Support for arbitrary arity for step circuit's IO (#107)

* support for arbitrary arity for F

* revive MinRoot example

* revive tests

* revive ecdsa

* remove unused code

* use None instead of Some(1u32)

* revive benches

* fix clippy warning

benches/compressed-snark.rs

@@ -57,8 +57,8 @@ fn bench_compressed_snark(c: &mut Criterion) {
         recursive_snark,
         NonTrivialTestCircuit::new(num_cons),
         TrivialTestCircuit::default(),
-        <G1 as Group>::Scalar::from(2u64),
-        <G2 as Group>::Scalar::from(2u64),
+        vec![<G1 as Group>::Scalar::from(2u64)],
+        vec![<G2 as Group>::Scalar::from(2u64)],
       );
       assert!(res.is_ok());
       let recursive_snark_unwrapped = res.unwrap();
@@ -67,8 +67,8 @@ fn bench_compressed_snark(c: &mut Criterion) {
       let res = recursive_snark_unwrapped.verify(
         &pp,
         i + 1,
-        <G1 as Group>::Scalar::from(2u64),
-        <G2 as Group>::Scalar::from(2u64),
+        vec![<G1 as Group>::Scalar::from(2u64)],
+        vec![<G2 as Group>::Scalar::from(2u64)],
       );
       assert!(res.is_ok());
 
@@ -98,8 +98,8 @@ fn bench_compressed_snark(c: &mut Criterion) {
           .verify(
             black_box(&pp),
             black_box(num_steps),
-            black_box(<G1 as Group>::Scalar::from(2u64)),
-            black_box(<G2 as Group>::Scalar::from(2u64)),
+            black_box(vec![<G1 as Group>::Scalar::from(2u64)]),
+            black_box(vec![<G2 as Group>::Scalar::from(2u64)]),
           )
           .is_ok());
       })
@@ -130,28 +130,32 @@ impl StepCircuit for NonTrivialTestCircuit
 where
   F: PrimeField,
 {
+  fn arity(&self) -> usize {
+    1
+  }
+
   fn synthesize<CS: ConstraintSystem<F>>(
     &self,
     cs: &mut CS,
-    z: AllocatedNum<F>,
-  ) -> Result<AllocatedNum<F>, SynthesisError> {
+    z: & class="p">[AllocatedNum<F>],
+  ) -> Result<Vec<AllocatedNum<F>>, SynthesisError> {
     // Consider a an equation: `x^2 = y`, where `x` and `y` are respectively the input and output.
-    let mut x = z;
+    let mut x = z[0].clone();
     let mut y = x.clone();
     for i in 0..self.num_cons {
       y = x.square(cs.namespace(|| format!("x_sq_{}", i)))?;
       x = y.clone();
     }
-    Ok(y)
+    Ok(vec![y])
   }
 
-  fn output(&self, z: &F) -> F {
-    let mut x = *z;
+  fn output(&self, z: &[F]) -> Vec<F> {
+    let mut x = z[0];
     let mut y = x;
     for _i in 0..self.num_cons {
       y = x * x;
       x = y;
     }
-    y
+    vec![y]
   }
 }

benches/recursive-snark.rs

@@ -57,8 +57,8 @@ fn bench_recursive_snark(c: &mut Criterion) {
         recursive_snark,
         NonTrivialTestCircuit::new(num_cons),
         TrivialTestCircuit::default(),
-        <G1 as Group>::Scalar::from(2u64),
-        <G2 as Group>::Scalar::from(2u64),
+        vec![<G1 as Group>::Scalar::from(2u64)],
+        vec![<G2 as Group>::Scalar::from(2u64)],
       );
       assert!(res.is_ok());
       let recursive_snark_unwrapped = res.unwrap();
@@ -67,8 +67,8 @@ fn bench_recursive_snark(c: &mut Criterion) {
       let res = recursive_snark_unwrapped.verify(
         &pp,
         i + 1,
-        <G1 as Group>::Scalar::from(2u64),
-        <G2 as Group>::Scalar::from(2u64),
+        vec![<G1 as Group>::Scalar::from(2u64)],
+        vec![<G2 as Group>::Scalar::from(2u64)],
       );
       assert!(res.is_ok());
 
@@ -84,8 +84,8 @@ fn bench_recursive_snark(c: &mut Criterion) {
           black_box(recursive_snark.clone()),
           black_box(NonTrivialTestCircuit::new(num_cons)),
           black_box(TrivialTestCircuit::default()),
-          black_box(<G1 as Group>::Scalar::from(2u64)),
-          black_box(<G2 as Group>::Scalar::from(2u64)),
+          black_box(vec![<G1 as Group>::Scalar::from(2u64)]),
+          black_box(vec![<G2 as Group>::Scalar::from(2u64)]),
         )
         .is_ok());
       })
@@ -100,8 +100,8 @@ fn bench_recursive_snark(c: &mut Criterion) {
           .verify(
             black_box(&pp),
             black_box(num_warmup_steps),
-            black_box(<G1 as Group>::Scalar::from(2u64)),
-            black_box(<G2 as Group>::Scalar::from(2u64)),
+            black_box(vec![<G1 as Group>::Scalar::from(2u64)]),
+            black_box(vec![<G2 as Group>::Scalar::from(2u64)]),
           )
           .is_ok());
       });
@@ -131,28 +131,32 @@ impl StepCircuit for NonTrivialTestCircuit
 where
   F: PrimeField,
 {
+  fn arity(&self) -> usize {
+    1
+  }
+
   fn synthesize<CS: ConstraintSystem<F>>(
     &self,
     cs: &mut CS,
-    z: AllocatedNum<F>,
-  ) -> Result<AllocatedNum<F>, SynthesisError> {
+    z: & class="p">[AllocatedNum<F>],
+  ) -> Result<Vec<AllocatedNum<F>>, SynthesisError> {
     // Consider a an equation: `x^2 = y`, where `x` and `y` are respectively the input and output.
-    let mut x = z;
+    let mut x = z[0].clone();
     let mut y = x.clone();
     for i in 0..self.num_cons {
       y = x.square(cs.namespace(|| format!("x_sq_{}", i)))?;
       x = y.clone();
     }
-    Ok(y)
+    Ok(vec![y])
   }
 
-  fn output(&self, z: &F) -> F {
-    let mut x = *z;
+  fn output(&self, z: &[F]) -> Vec<F> {
+    let mut x = z[0];
     let mut y = x;
     for _i in 0..self.num_cons {
       y = x * x;
       x = y;
     }
-    y
+    vec![y]
   }
 }

examples/ecdsa/circuit.rs

@@ -3,11 +3,6 @@ use bellperson::{
   ConstraintSystem, SynthesisError,
 };
 use ff::{PrimeField, PrimeFieldBits};
-use generic_array::typenum::U8;
-use neptune::{
-  circuit::poseidon_hash,
-  poseidon::{Poseidon, PoseidonConstants},
-};
 use nova_snark::{gadgets::ecc::AllocatedPoint, traits::circuit::StepCircuit};
 use subtle::Choice;
 
@@ -66,11 +61,6 @@ pub struct EcdsaCircuit
 where
   F: PrimeField<Repr = [u8; 32]>,
 {
-  pub z_r: Coordinate<F>,
-  pub z_g: Coordinate<F>,
-  pub z_pk: Coordinate<F>,
-  pub z_c: F,
-  pub z_s: F,
   pub r: Coordinate<F>,
   pub g: Coordinate<F>,
   pub pk: Coordinate<F>,
@@ -78,7 +68,6 @@ where
   pub s: F,
   pub c_bits: Vec<Choice>,
   pub s_bits: Vec<Choice>,
-  pub pc: PoseidonConstants<F, U8>,
 }
 
 impl<F> EcdsaCircuit<F>
@@ -88,42 +77,14 @@ where
   // Creates a new [`EcdsaCircuit<Fb, Fs>`]. The base and scalar field elements from the curve
   // field used by the signature are converted to scalar field elements from the cyclic curve
   // field used by the circuit.
-  pub fn new<Fb, Fs>(
-    num_steps: usize,
-    signatures: &[EcdsaSignature<Fb, Fs>],
-    pc: &PoseidonConstants<F, U8>,
-  ) -> (F, Vec<Self>)
+  pub fn new<Fb, Fs>(num_steps: usize, signatures: &[EcdsaSignature<Fb, Fs>]) -> (Vec<F>, Vec<Self>)
   where
     Fb: PrimeField<Repr = [u8; 32]>,
     Fs: PrimeField<Repr = [u8; 32]> + PrimeFieldBits,
   {
-    let mut z0 = F::zero();
+    let mut z0 = Vec::new();
     let mut circuits = Vec::new();
-    for i in 0..num_steps {
-      let mut j = i;
-      if i > 0 {
-        j = i - 1
-      };
-      let z_signature = &signatures[j];
-      let z_r = Coordinate::new(
-        F::from_repr(z_signature.r.x.to_repr()).unwrap(),
-        F::from_repr(z_signature.r.y.to_repr()).unwrap(),
-      );
-
-      let z_g = Coordinate::new(
-        F::from_repr(z_signature.g.x.to_repr()).unwrap(),
-        F::from_repr(z_signature.g.y.to_repr()).unwrap(),
-      );
-
-      let z_pk = Coordinate::new(
-        F::from_repr(z_signature.pk.x.to_repr()).unwrap(),
-        F::from_repr(z_signature.pk.y.to_repr()).unwrap(),
-      );
-
-      let z_c = F::from_repr(z_signature.c.to_repr()).unwrap();
-      let z_s = F::from_repr(z_signature.s.to_repr()).unwrap();
-
-      let signature = &signatures[i];
+    for (i, signature) in signatures.iter().enumerate().take(num_steps) {
       let r = Coordinate::new(
         F::from_repr(signature.r.x.to_repr()).unwrap(),
         F::from_repr(signature.r.y.to_repr()).unwrap(),
@@ -145,11 +106,6 @@ where
       let s = F::from_repr(signature.s.to_repr()).unwrap();
 
       let circuit = EcdsaCircuit {
-        z_r,
-        z_g,
-        z_pk,
-        z_c,
-        z_s,
         r,
         g,
         pk,
@@ -157,13 +113,11 @@ where
         s,
         c_bits,
         s_bits,
-        pc: pc.clone(),
       };
       circuits.push(circuit);
 
       if i == 0 {
-        z0 =
-          Poseidon::<F, U8>::new_with_preimage(&[r.x, r.y, g.x, g.y, pk.x, pk.y, c, s], pc).hash();
+        z0 = vec![r.x, r.y, g.x, g.y, pk.x, pk.y, c, s];
       }
     }
 
@@ -208,36 +162,18 @@ impl StepCircuit for EcdsaCircuit
 where
   F: PrimeField<Repr = [u8; 32]> + PrimeFieldBits,
 {
+  fn arity(&self) -> usize {
+    8
+  }
+
   // Prove knowledge of the sk used to generate the Ecdsa signature (R,s)
   // with public key PK and message commitment c.
   // [s]G == R + [c]PK
   fn synthesize<CS: ConstraintSystem<F>>(
     &self,
     cs: &mut CS,
-    z: AllocatedNum<F>,
-  ) -> Result<AllocatedNum<F>, SynthesisError> {
-    let z_rx = AllocatedNum::alloc(cs.namespace(|| "z_rx"), || Ok(self.z_r.x))?;
-    let z_ry = AllocatedNum::alloc(cs.namespace(|| "z_ry"), || Ok(self.z_r.y))?;
-    let z_gx = AllocatedNum::alloc(cs.namespace(|| "z_gx"), || Ok(self.z_g.x))?;
-    let z_gy = AllocatedNum::alloc(cs.namespace(|| "z_gy"), || Ok(self.z_g.y))?;
-    let z_pkx = AllocatedNum::alloc(cs.namespace(|| "z_pkx"), || Ok(self.z_pk.x))?;
-    let z_pky = AllocatedNum::alloc(cs.namespace(|| "z_pky"), || Ok(self.z_pk.y))?;
-    let z_c = AllocatedNum::alloc(cs.namespace(|| "z_c"), || Ok(self.z_c))?;
-    let z_s = AllocatedNum::alloc(cs.namespace(|| "z_s"), || Ok(self.z_s))?;
-
-    let z_hash = poseidon_hash(
-      cs.namespace(|| "input hash"),
-      vec![z_rx, z_ry, z_gx, z_gy, z_pkx, z_pky, z_c, z_s],
-      &self.pc,
-    )?;
-
-    cs.enforce(
-      || "z == z1",
-      |lc| lc + z.get_variable(),
-      |lc| lc + CS::one(),
-      |lc| lc + z_hash.get_variable(),
-    );
-
+    _z: &[AllocatedNum<F>],
+  ) -> Result<Vec<AllocatedNum<F>>, SynthesisError> {
     let g = AllocatedPoint::alloc(
       cs.namespace(|| "G"),
       Some((self.g.x, self.g.y, self.g.is_infinity)),
@@ -282,36 +218,12 @@ where
     let c = AllocatedNum::alloc(cs.namespace(|| "c"), || Ok(self.c))?;
     let s = AllocatedNum::alloc(cs.namespace(|| "s"), || Ok(self.s))?;
 
-    poseidon_hash(
-      cs.namespace(|| "output hash"),
-      vec![rx, ry, gx, gy, pkx, pky, c, s],
-      &self.pc,
-    )
+    Ok(vec![rx, ry, gx, gy, pkx, pky, c, s])
   }
 
-  fn output(&self, z: &F) -> F {
-    let z_hash = Poseidon::<F, U8>::new_with_preimage(
-      &[
-        self.z_r.x,
-        self.z_r.y,
-        self.z_g.x,
-        self.z_g.y,
-        self.z_pk.x,
-        self.z_pk.y,
-        self.z_c,
-        self.z_s,
-      ],
-      &self.pc,
-    )
-    .hash();
-    debug_assert_eq!(z, &z_hash);
-
-    Poseidon::<F, U8>::new_with_preimage(
-      &[
-        self.r.x, self.r.y, self.g.x, self.g.y, self.pk.x, self.pk.y, self.c, self.s,
-      ],
-      &self.pc,
-    )
-    .hash()
+  fn output(&self, _z: &[F]) -> Vec<F> {
+    vec![
+      self.r.x, self.r.y, self.g.x, self.g.y, self.pk.x, self.pk.y, self.c, self.s,
+    ]
   }
 }

examples/ecdsa/main.rs

@@ -9,8 +9,6 @@ use ff::{
   derive::byteorder::{ByteOrder, LittleEndian},
   Field, PrimeField, PrimeFieldBits,
 };
-use generic_array::typenum::U8;
-use neptune::{poseidon::PoseidonConstants, Strength};
 use nova_snark::{
   traits::{circuit::TrivialTestCircuit, Group as Nova_Group},
   CompressedSNARK, PublicParams, RecursiveSNARK,
@@ -165,22 +163,7 @@ fn main() {
   // produce public parameters
   println!("Generating public parameters...");
 
-  let pc = PoseidonConstants::<<G2 as Group>::Scalar, U8>::new_with_strength(Strength::Standard);
   let circuit_primary = EcdsaCircuit::<<G2 as Nova_Group>::Scalar> {
-    z_r: Coordinate::new(
-      <G2 as Nova_Group>::Scalar::zero(),
-      <G2 as Nova_Group>::Scalar::zero(),
-    ),
-    z_g: Coordinate::new(
-      <G2 as Nova_Group>::Scalar::zero(),
-      <G2 as Nova_Group>::Scalar::zero(),
-    ),
-    z_pk: Coordinate::new(
-      <G2 as Nova_Group>::Scalar::zero(),
-      <G2 as Nova_Group>::Scalar::zero(),
-    ),
-    z_c: <G2 as Nova_Group>::Scalar::zero(),
-    z_s: <G2 as Nova_Group>::Scalar::zero(),
     r: Coordinate::new(
       <G2 as Nova_Group>::Scalar::zero(),
       <G2 as Nova_Group>::Scalar::zero(),
@@ -197,7 +180,6 @@ fn main() {
     s: <G2 as Nova_Group>::Scalar::zero(),
     c_bits: vec![Choice::from(0u8); 256],
     s_bits: vec![Choice::from(0u8); 256],
-    pc: pc.clone(),
   };
 
   let circuit_secondary = TrivialTestCircuit::default();
@@ -258,10 +240,10 @@ fn main() {
   let (z0_primary, circuits_primary) = EcdsaCircuit::<<G2 as Nova_Group>::Scalar>::new::<
     <G1 as Nova_Group>::Base,
     <G1 as Nova_Group>::Scalar,
-  >(num_steps, &signatures(), &pc);
+  >(num_steps, &signatures());
 
   // Secondary circuit
-  let z0_secondary = <G1 as Group>::Scalar::zero();
+  let z0_secondary = vec![<G1 as Group>::Scalar::zero()];
 
   // produce a recursive SNARK
   println!("Generating a RecursiveSNARK...");
@@ -277,8 +259,8 @@ fn main() {
       recursive_snark,
       circuit_primary.clone(),
       circuit_secondary.clone(),
-      z0_primary,
-      z0_secondary,
+      z0_primary.clone(),
+      z0_secondary.clone(),
     );
     assert!(result.is_ok());
     println!("RecursiveSNARK::prove_step {}: {:?}", i, result.is_ok());
@@ -290,7 +272,7 @@ fn main() {
 
   // verify the recursive SNARK
   println!("Verifying the RecursiveSNARK...");
-  let res = recursive_snark.verify(&pp, num_steps, z0_primary, z0_secondary);
+  let res = recursive_snark.verify(&pp, num_steps, z0_primary.clone(), z0_secondary.clone());
   println!("RecursiveSNARK::verify: {:?}", res.is_ok());
   assert!(res.is_ok());
 

examples/minroot.rs

@@ -5,12 +5,6 @@ type G1 = pasta_curves::pallas::Point;
 type G2 = pasta_curves::vesta::Point;
 use ::bellperson::{gadgets::num::AllocatedNum, ConstraintSystem, SynthesisError};
 use ff::PrimeField;
-use generic_array::typenum::U2;
-use neptune::{
-  circuit::poseidon_hash,
-  poseidon::{Poseidon, PoseidonConstants},
-  Strength,
-};
 use nova_snark::{
   traits::{
     circuit::{StepCircuit, TrivialTestCircuit},
@@ -31,7 +25,7 @@ struct MinRootIteration {
 
 impl<F: PrimeField> MinRootIteration<F> {
   // produces a sample non-deterministic advice, executing one invocation of MinRoot per step
-  fn new(num_iters: usize, x_0: &F, y_0: &F, pc: &PoseidonConstants<F, U2>) -> (F, Vec<Self>) {
+  fn new(num_iters: usize, x_0: &F, y_0: &F) -> (Vec<F>, Vec<Self>) {
     // although this code is written generically, it is tailored to Pallas' scalar field
     // (p - 3 / 5)
     let exp = BigUint::parse_bytes(
@@ -65,7 +59,7 @@ impl MinRootIteration {
       y_i = y_i_plus_1;
     }
 
-    let z0 = Poseidon::<F, U2>::new_with_preimage(&[*x_0, *y_0], pc).hash();
+    let z0 = vec![*x_0, *y_0];
 
     (z0, res)
   }
@@ -74,23 +68,27 @@ impl MinRootIteration {
 #[derive(Clone, Debug)]
 struct MinRootCircuit<F: PrimeField> {
   seq: Vec<MinRootIteration<F>>,
-  pc: PoseidonConstants<F, U2>,
 }
 
 impl<F> StepCircuit<F> for MinRootCircuit<F>
 where
   F: PrimeField,
 {
+  fn arity(&self) -> usize {
+    2
+  }
+
   fn synthesize<CS: ConstraintSystem<F>>(
     &self,
     cs: &mut CS,
-    z: AllocatedNum<F>,
-  ) -> Result<AllocatedNum<F>, SynthesisError> {
-    let mut z_out: Result<AllocatedNum<F>, SynthesisError> = Err(SynthesisError::AssignmentMissing);
+    z: &[AllocatedNum<F>],
+  ) -> Result<Vec<AllocatedNum<F>>, SynthesisError> {
+    let mut z_out: Result<Vec<AllocatedNum<F>>, SynthesisError> =
+      Err(SynthesisError::AssignmentMissing);
 
-    // allocate variables to hold x_0 and y_0
-    let x_0 = AllocatedNum::alloc(cs.namespace(|| "x_0"), || Ok(self.seq[0].x_i))?;
-    let y_0 = AllocatedNum::alloc(cs.namespace(|| "y_0"), || Ok(self.seq[0].y_i))?;
+    // use the provided inputs
+    let x_0 = z[0].clone();
+    let y_0 = z[1].clone();
 
     // variables to hold running x_i and y_i
     let mut x_i = x_0;
@@ -102,21 +100,6 @@ where
           Ok(self.seq[i].x_i_plus_1)
         })?;
 
-      // check that z = hash(x_i, y_i), where z is an output from the prior step
-      if i == 0 {
-        let z_hash = poseidon_hash(
-          cs.namespace(|| "input hash"),
-          vec![x_i.clone(), y_i.clone()],
-          &self.pc,
-        )?;
-        cs.enforce(
-          || "z =? z_hash",
-          |lc| lc + z_hash.get_variable(),
-          |lc| lc + CS::one(),
-          |lc| lc + z.get_variable(),
-        );
-      }
-
       // check the following conditions hold:
       // (i) x_i_plus_1 = (x_i + y_i)^{1/5}, which can be more easily checked with x_i_plus_1^5 = x_i + y_i
       // (ii) y_i_plus_1 = x_i
@@ -135,11 +118,7 @@ where
 
       // return hash(x_i_plus_1, y_i_plus_1) since Nova circuits expect a single output
       if i == self.seq.len() - 1 {
-        z_out = poseidon_hash(
-          cs.namespace(|| "output hash"),
-          vec![x_i_plus_1.clone(), x_i.clone()],
-          &self.pc,
-        );
+        z_out = Ok(vec![x_i_plus_1.clone(), x_i.clone()]);
       }
 
       // update x_i and y_i for the next iteration
@@ -150,22 +129,16 @@ where
     z_out
   }
 
-  fn output(&self, z: &F) -> F {
+  fn output(&self, z: &[F]) -> Vec<F> {
     // sanity check
-    let z_hash =
-      Poseidon::<F, U2>::new_with_preimage(&[self.seq[0].x_i, self.seq[0].y_i], &self.pc).hash();
-    debug_assert_eq!(z, &z_hash);
-
-    // compute output hash using advice
-    let iters = self.seq.len();
-    Poseidon::<F, U2>::new_with_preimage(
-      &[
-        self.seq[iters - 1].x_i_plus_1,
-        self.seq[iters - 1].y_i_plus_1,
-      ],
-      &self.pc,
-    )
-    .hash()
+    debug_assert_eq!(z[0], self.seq[0].x_i);
+    debug_assert_eq!(z[1], self.seq[0].y_i);
+
+    // compute output using advice
+    vec![
+      self.seq[self.seq.len() - 1].x_i_plus_1,
+      self.seq[self.seq.len() - 1].y_i_plus_1,
+    ]
   }
 }
 
@@ -176,7 +149,6 @@ fn main() {
   let num_steps = 10;
   for num_iters_per_step in [1024, 2048, 4096, 8192, 16384, 32768, 65535] {
     // number of iterations of MinRoot per Nova's recursive step
-    let pc = PoseidonConstants::<<G1 as Group>::Scalar, U2>::new_with_strength(Strength::Standard);
     let circuit_primary = MinRootCircuit {
       seq: vec![
         MinRootIteration {
@@ -187,7 +159,6 @@ fn main() {
         };
         num_iters_per_step
       ],
-      pc: pc.clone(),
     };
 
     let circuit_secondary = TrivialTestCircuit::default();
@@ -228,7 +199,6 @@ fn main() {
       num_iters_per_step * num_steps,
       &<G1 as Group>::Scalar::zero(),
       &<G1 as Group>::Scalar::one(),
-      &pc,
     );
     let minroot_circuits = (0..num_steps)
       .map(|i| MinRootCircuit {
@@ -240,11 +210,10 @@ fn main() {
             y_i_plus_1: minroot_iterations[i * num_iters_per_step + j].y_i_plus_1,
           })
           .collect::<Vec<_>>(),
-        pc: pc.clone(),
       })
       .collect::<Vec<_>>();
 
-    let z0_secondary = <G2 as Group>::Scalar::zero();
+    let z0_secondary = vec![<G2 as Group>::Scalar::zero()];
 
     type C1 = MinRootCircuit<<G1 as Group>::Scalar>;
     type C2 = TrivialTestCircuit<<G2 as Group>::Scalar>;
@@ -259,8 +228,8 @@ fn main() {
         recursive_snark,
         circuit_primary.clone(),
         circuit_secondary.clone(),
-        z0_primary,
-        z0_secondary,
+        z0_primary.clone(),
+        z0_secondary.clone(),
       );
       assert!(res.is_ok());
       println!(
@@ -278,7 +247,7 @@ fn main() {
     // verify the recursive SNARK
     println!("Verifying a RecursiveSNARK...");
     let start = Instant::now();
-    let res = recursive_snark.verify(&pp, num_steps, z0_primary, z0_secondary);
+    let res = recursive_snark.verify(&pp, num_steps, z0_primary.clone(), z0_secondary.clone());
     println!(
       "RecursiveSNARK::verify: {:?}, took {:?}",
       res.is_ok(),

src/circuit.rs

@@ -8,12 +8,12 @@
 
 use super::{
   commitments::Commitment,
-  constants::{NUM_FE_FOR_HASH, NUM_HASH_BITS},
+  constants::{NUM_FE_WITHOUT_IO_FOR_CRHF, NUM_HASH_BITS},
   gadgets::{
     ecc::AllocatedPoint,
     r1cs::{AllocatedR1CSInstance, AllocatedRelaxedR1CSInstance},
     utils::{
-      alloc_num_equals, alloc_scalar_as_base, alloc_zero, conditionally_select, le_bits_to_num,
+      alloc_num_equals, alloc_scalar_as_base, alloc_zero, conditionally_select_vec, le_bits_to_num,
     },
   },
   r1cs::{R1CSInstance, RelaxedR1CSInstance},
@@ -50,8 +50,8 @@ impl NovaAugmentedCircuitParams {
 pub struct NovaAugmentedCircuitInputs<G: Group> {
   params: G::Scalar, // Hash(Shape of u2, Gens for u2). Needed for computing the challenge.
   i: G::Base,
-  z0: G::Base,
-  zi: Option<G::Base>,
+  z0: Vec<G::Base>,
+  zi: Option<Vec<G::Base>>,
   U: Option<RelaxedR1CSInstance<G>>,
   u: Option<R1CSInstance<G>>,
   T: Option<Commitment<G>>,
@@ -66,8 +66,8 @@ where
   pub fn new(
     params: G::Scalar,
     i: G::Base,
-    z0: G::Base,
-    zi: Option<G::Base>,
+    z0: Vec<G::Base>,
+    zi: Option<Vec<G::Base>>,
     U: Option<RelaxedR1CSInstance<G>>,
     u: Option<R1CSInstance<G>>,
     T: Option<Commitment<G>>,
@@ -121,12 +121,13 @@ where
   fn alloc_witness<CS: ConstraintSystem<<G as Group>::Base>>(
     &self,
     mut cs: CS,
+    arity: usize,
   ) -> Result<
     (
       AllocatedNum<G::Base>,
       AllocatedNum<G::Base>,
-      AllocatedNum<G::Base>,
-      AllocatedNum<G::Base>,
+      Vec<AllocatedNum<G::Base>>,
+      Vec<AllocatedNum<G::Base>>,
       AllocatedRelaxedR1CSInstance<G>,
       AllocatedR1CSInstance<G>,
       AllocatedPoint<G::Base>,
@@ -143,12 +144,23 @@ where
     let i = AllocatedNum::alloc(cs.namespace(|| "i"), || Ok(self.inputs.get()?.i))?;
 
     // Allocate z0
-    let z_0 = AllocatedNum::alloc(cs.namespace(|| "z0"), || Ok(self.inputs.get()?.z0))?;
+    let z_0 = (0..arity)
+      .map(|i| {
+        AllocatedNum::alloc(cs.namespace(|| format!("z0_{}", i)), || {
+          Ok(self.inputs.get()?.z0[i])
+        })
+      })
+      .collect::<Result<Vec<AllocatedNum<G::Base>>, _>>()?;
 
     // Allocate zi. If inputs.zi is not provided (base case) allocate default value 0
-    let z_i = AllocatedNum::alloc(cs.namespace(|| "zi"), || {
-      Ok(self.inputs.get()?.zi.unwrap_or_else(G::Base::zero))
-    })?;
+    let zero = vec![G::Base::zero(); arity];
+    let z_i = (0..arity)
+      .map(|i| {
+        AllocatedNum::alloc(cs.namespace(|| format!("zi_{}", i)), || {
+          Ok(self.inputs.get()?.zi.as_ref().unwrap_or(&zero)[i])
+        })
+      })
+      .collect::<Result<Vec<AllocatedNum<G::Base>>, _>>()?;
 
     // Allocate the running instance
     let U: AllocatedRelaxedR1CSInstance<G> = AllocatedRelaxedR1CSInstance::alloc(
@@ -215,18 +227,26 @@ where
     mut cs: CS,
     params: AllocatedNum<G::Base>,
     i: AllocatedNum<G::Base>,
-    z_0: AllocatedNum<G::Base>,
-    z_i: AllocatedNum<G::Base>,
+    z_0: Vec<AllocatedNum<G::Base>>,
+    z_i: Vec<AllocatedNum<G::Base>>,
     U: AllocatedRelaxedR1CSInstance<G>,
     u: AllocatedR1CSInstance<G>,
     T: AllocatedPoint<G::Base>,
+    arity: usize,
   ) -> Result<(AllocatedRelaxedR1CSInstance<G>, AllocatedBit), SynthesisError> {
     // Check that u.x[0] = Hash(params, U, i, z0, zi)
-    let mut ro = G::ROCircuit::new(self.ro_consts.clone(), NUM_FE_FOR_HASH);
+    let mut ro = G::ROCircuit::new(
+      self.ro_consts.clone(),
+      NUM_FE_WITHOUT_IO_FOR_CRHF + 2 * arity,
+    );
     ro.absorb(params.clone());
     ro.absorb(i);
-    ro.absorb(z_0);
-    ro.absorb(z_i);
+    for e in z_0 {
+      ro.absorb(e);
+    }
+    for e in z_i {
+      ro.absorb(e);
+    }
     U.absorb_in_ro(cs.namespace(|| "absorb U"), &mut ro)?;
 
     let hash_bits = ro.squeeze(cs.namespace(|| "Input hash"), NUM_HASH_BITS)?;
@@ -261,9 +281,11 @@ where
     self,
     cs: &mut CS,
   ) -> Result<(), SynthesisError> {
+    let arity = self.step_circuit.arity();
+
     // Allocate all witnesses
     let (params, i, z_0, z_i, U, u, T) =
-      self.alloc_witness(cs.namespace(|| "allocate the circuit witness"))?;
+      self.alloc_witness(cs.namespace(|| "allocate the circuit witness"), arity)?;
 
     // Compute variable indicating if this is the base case
     let zero = alloc_zero(cs.namespace(|| "zero"))?;
@@ -283,6 +305,7 @@ where
       U,
       u.clone(),
       T,
+      arity,
     )?;
 
     // Either check_non_base_pass=true or we are in the base case
@@ -317,7 +340,7 @@ where
     );
 
     // Compute z_{i+1}
-    let z_input = conditionally_select(
+    let z_input = conditionally_select_vec(
       cs.namespace(|| "select input to F"),
       &z_0,
       &z_i,
@@ -326,14 +349,24 @@ where
 
     let z_next = self
       .step_circuit
-      .synthesize(&mut cs.namespace(|| "F"), z_input)?;
+      .synthesize(&mut cs.namespace(|| "F"), &z_input)?;
+
+    if z_next.len() != arity {
+      return Err(SynthesisError::IncompatibleLengthVector(
+        "z_next".to_string(),
+      ));
+    }
 
     // Compute the new hash H(params, Unew, i+1, z0, z_{i+1})
-    let mut ro = G::ROCircuit::new(self.ro_consts, NUM_FE_FOR_HASH);
+    let mut ro = G::ROCircuit::new(self.ro_consts, NUM_FE_WITHOUT_IO_FOR_CRHF + 2 * arity);
     ro.absorb(params);
     ro.absorb(i_new.clone());
-    ro.absorb(z_0);
-    ro.absorb(z_next);
+    for e in z_0 {
+      ro.absorb(e);
+    }
+    for e in z_next {
+      ro.absorb(e);
+    }
     Unew.absorb_in_ro(cs.namespace(|| "absorb U_new"), &mut ro)?;
     let hash_bits = ro.squeeze(cs.namespace(|| "output hash bits"), NUM_HASH_BITS)?;
     let hash = le_bits_to_num(cs.namespace(|| "convert hash to num"), hash_bits)?;
@@ -397,8 +430,15 @@ mod tests {
     // Execute the base case for the primary
     let zero1 = <<G2 as Group>::Base as Field>::zero();
     let mut cs1: SatisfyingAssignment<G1> = SatisfyingAssignment::new();
-    let inputs1: NovaAugmentedCircuitInputs<G2> =
-      NovaAugmentedCircuitInputs::new(shape2.get_digest(), zero1, zero1, None, None, None, None);
+    let inputs1: NovaAugmentedCircuitInputs<G2> = NovaAugmentedCircuitInputs::new(
+      shape2.get_digest(),
+      zero1,
+      vec![zero1],
+      None,
+      None,
+      None,
+      None,
+    );
     let circuit1: NovaAugmentedCircuit<G2, TrivialTestCircuit<<G2 as Group>::Base>> =
       NovaAugmentedCircuit::new(
         params1,
@@ -417,7 +457,7 @@ mod tests {
     let inputs2: NovaAugmentedCircuitInputs<G1> = NovaAugmentedCircuitInputs::new(
       shape1.get_digest(),
       zero2,
-      zero2,
+      vec![zero2],
       None,
       None,
       Some(inst1),

src/constants.rs

@@ -2,5 +2,5 @@ pub(crate) const NUM_CHALLENGE_BITS: usize = 128;
 pub(crate) const NUM_HASH_BITS: usize = 250;
 pub(crate) const BN_LIMB_WIDTH: usize = 64;
 pub(crate) const BN_N_LIMBS: usize = 4;
-pub(crate) const NUM_FE_FOR_HASH: usize = 19;
+pub(crate) const NUM_FE_WITHOUT_IO_FOR_CRHF: usize = 17;
 pub(crate) const NUM_FE_FOR_RO: usize = 24;

src/errors.rs

@@ -24,4 +24,8 @@ pub enum NovaError {
   InvalidIPA,
   /// returned when an invalid sum-check proof is provided
   InvalidSumcheckProof,
+  /// returned when the initial input to an incremental computation differs from a previously declared arity
+  InvalidInitialInputLength,
+  /// returned when the step execution produces an output whose length differs from a previously declared arity
+  InvalidStepOutputLength,
 }

src/gadgets/utils.rs

@@ -211,6 +211,22 @@ pub fn conditionally_select>(
   Ok(c)
 }
 
+/// If condition return a otherwise b
+pub fn conditionally_select_vec<F: PrimeField, CS: ConstraintSystem<F>>(
+  mut cs: CS,
+  a: &[AllocatedNum<F>],
+  b: &[AllocatedNum<F>],
+  condition: &Boolean,
+) -> Result<Vec<AllocatedNum<F>>, SynthesisError> {
+  a.iter()
+    .zip(b.iter())
+    .enumerate()
+    .map(|(i, (a, b))| {
+      conditionally_select(cs.namespace(|| format!("select_{}", i)), a, b, condition)
+    })
+    .collect::<Result<Vec<AllocatedNum<F>>, SynthesisError>>()
+}
+
 /// If condition return a otherwise b where a and b are BigNats
 pub fn conditionally_select_bignat<F: PrimeField, CS: ConstraintSystem<F>>(
   mut cs: CS,

src/lib.rs

@@ -26,7 +26,7 @@ use crate::bellperson::{
 };
 use ::bellperson::{Circuit, ConstraintSystem};
 use circuit::{NovaAugmentedCircuit, NovaAugmentedCircuitInputs, NovaAugmentedCircuitParams};
-use constants::{BN_LIMB_WIDTH, BN_N_LIMBS, NUM_FE_FOR_HASH, NUM_HASH_BITS};
+use constants::{BN_LIMB_WIDTH, BN_N_LIMBS, NUM_FE_WITHOUT_IO_FOR_CRHF, NUM_HASH_BITS};
 use core::marker::PhantomData;
 use errors::NovaError;
 use ff::Field;
@@ -48,6 +48,8 @@ where
   C1: StepCircuit<G1::Scalar>,
   C2: StepCircuit<G2::Scalar>,
 {
+  F_arity_primary: usize,
+  F_arity_secondary: usize,
   ro_consts_primary: ROConstants<G1>,
   ro_consts_circuit_primary: ROConstantsCircuit<G2>,
   r1cs_gens_primary: R1CSGens<G1>,
@@ -81,6 +83,9 @@ where
     let ro_consts_primary: ROConstants<G1> = ROConstants::<G1>::new();
     let ro_consts_secondary: ROConstants<G2> = ROConstants::<G2>::new();
 
+    let F_arity_primary = c_primary.arity();
+    let F_arity_secondary = c_secondary.arity();
+
     // ro_consts_circuit_primart are parameterized by G2 because the type alias uses G2::Base = G1::Scalar
     let ro_consts_circuit_primary: ROConstantsCircuit<G2> = ROConstantsCircuit::<G2>::new();
     let ro_consts_circuit_secondary: ROConstantsCircuit<G1> = ROConstantsCircuit::<G1>::new();
@@ -110,6 +115,8 @@ where
     let r1cs_shape_padded_secondary = r1cs_shape_secondary.pad();
 
     Self {
+      F_arity_primary,
+      F_arity_secondary,
       ro_consts_primary,
       ro_consts_circuit_primary,
       r1cs_gens_primary,
@@ -162,8 +169,8 @@ where
   l_w_secondary: R1CSWitness<G2>,
   l_u_secondary: R1CSInstance<G2>,
   i: usize,
-  zi_primary: G1::Scalar,
-  zi_secondary: G2::Scalar,
+  zi_primary: Vec<G1::Scalar>,
+  zi_secondary: Vec<G2::Scalar>,
   _p_c1: PhantomData<C1>,
   _p_c2: PhantomData<C2>,
 }
@@ -182,9 +189,13 @@ where
     recursive_snark: Option<Self>,
     c_primary: C1,
     c_secondary: C2,
-    z0_primary: G1::Scalar,
-    z0_secondary: G2::Scalar,
+    z0_primary: Vec<G1::Scalar>,
+    z0_secondary: Vec<G2::Scalar>,
   ) -> Result<Self, NovaError> {
+    if z0_primary.len() != pp.F_arity_primary || z0_secondary.len() != pp.F_arity_secondary {
+      return Err(NovaError::InvalidInitialInputLength);
+    }
+
     match recursive_snark {
       None => {
         // base case for the primary
@@ -192,7 +203,7 @@ where
         let inputs_primary: NovaAugmentedCircuitInputs<G2> = NovaAugmentedCircuitInputs::new(
           pp.r1cs_shape_secondary.get_digest(),
           G1::Scalar::zero(),
-          z0_primary,
+          z0_primary.clone(),
           None,
           None,
           None,
@@ -215,7 +226,7 @@ where
         let inputs_secondary: NovaAugmentedCircuitInputs<G1> = NovaAugmentedCircuitInputs::new(
           pp.r1cs_shape_primary.get_digest(),
           G2::Scalar::zero(),
-          z0_secondary,
+          z0_secondary.clone(),
           None,
           None,
           Some(u_primary.clone()),
@@ -254,6 +265,10 @@ where
         let zi_primary = c_primary.output(&z0_primary);
         let zi_secondary = c_secondary.output(&z0_secondary);
 
+        if z0_primary.len() != pp.F_arity_primary || z0_secondary.len() != pp.F_arity_secondary {
+          return Err(NovaError::InvalidStepOutputLength);
+        }
+
         Ok(Self {
           r_W_primary,
           r_U_primary,
@@ -287,7 +302,7 @@ where
           pp.r1cs_shape_secondary.get_digest(),
           G1::Scalar::from(r_snark.i as u64),
           z0_primary,
-          Some(r_snark.zi_primary),
+          Some(r_snark.zi_primary.clone()),
           Some(r_snark.r_U_secondary.clone()),
           Some(r_snark.l_u_secondary.clone()),
           Some(nifs_secondary.comm_T.decompress()?),
@@ -321,7 +336,7 @@ where
           pp.r1cs_shape_primary.get_digest(),
           G2::Scalar::from(r_snark.i as u64),
           z0_secondary,
-          Some(r_snark.zi_secondary),
+          Some(r_snark.zi_secondary.clone()),
           Some(r_snark.r_U_primary.clone()),
           Some(l_u_primary.clone()),
           Some(nifs_primary.comm_T.decompress()?),
@@ -367,9 +382,9 @@ where
     &self,
     pp: &PublicParams<G1, G2, C1, C2>,
     num_steps: usize,
-    z0_primary: G1::Scalar,
-    z0_secondary: G2::Scalar,
-  ) -> Result<(G1::Scalar, G2::Scalar), NovaError> {
+    z0_primary: Vec<G1::Scalar>,
+    z0_secondary: Vec<G2::Scalar>,
+  ) -> Result<(Vec<G1::Scalar>, Vec<G2::Scalar>), NovaError> {
     // number of steps cannot be zero
     if num_steps == 0 {
       return Err(NovaError::ProofVerifyError);
@@ -391,18 +406,32 @@ where
 
     // check if the output hashes in R1CS instances point to the right running instances
     let (hash_primary, hash_secondary) = {
-      let mut hasher = <G2 as Group>::RO::new(pp.ro_consts_secondary.clone(), NUM_FE_FOR_HASH);
+      let mut hasher = <G2 as Group>::RO::new(
+        pp.ro_consts_secondary.clone(),
+        NUM_FE_WITHOUT_IO_FOR_CRHF + 2 * pp.F_arity_primary,
+      );
       hasher.absorb(scalar_as_base::<G2>(pp.r1cs_shape_secondary.get_digest()));
       hasher.absorb(G1::Scalar::from(num_steps as u64));
-      hasher.absorb(z0_primary);
-      hasher.absorb(self.zi_primary);
+      for e in &z0_primary {
+        hasher.absorb(*e);
+      }
+      for e in &self.zi_primary {
+        hasher.absorb(*e);
+      }
       self.r_U_secondary.absorb_in_ro(&mut hasher);
 
-      let mut hasher2 = <G1 as Group>::RO::new(pp.ro_consts_primary.clone(), NUM_FE_FOR_HASH);
+      let mut hasher2 = <G1 as Group>::RO::new(
+        pp.ro_consts_primary.clone(),
+        NUM_FE_WITHOUT_IO_FOR_CRHF + 2 * pp.F_arity_secondary,
+      );
       hasher2.absorb(scalar_as_base::<G1>(pp.r1cs_shape_primary.get_digest()));
       hasher2.absorb(G2::Scalar::from(num_steps as u64));
-      hasher2.absorb(z0_secondary);
-      hasher2.absorb(self.zi_secondary);
+      for e in &z0_secondary {
+        hasher2.absorb(*e);
+      }
+      for e in &self.zi_secondary {
+        hasher2.absorb(*e);
+      }
       self.r_U_primary.absorb_in_ro(&mut hasher2);
 
       (
@@ -463,7 +492,7 @@ where
     res_r_secondary?;
     res_l_secondary?;
 
-    Ok((self.zi_primary, self.zi_secondary))
+    Ok((self.zi_primary.clone(), self.zi_secondary.clone()))
   }
 }
 
@@ -488,8 +517,8 @@ where
   nifs_secondary: NIFS<G2>,
   f_W_snark_secondary: S2,
 
-  zn_primary: G1::Scalar,
-  zn_secondary: G2::Scalar,
+  zn_primary: Vec<G1::Scalar>,
+  zn_secondary: Vec<G2::Scalar>,
 
   _p_c1: PhantomData<C1>,
   _p_c2: PhantomData<C2>,
@@ -574,8 +603,8 @@ where
       nifs_secondary,
       f_W_snark_secondary: f_W_snark_secondary?,
 
-      zn_primary: recursive_snark.zi_primary,
-      zn_secondary: recursive_snark.zi_secondary,
+      zn_primary: recursive_snark.zi_primary.clone(),
+      zn_secondary: recursive_snark.zi_secondary.clone(),
 
       _p_c1: Default::default(),
       _p_c2: Default::default(),
@@ -587,9 +616,9 @@ where
     &self,
     pp: &PublicParams<G1, G2, C1, C2>,
     num_steps: usize,
-    z0_primary: G1::Scalar,
-    z0_secondary: G2::Scalar,
-  ) -> Result<(G1::Scalar, G2::Scalar), NovaError> {
+    z0_primary: Vec<G1::Scalar>,
+    z0_secondary: Vec<G2::Scalar>,
+  ) -> Result<(Vec<G1::Scalar>, Vec<G2::Scalar>), NovaError> {
     // number of steps cannot be zero
     if num_steps == 0 {
       return Err(NovaError::ProofVerifyError);
@@ -606,18 +635,32 @@ where
 
     // check if the output hashes in R1CS instances point to the right running instances
     let (hash_primary, hash_secondary) = {
-      let mut hasher = <G2 as Group>::RO::new(pp.ro_consts_secondary.clone(), NUM_FE_FOR_HASH);
+      let mut hasher = <G2 as Group>::RO::new(
+        pp.ro_consts_secondary.clone(),
+        NUM_FE_WITHOUT_IO_FOR_CRHF + 2 * pp.F_arity_primary,
+      );
       hasher.absorb(scalar_as_base::<G2>(pp.r1cs_shape_secondary.get_digest()));
       hasher.absorb(G1::Scalar::from(num_steps as u64));
-      hasher.absorb(z0_primary);
-      hasher.absorb(self.zn_primary);
+      for e in z0_primary {
+        hasher.absorb(e);
+      }
+      for e in &self.zn_primary {
+        hasher.absorb(*e);
+      }
       self.r_U_secondary.absorb_in_ro(&mut hasher);
 
-      let mut hasher2 = <G1 as Group>::RO::new(pp.ro_consts_primary.clone(), NUM_FE_FOR_HASH);
+      let mut hasher2 = <G1 as Group>::RO::new(
+        pp.ro_consts_primary.clone(),
+        NUM_FE_WITHOUT_IO_FOR_CRHF + 2 * pp.F_arity_secondary,
+      );
       hasher2.absorb(scalar_as_base::<G1>(pp.r1cs_shape_primary.get_digest()));
       hasher2.absorb(G2::Scalar::from(num_steps as u64));
-      hasher2.absorb(z0_secondary);
-      hasher2.absorb(self.zn_secondary);
+      for e in z0_secondary {
+        hasher2.absorb(e);
+      }
+      for e in &self.zn_secondary {
+        hasher2.absorb(*e);
+      }
       self.r_U_primary.absorb_in_ro(&mut hasher2);
 
       (
@@ -665,7 +708,7 @@ where
     res_primary?;
     res_secondary?;
 
-    Ok((self.zn_primary, self.zn_secondary))
+    Ok((self.zn_primary.clone(), self.zn_secondary.clone()))
   }
 }
 
@@ -690,15 +733,19 @@ mod tests {
   where
     F: PrimeField,
   {
+    fn arity(&self) -> usize {
+      1
+    }
+
     fn synthesize<CS: ConstraintSystem<F>>(
       &self,
       cs: &mut CS,
-      z: AllocatedNum<F>,
-    ) -> Result<AllocatedNum<F>, SynthesisError> {
+      z: & class="p">[AllocatedNum<F>],
+    ) -> Result<Vec<AllocatedNum<F>>, SynthesisError> {
       // Consider a cubic equation: `x^3 + x + 5 = y`, where `x` and `y` are respectively the input and output.
-      let x = z;
+      let x = &z[0];
       let x_sq = x.square(cs.namespace(|| "x_sq"))?;
-      let x_cu = x_sq.mul(cs.namespace(|| "x_cu"), &x)?;
+      let x_cu = x_sq.mul(cs.namespace(|| "x_cu"), x)?;
       let y = AllocatedNum::alloc(cs.namespace(|| "y"), || {
         Ok(x_cu.get_value().unwrap() + x.get_value().unwrap() + F::from(5u64))
       })?;
@@ -718,11 +765,11 @@ mod tests {
         |lc| lc + y.get_variable(),
       );
 
-      Ok(y)
+      Ok(vec![y])
     }
 
-    fn output(&self, z: &F) -> F {
-      *z * *z * *z + z + F::from(5u64)
+    fn output(&self, z: &[F]) -> Vec<F> {
+      vec![z[0] * z[0] * z[0] + z[0] + F::from(5u64)]
     }
   }
 
@@ -744,8 +791,8 @@ mod tests {
       None,
       TrivialTestCircuit::default(),
       TrivialTestCircuit::default(),
-      <G1 as Group>::Scalar::zero(),
-      <G2 as Group>::Scalar::zero(),
+      vec![<G1 as Group>::Scalar::zero()],
+      vec![<G2 as Group>::Scalar::zero()],
     );
     assert!(res.is_ok());
     let recursive_snark = res.unwrap();
@@ -754,8 +801,8 @@ mod tests {
     let res = recursive_snark.verify(
       &pp,
       num_steps,
-      <G1 as Group>::Scalar::zero(),
-      <G2 as Group>::Scalar::zero(),
+      vec![<G1 as Group>::Scalar::zero()],
+      vec![<G2 as Group>::Scalar::zero()],
     );
     assert!(res.is_ok());
   }
@@ -791,8 +838,8 @@ mod tests {
         recursive_snark,
         circuit_primary.clone(),
         circuit_secondary.clone(),
-        <G1 as Group>::Scalar::one(),
-        <G2 as Group>::Scalar::zero(),
+        vec![<G1 as Group>::Scalar::one()],
+        vec![<G2 as Group>::Scalar::zero()],
       );
       assert!(res.is_ok());
       let recursive_snark_unwrapped = res.unwrap();
@@ -801,8 +848,8 @@ mod tests {
       let res = recursive_snark_unwrapped.verify(
         &pp,
         i + 1,
-        <G1 as Group>::Scalar::one(),
-        <G2 as Group>::Scalar::zero(),
+        vec![<G1 as Group>::Scalar::one()],
+        vec![<G2 as Group>::Scalar::zero()],
       );
       assert!(res.is_ok());
 
@@ -817,21 +864,21 @@ mod tests {
     let res = recursive_snark.verify(
       &pp,
       num_steps,
-      <G1 as Group>::Scalar::one(),
-      <G2 as Group>::Scalar::zero(),
+      vec![<G1 as Group>::Scalar::one()],
+      vec![<G2 as Group>::Scalar::zero()],
     );
     assert!(res.is_ok());
 
     let (zn_primary, zn_secondary) = res.unwrap();
 
     // sanity: check the claimed output with a direct computation of the same
-    assert_eq!(zn_primary, <G1 as Group>::Scalar::one());
-    let mut zn_secondary_direct = <G2 as Group>::Scalar::zero();
+    assert_eq!(zn_primary, vec![<G1 as Group>::Scalar::one()]);
+    let mut zn_secondary_direct = vec![<G2 as Group>::Scalar::zero()];
     for _i in 0..num_steps {
       zn_secondary_direct = CubicCircuit::default().output(&zn_secondary_direct);
     }
     assert_eq!(zn_secondary, zn_secondary_direct);
-    assert_eq!(zn_secondary, <G2 as Group>::Scalar::from(2460515u64));
+    assert_eq!(zn_secondary, vec![<G2 as Group>::Scalar::from(2460515u64)]);
   }
 
   #[test]
@@ -865,8 +912,8 @@ mod tests {
         recursive_snark,
         circuit_primary.clone(),
         circuit_secondary.clone(),
-        <G1 as Group>::Scalar::one(),
-        <G2 as Group>::Scalar::zero(),
+        vec![<G1 as Group>::Scalar::one()],
+        vec![<G2 as Group>::Scalar::zero()],
       );
       assert!(res.is_ok());
       recursive_snark = Some(res.unwrap());
@@ -879,21 +926,21 @@ mod tests {
     let res = recursive_snark.verify(
       &pp,
       num_steps,
-      <G1 as Group>::Scalar::one(),
-      <G2 as Group>::Scalar::zero(),
+      vec![<G1 as Group>::Scalar::one()],
+      vec![<G2 as Group>::Scalar::zero()],
     );
     assert!(res.is_ok());
 
     let (zn_primary, zn_secondary) = res.unwrap();
 
     // sanity: check the claimed output with a direct computation of the same
-    assert_eq!(zn_primary, <G1 as Group>::Scalar::one());
-    let mut zn_secondary_direct = <G2 as Group>::Scalar::zero();
+    assert_eq!(zn_primary, vec![<G1 as Group>::Scalar::one()]);
+    let mut zn_secondary_direct = vec![<G2 as Group>::Scalar::zero()];
     for _i in 0..num_steps {
       zn_secondary_direct = CubicCircuit::default().output(&zn_secondary_direct);
     }
     assert_eq!(zn_secondary, zn_secondary_direct);
-    assert_eq!(zn_secondary, <G2 as Group>::Scalar::from(2460515u64));
+    assert_eq!(zn_secondary, vec![<G2 as Group>::Scalar::from(2460515u64)]);
 
     // produce a compressed SNARK
     let res = CompressedSNARK::<_, _, _, _, S1, S2>::prove(&pp, &recursive_snark);
@@ -904,8 +951,8 @@ mod tests {
     let res = compressed_snark.verify(
       &pp,
       num_steps,
-      <G1 as Group>::Scalar::one(),
-      <G2 as Group>::Scalar::zero(),
+      vec![<G1 as Group>::Scalar::one()],
+      vec![<G2 as Group>::Scalar::zero()],
     );
     assert!(res.is_ok());
   }
@@ -922,7 +969,7 @@ mod tests {
     where
       F: PrimeField,
     {
-      fn new(num_steps: usize) -> (F, Vec<Self>) {
+      fn new(num_steps: usize) -> (Vec<F>, Vec<Self>) {
         let mut powers = Vec::new();
         let rng = &mut rand::rngs::OsRng;
         let mut seed = F::random(rng);
@@ -939,7 +986,7 @@ mod tests {
 
         // reverse the powers to get roots
         let roots = powers.into_iter().rev().collect::<Vec<Self>>();
-        (roots[0].y, roots[1..].to_vec())
+        (vec![roots[0].y], roots[1..].to_vec())
       }
     }
 
@@ -947,12 +994,16 @@ mod tests {
     where
       F: PrimeField,
     {
+      fn arity(&self) -> usize {
+        1
+      }
+
       fn synthesize<CS: ConstraintSystem<F>>(
         &self,
         cs: &mut CS,
-        z: AllocatedNum<F>,
-      ) -> Result<AllocatedNum<F>, SynthesisError> {
-        let x = z;
+        z: & class="p">[AllocatedNum<F>],
+      ) -> Result<Vec<AllocatedNum<F>>, SynthesisError> {
+        let x = &z[0];
 
         // we allocate a variable and set it to the provided non-derministic advice.
         let y = AllocatedNum::alloc(cs.namespace(|| "y"), || Ok(self.y))?;
@@ -969,12 +1020,12 @@ mod tests {
           |lc| lc + x.get_variable(),
         );
 
-        Ok(y)
+        Ok(vec![y])
       }
 
-      fn output(&self, z: &F) -> F {
+      fn output(&self, z: &[F]) -> Vec<F> {
         // sanity check
-        let x = *z;
+        let x = z[0];
         let y_pow_5 = {
           let y = self.y;
           let y_sq = y.square();
@@ -985,7 +1036,7 @@ mod tests {
 
         // return non-deterministic advice
         // as the output of the step
-        self.y
+        vec![self.y]
       }
     }
 
@@ -1007,7 +1058,7 @@ mod tests {
 
     // produce non-deterministic advice
     let (z0_primary, roots) = FifthRootCheckingCircuit::new(num_steps);
-    let z0_secondary = <G2 as Group>::Scalar::zero();
+    let z0_secondary = vec![<G2 as Group>::Scalar::zero()];
 
     // produce a recursive SNARK
     let mut recursive_snark: Option<
@@ -1025,8 +1076,8 @@ mod tests {
         recursive_snark,
         circuit_primary.clone(),
         circuit_secondary.clone(),
-        z0_primary,
-        z0_secondary,
+        z0_primary.clone(),
+        z0_secondary.clone(),
       );
       assert!(res.is_ok());
       recursive_snark = Some(res.unwrap());
@@ -1036,7 +1087,7 @@ mod tests {
     let recursive_snark = recursive_snark.unwrap();
 
     // verify the recursive SNARK
-    let res = recursive_snark.verify(&pp, num_steps, z0_primary, z0_secondary);
+    let res = recursive_snark.verify(&pp, num_steps, z0_primary.clone(), z0_secondary.clone());
     assert!(res.is_ok());
 
     // produce a compressed SNARK
@@ -1067,8 +1118,8 @@ mod tests {
       None,
       TrivialTestCircuit::default(),
       CubicCircuit::default(),
-      <G1 as Group>::Scalar::one(),
-      <G2 as Group>::Scalar::zero(),
+      vec![<G1 as Group>::Scalar::one()],
+      vec![<G2 as Group>::Scalar::zero()],
     );
     assert!(res.is_ok());
     let recursive_snark = res.unwrap();
@@ -1077,14 +1128,14 @@ mod tests {
     let res = recursive_snark.verify(
       &pp,
       num_steps,
-      <G1 as Group>::Scalar::one(),
-      <G2 as Group>::Scalar::zero(),
+      vec![<G1 as Group>::Scalar::one()],
+      vec![<G2 as Group>::Scalar::zero()],
     );
     assert!(res.is_ok());
 
     let (zn_primary, zn_secondary) = res.unwrap();
 
-    assert_eq!(zn_primary, <G1 as Group>::Scalar::one());
-    assert_eq!(zn_secondary, <G2 as Group>::Scalar::from(5u64));
+    assert_eq!(zn_primary, vec![<G1 as Group>::Scalar::one()]);
+    assert_eq!(zn_secondary, vec![<G2 as Group>::Scalar::from(5u64)]);
   }
 }

src/poseidon.rs

@@ -86,7 +86,7 @@ where
       SpongeOp::Squeeze(1u32),
     ]);
 
-    sponge.start(parameter, Some(1u32), acc);
+    sponge.start(parameter, None, acc);
     assert_eq!(self.num_absorbs, self.state.len());
     SpongeAPI::absorb(&mut sponge, self.num_absorbs as u32, &self.state, acc);
     let hash = SpongeAPI::squeeze(&mut sponge, 1, acc);
@@ -163,7 +163,7 @@ where
       let acc = &mut ns;
       assert_eq!(self.num_absorbs, self.state.len());
 
-      sponge.start(parameter, Some(1u32), acc);
+      sponge.start(parameter, None, acc);
       neptune::sponge::api::SpongeAPI::absorb(
         &mut sponge,
         self.num_absorbs as u32,

src/spartan_with_ipa_pc/mod.rs

@@ -323,8 +323,7 @@ impl RelaxedR1CSSNARKTrait for RelaxedR1CSSNARK {
             .map(|i| (i + 1, U.X[i]))
             .collect::<Vec<(usize, G::Scalar)>>(),
         );
-        SparsePolynomial::new((vk.S.num_vars as f64).log2() as usize, poly_X)
-          .evaluate(&r_y[1..].to_vec())
+        SparsePolynomial::new((vk.S.num_vars as f64).log2() as usize, poly_X).evaluate(&r_y[1..])
       };
       (G::Scalar::one() - r_y[0]) * self.eval_W + r_y[0] * eval_X
     };

src/traits/circuit.rs

@@ -5,16 +5,22 @@ use ff::PrimeField;
 
 /// A helper trait for a step of the incremental computation (i.e., circuit for F)
 pub trait StepCircuit<F: PrimeField>: Send + Sync + Clone {
+  /// Return the the number of inputs or outputs of each step
+  /// (this method is called only at circuit synthesis time)
+  /// `synthesize` and `output` methods are expected to take as
+  /// input a vector of size equal to arity and output a vector of size equal to arity  
+  fn arity(&self) -> usize;
+
   /// Sythesize the circuit for a computation step and return variable
   /// that corresponds to the output of the step z_{i+1}
   fn synthesize<CS: ConstraintSystem<F>>(
     &self,
     cs: &mut CS,
-    z: AllocatedNum<F>,
-  ) -> Result<AllocatedNum<F>, SynthesisError>;
+    z: & class="p">[AllocatedNum<F>],
+  ) -> Result<Vec<AllocatedNum<F>>, SynthesisError>;
 
   /// return the output of the step when provided with with the step's input
-  fn output(&self, z: &F) -> F;
+  fn output(&self, z: &[F]) -> Vec<F>;
 }
 
 /// A trivial step circuit that simply returns the input
@@ -27,15 +33,19 @@ impl StepCircuit for TrivialTestCircuit
 where
   F: PrimeField,
 {
+  fn arity(&self) -> usize {
+    1
+  }
+
   fn synthesize<CS: ConstraintSystem<F>>(
     &self,
     _cs: &mut CS,
-    z: AllocatedNum<F>,
-  ) -> Result<AllocatedNum<F>, SynthesisError> {
-    Ok(z)
+    z: & class="p">[AllocatedNum<F>],
+  ) -> Result<Vec<AllocatedNum<F>>, SynthesisError> {
+    Ok(z.to_vec())
   }
 
-  fn output(&self, z: &F) -> F {
-    *z
+  fn output(&self, z: &[F]) -> Vec<F> {
+    z.to_vec()
   }
 }