feat(backend): Use Schnorr signatures for signed PODs (#236)

* Implement non-native extension field arithmetic

* Schnorr signature verification (#221)

* Use Schnorr signatures for signed PODs

* add custom gates (#237)

* Clippy

* Formatting

* Apply suggestions from code review

Co-authored-by: Eduard S. <eduardsanou@posteo.net>

* Fix typo

* Fix tests

* Point -> PublicKey

* Remove default nnf_div implementation for clarity

* Code review & edits for clarity

* Remove suspicious mutation

* Simplify computation

* Fix division

* Fix

* Update src/backends/plonky2/primitives/ec/curve.rs

Co-authored-by: Eduard S. <eduardsanou@posteo.net>

* Update src/backends/plonky2/primitives/ec/curve.rs

Co-authored-by: Eduard S. <eduardsanou@posteo.net>

* Fixes

* Add public key to signed POD struct

* Style

* Elaborate on in-circuit field->biguint conversion

* Add missing gates

* Comments

* Add bits to biguint struct

* Comments

* Comment

---------

Co-authored-by: Daniel Gulotta <dgulotta@alum.mit.edu>
Co-authored-by: Eduard S. <eduardsanou@posteo.net>

src/backends/plonky2/primitives/ec/gates/curve.rs

@@ -0,0 +1,83 @@
+use plonky2::field::goldilocks_field::GoldilocksField;
+
+use crate::backends::plonky2::primitives::ec::{
+    curve::{add_homog_offset, ECFieldExt},
+    gates::{field::QuinticTensor, generic::SimpleGate},
+};
+
+/// Gate computing the addition of two elliptic curve points in
+/// homogeneous coordinates *minus* an offset in the `z` and `t`
+/// coordinates, viz. the extension field generator times `Point::B1`,
+/// cf. CircuitBuilderElliptic::add_point.
+#[derive(Debug, Clone)]
+pub struct ECAddHomogOffset;
+
+impl SimpleGate for ECAddHomogOffset {
+    type F = GoldilocksField;
+    const INPUTS_PER_OP: usize = 20;
+    const OUTPUTS_PER_OP: usize = 20;
+    const DEGREE: usize = 4;
+    const ID: &'static str = "ECAddHomog";
+    fn eval<const D: usize>(
+        wires: &[<Self::F as plonky2::field::extension::Extendable<D>>::Extension],
+    ) -> Vec<<Self::F as plonky2::field::extension::Extendable<D>>::Extension>
+    where
+        Self::F: plonky2::field::extension::Extendable<D>,
+    {
+        let mut ans = Vec::with_capacity(20);
+        let x1 = QuinticTensor::from_base(wires[0..5].try_into().unwrap());
+        let u1 = QuinticTensor::from_base(wires[5..10].try_into().unwrap());
+        let x2 = QuinticTensor::from_base(wires[10..15].try_into().unwrap());
+        let u2 = QuinticTensor::from_base(wires[15..20].try_into().unwrap());
+        let out = add_homog_offset(x1, u1, x2, u2);
+        for v in out {
+            ans.extend(v.to_base());
+        }
+        ans
+    }
+}
+
+#[cfg(test)]
+mod test {
+    use plonky2::{
+        gates::gate_testing::{test_eval_fns, test_low_degree},
+        plonk::{circuit_data::CircuitConfig, config::PoseidonGoldilocksConfig},
+    };
+
+    use crate::backends::plonky2::primitives::ec::gates::{
+        curve::ECAddHomogOffset, generic::GateAdapter,
+    };
+
+    #[test]
+    fn test_recursion() -> Result<(), anyhow::Error> {
+        let config = CircuitConfig::standard_recursion_config();
+        let gate = GateAdapter::<ECAddHomogOffset>::new_from_config(&config);
+
+        test_eval_fns::<_, PoseidonGoldilocksConfig, _, 2>(gate)
+    }
+
+    #[test]
+    fn test_low_degree_orig() -> Result<(), anyhow::Error> {
+        let config = CircuitConfig::standard_recursion_config();
+        let gate = GateAdapter::<ECAddHomogOffset>::new_from_config(&config);
+
+        test_low_degree::<_, _, 2>(gate);
+        Ok(())
+    }
+
+    #[test]
+    fn test_low_degree_recursive() -> Result<(), anyhow::Error> {
+        let config = CircuitConfig::standard_recursion_config();
+        let orig_gate = GateAdapter::<ECAddHomogOffset>::new_from_config(&config);
+
+        test_low_degree::<_, _, 2>(orig_gate.recursive_gate());
+        Ok(())
+    }
+
+    #[test]
+    fn test_double_recursion() -> Result<(), anyhow::Error> {
+        let config = CircuitConfig::standard_recursion_config();
+        let orig_gate = GateAdapter::<ECAddHomogOffset>::new_from_config(&config);
+        test_eval_fns::<_, PoseidonGoldilocksConfig, _, 2>(orig_gate.recursive_gate())
+    }
+}