Unify the cross-group DLEq challenges

This does reduce the strength of the challenges to that of the weaker 
field, yet that doesn't have any impact on whether or not this is ZK due 
to the key being shared across fields.

Saves ~8kb.

crypto/dleq/src/cross_group/mod.rs

@@ -8,7 +8,7 @@ use group::{ff::{Field, PrimeField, PrimeFieldBits}, prime::PrimeGroup};
 use crate::{Generators, challenge};
 
 pub mod scalar;
-use scalar::scalar_normalize;
+use scalar::{scalar_normalize, scalar_convert};
 
 pub(crate) mod schnorr;
 use schnorr::SchnorrPoK;
@@ -32,7 +32,7 @@ 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> {
   commitments: (G0, G1),
-  e: (G0::Scalar, G1::Scalar),
+  e: G0::Scalar,
   s: [(G0::Scalar, G1::Scalar); 2]
 }
 
@@ -41,8 +41,7 @@ impl<G0: PrimeGroup, G1: PrimeGroup> Bit<G0, G1> {
   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.0.to_repr().as_ref())?;
-    w.write_all(self.e.1.to_repr().as_ref())?;
+    w.write_all(self.e.to_repr().as_ref())?;
     for i in 0 .. 2 {
       w.write_all(self.s[i].0.to_repr().as_ref())?;
       w.write_all(self.s[i].1.to_repr().as_ref())?;
@@ -55,7 +54,7 @@ impl<G0: PrimeGroup, G1: PrimeGroup> Bit<G0, G1> {
     Ok(
       Bit {
         commitments: (read_point(r)?, read_point(r)?),
-        e: (read_scalar(r)?, read_scalar(r)?),
+        e: read_scalar(r)?,
         s: [
           (read_scalar(r)?, read_scalar(r)?),
           (read_scalar(r)?, read_scalar(r)?)
@@ -71,6 +70,8 @@ pub enum DLEqError {
   InvalidProofOfKnowledge,
   #[error("invalid proof length")]
   InvalidProofLength,
+  #[error("invalid challenge")]
+  InvalidChallenge,
   #[error("invalid proof")]
   InvalidProof
 }
@@ -117,7 +118,7 @@ impl<G0: PrimeGroup, G1: PrimeGroup> DLEqProof<G0, G1>
   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());
-    (challenge(&mut transcript, b"challenge_G"), challenge(&mut transcript, b"challenge_H"))
+    scalar_normalize(challenge(&mut transcript))
   }
 
   #[allow(non_snake_case)]
@@ -134,7 +135,6 @@ impl<G0: PrimeGroup, G1: PrimeGroup> DLEqProof<G0, G1>
     )
   }
 
-  // TODO: Use multiexp here after https://github.com/serai-dex/serai/issues/17
   fn reconstruct_key<G: PrimeGroup>(
     commitments: impl Iterator<Item = G>
   ) -> G where G::Scalar: PrimeFieldBits {
@@ -240,9 +240,9 @@ impl<G0: PrimeGroup, G1: PrimeGroup> DLEqProof<G0, G1>
 
       bits.push(
         if *bit {
-          Bit { commitments, e: e_0, s: [s_1, s_0] }
+          Bit { commitments, e: e_0.0, s: [s_1, s_0] }
         } else {
-          Bit { commitments, e: e_1, s: [s_0, s_1] }
+          Bit { commitments, e: e_1.0, s: [s_0, s_1] }
         }
       );
 
@@ -282,7 +282,8 @@ impl<G0: PrimeGroup, G1: PrimeGroup> DLEqProof<G0, G1>
     for (i, bit) in self.bits.iter().enumerate() {
       Self::transcript_bit(transcript, i, bit.commitments);
 
-      if bit.e != Self::R_nonces(
+      let bit_e = (bit.e, scalar_convert(bit.e).ok_or(DLEqError::InvalidChallenge)?);
+      if bit_e != Self::R_nonces(
         transcript.clone(),
         generators,
         bit.s[0],
@@ -295,7 +296,7 @@ impl<G0: PrimeGroup, G1: PrimeGroup> DLEqProof<G0, G1>
           generators,
           bit.s[1],
           bit.commitments,
-          bit.e
+          bit_e
         )
       ) {
         return Err(DLEqError::InvalidProof);

crypto/dleq/src/cross_group/schnorr.rs

@@ -28,7 +28,7 @@ impl<G: PrimeGroup> SchnorrPoK<G> {
     transcript.append_message(b"generator", generator.to_bytes().as_ref());
     transcript.append_message(b"nonce", R.to_bytes().as_ref());
     transcript.append_message(b"public_key", A.to_bytes().as_ref());
-    challenge(transcript, b"challenge")
+    challenge(transcript)
   }
 
   pub(crate) fn prove<R: RngCore + CryptoRng, T: Transcript>(

crypto/dleq/src/lib.rs

@@ -33,10 +33,7 @@ impl<G: PrimeGroup> Generators<G> {
   }
 }
 
-pub(crate) fn challenge<T: Transcript, F: PrimeField>(
-  transcript: &mut T,
-  label: &'static [u8]
-) -> F {
+pub(crate) fn challenge<T: Transcript, F: PrimeField>(transcript: &mut T) -> F {
   assert!(F::NUM_BITS <= 384);
 
   // From here, there are three ways to get a scalar under the ff/group API
@@ -44,7 +41,7 @@ pub(crate) fn challenge<T: Transcript, F: PrimeField>(
   // 2: Grabbing a UInt library to perform reduction by the modulus, then determining endianess
   //    and loading it in
   // 3: Iterating over each byte and manually doubling/adding. This is simplest
-  let challenge_bytes = transcript.challenge(label);
+  let challenge_bytes = transcript.challenge(b"challenge");
   assert!(challenge_bytes.as_ref().len() == 64);
 
   let mut challenge = F::zero();
@@ -94,7 +91,7 @@ impl<G: PrimeGroup> DLEqProof<G> {
     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, b"challenge")
+    challenge(transcript)
   }
 
   pub fn prove<R: RngCore + CryptoRng, T: Transcript>(