Schnorr's DLog PoK (#11)

* init: Schnorr's identification protocol

* update: REAME

* checkpoint

* add: Schnorr README

* schnorr signatures workgit status

* Update [Sch91]schnorr-discrete-log-proof-of-knowledge/README.md

Co-authored-by: coderabbitai[bot] <136622811+coderabbitai[bot]@users.noreply.github.com>

* Update [Sch91]schnorr-discrete-log-proof-of-knowledge/README.md

Co-authored-by: coderabbitai[bot] <136622811+coderabbitai[bot]@users.noreply.github.com>

---------

Co-authored-by: coderabbitai[bot] <136622811+coderabbitai[bot]@users.noreply.github.com>

Cargo.toml

@@ -5,6 +5,7 @@ members = [
   "halo2-trials",
   "[Sha97]shamir-secret-sharing",
   "[Fel87]feldman-verifiable-secret-sharing",
+  "[Sch91]schnorr-discrete-log-proof-of-knowledge",
 ]
 resolver = "2"
 
@@ -18,6 +19,7 @@ ark-bls12-381 = "0.4.0"
 ark-ec = "0.4.2"
 ark-ff = "0.4.2"
 ark-std = "0.4.0"
+ark-crypto-primitives = "0.4.0"
 num-traits = "0.2.18"
 nalgebra = "0.32.5"
 halo2 = "0.0.0"

README.md

@@ -13,3 +13,4 @@ cd Cryptography-Research && cargo test
 ## References
 - **\[Sha97\]**: Shamir's secret sharing https://apps.dtic.mil/sti/pdfs/ADA069397.pdf.
 - **\[Fel87\]**: Feldman's verifiable secret sharing https://www.zkdocs.com/docs/zkdocs/protocol-primitives/verifiable-secret-sharing/
+- **\[Sch91\]**: Schnorr's DLog PoK https://www.zkdocs.com/docs/zkdocs/zero-knowledge-protocols/schnorr/

ToDo.md

@@ -0,0 +1,10 @@
+## Further implementations
+- Reckle trees: http://lagrange.dev/reckle-trees
+- Verkle trees:
+- Caulk lookup argument
+- Caulk+ lookup argument
+- Halo2 lookup argument
+- Plonk lookup argument
+- Lagrange interpolation
+- Multivariate polynomial domain extension
+- Weak fiat shamir: https://eprint.iacr.org/2023/691.pdf

[Sch91]schnorr-discrete-log-proof-of-knowledge/Cargo.toml

@@ -0,0 +1,11 @@
+[package]
+edition = "2021"
+name = "schnorr-discrete-log-proof-of-knowledge"
+version = "0.1.0"
+
+[dependencies]
+ark-bls12-381 = { workspace = true }
+ark-crypto-primitives = { workspace = true }
+ark-ec = { workspace = true }
+ark-ff = { workspace = true }
+ark-std = { workspace = true }

[Sch91]schnorr-discrete-log-proof-of-knowledge/README.md

@@ -0,0 +1,14 @@
+# Schnorr's Identification Protocol
+Schnorr's Identification Protocol represented in this repo via \[Sch91\] is a simple zero-knowledge protocol wherein a prover $P$ can convince a verifier $V$ that they know of some private value $x$ under $G$ (where discrete-log problem is hard), against a securely held value $h=g^x$ (where $g$ is a generator of $G$) at $V$ without revealing $x$ itself.
+
+In the interactive format, the protocol runs as follows:
+![Interactive Schnorr Protocol](assets/interactive_schnorr.png)
+
+Here, generation of $u$ is necessary to maintain secrecy of x (the additive blinding factor). As without the value $r$, $z = x c$, and $x$ can be retrieved by operation $x = z c^{-1}$ where $c^{-1}$ is the inverse of element $c$ in the group.
+
+## References
+## References
+
+[Stanford CS355 Lecture 5](https://crypto.stanford.edu/cs355/19sp/lec5.pdf)
+
+- https://crypto.stackexchange.com/questions/58954/multiplication-of-two-points-belong-to-elliptic-curve

[Sch91]schnorr-discrete-log-proof-of-knowledge/src/lib.rs

@@ -0,0 +1,83 @@
+// Much reference borrowed from
+// https://github.com/arkworks-rs/r1cs-tutorial/blob/main/simple-payments/src/signature/schnorr/mod.rs#L53
+
+use std::marker::PhantomData;
+
+use ark_crypto_primitives::Error;
+use ark_ec::{AffineRepr, CurveGroup, Group};
+use ark_ff::PrimeField;
+
+pub struct Schnorr<C: CurveGroup> {
+    _group: PhantomData<C>,
+}
+
+pub type PublicKey<C> = <C as CurveGroup>::Affine;
+
+#[derive(Clone, Default, Debug)]
+pub struct SecretKey<C: CurveGroup + Group> {
+    pub secret_key: C::ScalarField,
+    pub public_key: PublicKey<C>,
+}
+
+pub struct Parameters<C: CurveGroup> {
+    pub generator: C::Affine,
+    pub salt: Option<[u8; 32]>,
+}
+
+impl<C: CurveGroup + Group> Schnorr<C>
+where
+    C::ScalarField: PrimeField,
+{
+    fn setup() -> Result<Parameters<C>, Error> {
+        Ok(Parameters::<C> {
+            generator: C::generator().into(),
+            salt: Default::default(),
+        })
+    }
+
+    fn from_secret(parameters: &Parameters<C>, secret: u64) -> Result<SecretKey<C>, Error> {
+        let secret_key = C::ScalarField::from_bigint(secret.into()).unwrap();
+        let public_key = parameters.generator.mul_bigint(&[secret]).into();
+        Ok(SecretKey::<C> {
+            secret_key,
+            public_key,
+        })
+    }
+}
+
+#[cfg(test)]
+mod tests {
+    use ark_bls12_381::G1Projective;
+    use ark_ec::{AffineRepr, Group};
+    use ark_ff::PrimeField;
+
+    use super::Schnorr;
+    use crate::Parameters;
+
+    #[test]
+    fn schnorr_dlog_pok() {
+        let secret_x = 541;
+        let setup_params: Parameters<G1Projective> = Schnorr::setup().unwrap();
+        let secret = Schnorr::from_secret(&setup_params, secret_x).unwrap();
+
+        // initiate protocol
+        let prover_random_r = <G1Projective as Group>::ScalarField::from(412);
+        let verifier_random_c = <G1Projective as Group>::ScalarField::from(31981);
+        let prover_computed_z = prover_random_r
+            + <G1Projective as Group>::ScalarField::from(secret_x) * verifier_random_c;
+
+        let prover_generated_u = setup_params
+            .generator
+            .mul_bigint(prover_random_r.into_bigint());
+
+        let verifier_lhs = setup_params
+            .generator
+            .mul_bigint(prover_computed_z.into_bigint());
+        let verifier_rhs = prover_generated_u
+            + secret
+                .public_key
+                .mul_bigint(verifier_random_c.into_bigint());
+
+        assert_eq!(verifier_lhs, verifier_rhs);
+    }
+}