Feature/traits (#3)

* feat: draft traits `FoldingScheme` and `Decider`

Co-authored-by: arnaucube <root@arnaucube.com>

* Add Transcript trait, with PoseidonTranscript impl (#1)

Add also the PoseidonTranscriptVar (gadget).

* Update FoldingScheme trait to take C1 & C2 as params (#2)

* Update FoldingScheme trait to take C1 & C2 as params

Update FoldingScheme trait to take C1 & C2 as params which are used by
the diverse folding schemes as a cycle of curves.

* Add constraint to FoldingScheme C1,C2 fields swap.

Co-authored-by: Han <tinghan0110@gmail.com>

---------

Co-authored-by: Han <tinghan0110@gmail.com>

* move transcript to it's own mod

---------

Co-authored-by: han0110 <tinghan0110@gmail.com>

src/lib.rs

@@ -0,0 +1,85 @@
+#![allow(non_snake_case)]
+#![allow(non_upper_case_globals)]
+
+use ark_ec::CurveGroup;
+use ark_ff::PrimeField;
+use ark_std::{fmt::Debug, rand::RngCore};
+use thiserror::Error;
+
+pub mod transcript;
+use transcript::Transcript;
+
+#[derive(Debug, Error)]
+pub enum Error {
+    #[error("Relation not satisfied")]
+    NotSatisfied,
+}
+
+/// FoldingScheme defines trait that is implemented by the diverse folding schemes. It is defined
+/// over a cycle of curves (C1, C2), where:
+/// - C1 is the main curve, which ScalarField we use as our F for al the field operations
+/// - C2 is the auxiliary curve, which we use for the commitments, whose BaseField (for point
+/// coordinates) are in the C1::ScalarField
+pub trait FoldingScheme<C1: CurveGroup, C2: CurveGroup>: Clone + Debug
+where
+    C1: CurveGroup<BaseField = C2::ScalarField, ScalarField = C2::BaseField>,
+    C2::BaseField: PrimeField,
+{
+    // type PCS: PolynomialCommitmentScheme<C>; // maybe not needed, just PedersenCommitment
+    type PreprocessorParam: Debug;
+    type ProverParam: Debug;
+    type VerifierParam: Debug;
+    type FreshInstance: Debug;
+    type PublicInput: Debug;
+    type CommittedInstanceWithWitness: Debug;
+    type CommittedInstance: Clone + Debug;
+
+    fn preprocess(
+        // pcs_param: &<Self::CS as PolynomialCommitmentScheme<C>>::Param,
+        prep_param: &Self::PreprocessorParam,
+    ) -> Result<(Self::ProverParam, Self::VerifierParam), Error>;
+
+    fn init_accumulator(
+        pp: &Self::ProverParam,
+    ) -> Result<Self::CommittedInstanceWithWitness, Error>;
+
+    fn prove(
+        pp: &Self::ProverParam,
+        running_instance: &mut Self::CommittedInstanceWithWitness,
+        incomming_instances: &[Self::FreshInstance],
+        transcript: &mut impl Transcript<C1::ScalarField>,
+        rng: impl RngCore,
+    ) -> Result<(), Error>;
+
+    fn verify(
+        vp: &Self::VerifierParam,
+        running_instance: &mut Self::CommittedInstance,
+        incomming_instances: &[Self::PublicInput],
+        transcript: &mut impl Transcript<C1::ScalarField>,
+        rng: impl RngCore,
+    ) -> Result<(), Error>;
+}
+
+pub trait Decider<C: CurveGroup>: Clone + Debug {
+    type PreprocessorParam: Debug;
+    type ProverParam: Debug;
+    type VerifierParam: Debug;
+    type FreshInstance: Debug;
+    type PublicInput: Debug;
+    type CommittedInstanceWithWitness: Debug;
+    type CommittedInstance: Clone + Debug;
+
+    fn prove(
+        pp: &Self::ProverParam,
+        running_instance: &Self::CommittedInstanceWithWitness,
+        transcript: &mut impl Transcript<C::ScalarField>,
+        rng: impl RngCore,
+    ) -> Result<(), Error>;
+
+    fn verify(
+        vp: &Self::VerifierParam,
+        running_instance: &Self::CommittedInstance,
+        transcript: &mut impl Transcript<C::ScalarField>,
+        rng: impl RngCore,
+    ) -> Result<(), Error>;
+}