chore: add example with private state

src/usage/frontend.md

@@ -1,6 +1,6 @@
 # Frontend
 
-The frontend interface allows to define the circuit to be folded. The available frontends are [`circom`](https://github.com/iden3/circom) or [arkworks](https://github.com/arkworks-rs/r1cs-std). We will show here how to define a circuit using `arkworks`.
+The frontend interface allows to define the circuit to be folded. The currently available frontends are [`circom`](https://github.com/iden3/circom) or [arkworks](https://github.com/arkworks-rs/r1cs-std). We will show here how to define a circuit using `arkworks`.
 
 # The `FCircuit` trait
 
@@ -14,14 +14,14 @@ To be folded with sonobe, a circuit needs to implement the [`FCircuit` trait](ht
 pub trait FCircuit<F: PrimeField>: Clone + Debug {
     type Params: Debug;
 
-    /// returns a new FCircuit instance
+    /// Returns a new FCircuit instance
     fn new(params: Self::Params) -> Self;
 
-    /// returns the number of elements in the state of the FCircuit, which corresponds to the
+    /// Returns the number of elements in the state of the FCircuit, which corresponds to the
     /// FCircuit inputs.
     fn state_len(&self) -> usize;
 
-    /// computes the next state values in place, assigning z_{i+1} into z_i, and computing the new
+    /// Computes the next state values in place, assigning z_{i+1} into z_i, and computing the new
     /// z_{i+1}
     fn step_native(
         // this method uses self, so that each FCircuit implementation (and different frontends)
@@ -31,7 +31,7 @@ pub trait FCircuit<F: PrimeField>: Clone + Debug {
         z_i: Vec<F>,
     ) -> Result<Vec<F>, Error>;
 
-    /// generates the constraints for the step of F for the given z_i
+    /// Generates the constraints for the step of F for the given z_i
     fn generate_step_constraints(
         // this method uses self, so that each FCircuit implementation (and different frontends)
         // can hold a state if needed to store data to generate the constraints.
@@ -45,7 +45,7 @@ pub trait FCircuit<F: PrimeField>: Clone + Debug {
 
 # Example
 
-Let's walk through different simple examples implementing the `FCircuit` trait. By the end, you will hopefully be familiar with how to integrate an `arkworks` circuit into sonobe.
+Let's walk through different simple examples implementing the `FCircuit` trait. By the end of this section, you will hopefully be familiar with how to integrate an `arkworks` circuit into sonobe.
 
 ## Folding a simple circuit
 
@@ -90,7 +90,7 @@ impl<F: PrimeField> FCircuit<F> for MultiInputsFCircuit<F> {
         Ok(vec![a, b, c, d, e]) // The length of the returned vector should match `state_len`
     }
 
-    /// generates R1CS constraints for the step of F for the given z_i
+    /// Generates R1CS constraints for the step of F for the given z_i
     fn generate_step_constraints(
         &self,
         cs: ConstraintSystemRef<F>,
@@ -110,18 +110,21 @@ impl<F: PrimeField> FCircuit<F> for MultiInputsFCircuit<F> {
         Ok(vec![a, b, c, d, e]) // The length of the returned vector should match `state_len`
     }
 }
-
 ```
 
 ## Folding a `Sha256` circuit
 
-We will fold a simple `Sha256` circuit. The circuit has a state of 1 public element. At each step, we will want the circuit to compute the next state by applying the `Sha256` function to the current state.
+We will fold a simple `Sha256` circuit. The circuit has a state of 1 public element. At each step, we will want the circuit to compute the next state by applying the `Sha256` function to the current state. 
+
+Note that the logic here is also very similar to the previous example: write a struct that will hold the circuit, implement the `FCircuit` trait for the struct, ensure that the length of the state is correct, and implement the `step_native` and `generate_step_constraints` methods.
 
 ```rust
+// Define a struct that will be our circuit. This struct will implement the FCircuit trait.
 #[derive(Clone, Copy, Debug)]
 pub struct Sha256FCircuit<F: PrimeField> {
     _f: PhantomData<F>,
 }
+
 impl<F: PrimeField> FCircuit<F> for Sha256FCircuit<F> {
     type Params = ();
 
@@ -132,7 +135,7 @@ impl<F: PrimeField> FCircuit<F> for Sha256FCircuit<F> {
         1
     }
 
-    /// computes the next state values in place, assigning z_{i+1} into z_i, and computing the new
+    /// Computes the next state values in place, assigning z_{i+1} into z_i, and computing the new
     /// z_{i+1}
     fn step_native(&self, _i: usize, z_i: Vec<F>) -> Result<Vec<F>, Error> {
         let out_bytes = Sha256::evaluate(&(), z_i[0].into_bigint().to_bytes_le()).unwrap();
@@ -141,7 +144,7 @@ impl<F: PrimeField> FCircuit<F> for Sha256FCircuit<F> {
         Ok(vec![out[0]])
     }
 
-    /// generates the constraints for the step of F for the given z_i
+    /// Generates the constraints for the step of F for the given z_i
     fn generate_step_constraints(
         &self,
         _cs: ConstraintSystemRef<F>,
@@ -156,4 +159,64 @@ impl<F: PrimeField> FCircuit<F> for Sha256FCircuit<F> {
 }
 ```
 
-## Folding a circuit with public and private inputs
+## Folding a circuit with public and private inputs
+
+Sometimes, the circuit to be folded will have private inputs. Let's see how we can setup such a circuit to be folded with sonobe. Again, the logic here is also very similar to our previous examples. The main difference is that the `struct` which will hold the circuit also holds a `Vec` of private inputs.
+
+```rust
+#[derive(Clone, Debug)]
+pub struct ACircuitWithPrivateState<F: PrimeField> {
+    _f: PhantomData<F>,
+    private_state: Vec<F>, // private inputs, here a `Vec` of field elements, but you can specify whatever type you prefer here
+}
+
+impl<F: PrimeField> FCircuit<F> for ACircuitWithPrivateState<F> {
+    type Params = Vec<F>;
+
+    fn new(params: Self::Params) -> Self {
+        Self {
+            _f: PhantomData,
+            private_state: params,
+        }
+    }
+
+    fn state_len(&self) -> usize {
+        3 // the length of the state should match the size of the public inputs, not including the private inputs
+    }
+
+    fn step_native(&self, i: usize, z_i: Vec<F>) -> Result<Vec<F>, folding_schemes::Error> {
+        let a = z_i[0] + F::from(4_u32);
+        let b = z_i[1] + F::from(40_u32);
+        let c = z_i[2] * F::from(4_u32);
+        let d = self.private_state[0] * a;
+        let e = self.private_state[1] * c;
+
+        Ok(vec![a, b, c])
+        Ok(new_z_i)
+    }
+
+    fn generate_step_constraints(
+        &self,
+        cs: ConstraintSystemRef<F>,
+        i: usize,
+        z_i: Vec<ark_r1cs_std::fields::fp::FpVar<F>>,
+    ) -> Result<Vec<ark_r1cs_std::fields::fp::FpVar<F>>, SynthesisError> {
+        let four = FpVar::<F>::new_constant(cs.clone(), F::from(4u32))?;
+        let forty = FpVar::<F>::new_constant(cs.clone(), F::from(40u32))?;
+        let onehundred = FpVar::<F>::new_constant(cs.clone(), F::from(100u32))?;
+
+        // we still need to allocate our private state as witnesses
+        let priv_var_0 = FpVar::<F>::new_witness(cs.clone(), || Ok(self.private_state[0].clone()))?;
+        let priv_var_1 = FpVar::<F>::new_witness(cs.clone(), || Ok(self.private_state[1].clone()))?;
+
+        let a = z_i[0].clone() + four.clone();
+        let b = z_i[1].clone() + forty.clone();
+        let c = z_i[2].clone() * four;
+        let d = priv_var_0 * a;
+        let e = priv_var_1 * c;
+
+        Ok(vec![a, b, c])
+    }
+}
+
+```