eq, Debug)]
 pub enum DLEqError {
   #[error("invalid proof of knowledge")]

crypto/dleq/src/lib.rs

@@ -1,4 +1,5 @@
-use thiserror::Error;
+#![cfg_attr(not(feature = "std"), no_std)]
+
 use rand_core::{RngCore, CryptoRng};
 
 use transcript::Transcript;
@@ -15,24 +16,6 @@ pub mod cross_group;
 #[cfg(test)]
 mod tests;
 
-#[derive(Clone, Copy, PartialEq, Eq)]
-pub struct Generators<G: PrimeGroup> {
-  primary: G,
-  alt: G
-}
-
-impl<G: PrimeGroup> Generators<G> {
-  pub fn new(primary: G, alt: G) -> Generators<G> {
-    Generators { primary, alt }
-  }
-
-  fn transcript<T: Transcript>(&self, transcript: &mut T) {
-    transcript.domain_separate(b"generators");
-    transcript.append_message(b"primary", self.primary.to_bytes().as_ref());
-    transcript.append_message(b"alternate", self.alt.to_bytes().as_ref());
-  }
-}
-
 pub(crate) fn challenge<T: Transcript, F: PrimeField>(transcript: &mut T) -> F {
   // From here, there are three ways to get a scalar under the ff/group API
   // 1: Scalar::random(ChaCha12Rng::from_seed(self.transcript.rng_seed(b"challenge")))
@@ -70,9 +53,8 @@ fn read_scalar<R: Read, F: PrimeField>(r: &mut R) -> io::Result<F> {
   Ok(scalar.unwrap())
 }
 
-#[derive(Error, Debug)]
+#[derive(Debug)]
 pub enum DLEqError {
-  #[error("invalid proof")]
   InvalidProof
 }
 
@@ -84,34 +66,26 @@ pub struct DLEqProof<G: PrimeGroup> {
 
 #[allow(non_snake_case)]
 impl<G: PrimeGroup> DLEqProof<G> {
-  fn challenge<T: Transcript>(
+  fn transcript<T: Transcript>(transcript: &mut T, generator: G, nonce: G, point: G) {
-    transcript: &mut T,
+    transcript.append_message(b"generator", generator.to_bytes().as_ref());
-    generators: Generators<G>,
+    transcript.append_message(b"nonce", nonce.to_bytes().as_ref());
-    nonces: (G, G),
+    transcript.append_message(b"point", point.to_bytes().as_ref());
-    points: (G, G)
-  ) -> G::Scalar {
-    generators.transcript(transcript);
-    transcript.domain_separate(b"dleq");
-    transcript.append_message(b"nonce_primary", nonces.0.to_bytes().as_ref());
-    transcript.append_message(b"nonce_alternate", nonces.1.to_bytes().as_ref());
-    transcript.append_message(b"point_primary", points.0.to_bytes().as_ref());
-    transcript.append_message(b"point_alternate", points.1.to_bytes().as_ref());
-    challenge(transcript)
   }
 
   pub fn prove<R: RngCore + CryptoRng, T: Transcript>(
     rng: &mut R,
     transcript: &mut T,
-    generators: Generators<G>,
+    generators: &[G],
     scalar: G::Scalar
   ) -> DLEqProof<G> {
     let r = G::Scalar::random(rng);
-    let c = Self::challenge(
+
-      transcript,
+    transcript.domain_separate(b"dleq");
-      generators,
+    for generator in generators {
-      (generators.primary * r, generators.alt * r),
+      Self::transcript(transcript, *generator, *generator * r, *generator * scalar);
-      (generators.primary * scalar, generators.alt * scalar)
+    }
-    );
+
+    let c = challenge(transcript);
     let s = r + (c * scalar);
 
     DLEqProof { c, s }
@@ -120,18 +94,19 @@ impl<G: PrimeGroup> DLEqProof<G> {
   pub fn verify<T: Transcript>(
     &self,
     transcript: &mut T,
-    generators: Generators<G>,
+    generators: &[G],
-    points: (G, G)
+    points: &[G]
   ) -> Result<(), DLEqError> {
-    if self.c != Self::challenge(
+    if generators.len() != points.len() {
-      transcript,
+      Err(DLEqError::InvalidProof)?;
-      generators,
+    }
-      (
+
-        (generators.primary * self.s) - (points.0 * self.c),
+    transcript.domain_separate(b"dleq");
-        (generators.alt * self.s) -  (points.1 * self.c)
+    for (generator, point) in generators.iter().zip(points) {
-      ),
+      Self::transcript(transcript, *generator, (*generator * self.s) - (*point * self.c), *point);
-      points
+    }
-    ) {
+
+    if self.c != challenge(transcript) {
       Err(DLEqError::InvalidProof)?;
     }
 

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

@@ -12,10 +12,9 @@ use dalek_ff_group::{self as dfg, EdwardsPoint};
 use transcript::{Transcript, RecommendedTranscript};
 
 use crate::{
-  Generators,
   cross_group::{
     scalar::mutual_scalar_from_bytes,
-    ClassicLinearDLEq, EfficientLinearDLEq, ConciseLinearDLEq, CompromiseLinearDLEq
+    Generators, ClassicLinearDLEq, EfficientLinearDLEq, ConciseLinearDLEq, CompromiseLinearDLEq
   }
 };
 

crypto/dleq/src/tests/mod.rs

@@ -11,33 +11,47 @@ use k256::{Scalar, ProjectivePoint};
 
 use transcript::{Transcript, RecommendedTranscript};
 
-use crate::{Generators, DLEqProof};
+use crate::DLEqProof;
 
 #[test]
 fn test_dleq() {
   let transcript = || RecommendedTranscript::new(b"DLEq Proof Test");
 
-  let generators = Generators::new(
+  let generators = [
     ProjectivePoint::GENERATOR,
     ProjectivePoint::from_bytes(
       &(hex!("0250929b74c1a04954b78b4b6035e97a5e078a5a0f28ec96d547bfee9ace803ac0").into())
+    ).unwrap(),
+    // Just an increment of the last byte from the previous, where the previous two are valid
+    ProjectivePoint::from_bytes(
+      &(hex!("0250929b74c1a04954b78b4b6035e97a5e078a5a0f28ec96d547bfee9ace803ac4").into())
+    ).unwrap(),
+    ProjectivePoint::from_bytes(
+      &(hex!("0250929b74c1a04954b78b4b6035e97a5e078a5a0f28ec96d547bfee9ace803aca").into())
+    ).unwrap(),
+    ProjectivePoint::from_bytes(
+      &(hex!("0250929b74c1a04954b78b4b6035e97a5e078a5a0f28ec96d547bfee9ace803acb").into())
     ).unwrap()
-  );
+  ];
 
-  let key = Scalar::random(&mut OsRng);
+  for i in 0 .. 5 {
-  let proof = DLEqProof::prove(&mut OsRng, &mut transcript(), generators, key);
+    let key = Scalar::random(&mut OsRng);
+    let proof = DLEqProof::prove(&mut OsRng, &mut transcript(), &generators[.. i], key);
 
-  let keys = (generators.primary * key, generators.alt * key);
+    let mut keys = [ProjectivePoint::GENERATOR; 5];
-  proof.verify(&mut transcript(), generators, keys).unwrap();
+    for k in 0 .. 5 {
+      keys[k] = generators[k] * key;
+    }
+    proof.verify(&mut transcript(), &generators[.. i], &keys[.. i]).unwrap();
 
-  #[cfg(feature = "serialize")]
+    #[cfg(feature = "serialize")]
-  {
+    {
-    let mut buf = vec![];
+      let mut buf = vec![];
-    proof.serialize(&mut buf).unwrap();
+      proof.serialize(&mut buf).unwrap();
-    let deserialized = DLEqProof::<ProjectivePoint>::deserialize(
+      let deserialized = DLEqProof::<ProjectivePoint>::deserialize(
-      &mut std::io::Cursor::new(&buf)
+        &mut std::io::Cursor::new(&buf)
-    ).unwrap();
+      ).unwrap();
-    assert_eq!(proof, deserialized);
+      assert_eq!(proof, deserialized);
-    deserialized.verify(&mut transcript(), generators, keys).unwrap();
+    }
   }
 }

crypto/frost/src/sign.rs

@@ -8,7 +8,7 @@ use transcript::Transcript;
 use group::{ff::{Field, PrimeField}, Group, GroupEncoding};
 use multiexp::multiexp_vartime;
 
-use dleq::{Generators, DLEqProof};
+use dleq::DLEqProof;
 
 use crate::{
   curve::Curve,
@@ -88,8 +88,8 @@ fn preprocess<R: RngCore + CryptoRng, C: Curve, A: Algorithm<C>>(
   params: &mut Params<C, A>,
 ) -> (PreprocessPackage<C>, Vec<u8>) {
   let mut serialized = Vec::with_capacity(2 * C::G_len());
-  let (nonces, commitments) = params.algorithm.nonces().iter().cloned().map(
+  let (nonces, commitments) = params.algorithm.nonces().iter().map(
-    |mut generators| {
+    |generators| {
       let nonces = [
         C::random_nonce(params.view().secret_share(), &mut *rng),
         C::random_nonce(params.view().secret_share(), &mut *rng)
@@ -103,21 +103,23 @@ fn preprocess<R: RngCore + CryptoRng, C: Curve, A: Algorithm<C>>(
       };
 
       let mut commitments = Vec::with_capacity(generators.len());
-      let first = generators.remove(0);
-      commitments.push(commit(first, &mut serialized));
-
-      // Iterate over the rest
       for generator in generators.iter() {
         commitments.push(commit(*generator, &mut serialized));
-        // Provide a DLEq to verify these commitments are for the same nonce
+      }
-        // TODO: Provide a single DLEq. See https://github.com/serai-dex/serai/issues/34
+
+      // Provide a DLEq proof to verify these commitments are for the same nonce
+      if generators.len() >= 2 {
+        // Uses an independent transcript as each signer must do this now, yet we validate them
+        // sequentially by the global order. Avoids needing to clone and fork the transcript around
+        let mut transcript = nonce_transcript::<A::Transcript>();
+
+        // This could be further optimized with a multi-nonce proof.
+        // See https://github.com/serai-dex/serai/issues/38
         for nonce in nonces {
           DLEqProof::prove(
             &mut *rng,
-            // Uses an independent transcript as each signer must do this now, yet we validate them
+            &mut transcript,
-            // sequentially by the global order. Avoids needing to clone the transcript around
+            &generators,
-            &mut nonce_transcript::<A::Transcript>(),
-            Generators::new(first, *generator),
             nonce
           ).serialize(&mut serialized).unwrap();
         }
@@ -203,21 +205,20 @@ fn sign_with_share<Re: Read, C: Curve, A: Algorithm<C>>(
         let mut commitments = Vec::with_capacity(nonces.len());
         for (n, nonce_generators) in nonces.clone().iter_mut().enumerate() {
           commitments.push(Vec::with_capacity(nonce_generators.len()));
-
+          for _ in 0 .. nonce_generators.len() {
-          let first = nonce_generators.remove(0);
-          commitments[n].push(read_D_E::<_, C>(&mut cursor, *l)?);
-          transcript(params.algorithm.transcript(), commitments[n][0]);
-
-          for generator in nonce_generators {
             commitments[n].push(read_D_E::<_, C>(&mut cursor, *l)?);
             transcript(params.algorithm.transcript(), commitments[n][commitments[n].len() - 1]);
+          }
+
+          if nonce_generators.len() >= 2 {
+            let mut transcript = nonce_transcript::<A::Transcript>();
             for de in 0 .. 2 {
               DLEqProof::deserialize(
                 &mut cursor
               ).map_err(|_| FrostError::InvalidCommitment(*l))?.verify(
-                &mut nonce_transcript::<A::Transcript>(),
+                &mut transcript,
-                Generators::new(first, *generator),
+                &nonce_generators,
-                (commitments[n][0][de], commitments[n][commitments[n].len() - 1][de])
+                &commitments[n].iter().map(|commitments| commitments[de]).collect::<Vec<_>>(),
               ).map_err(|_| FrostError::InvalidCommitment(*l))?;
             }
           }