Optimize verifier eq (#102)

* wip we need to be able to do batch opening for different poly sizes or pad poly with zeros

* fix small public inputs. Only works for pow2 pubinput

Co-authored-by: Charles Chen <chancharles92@gmail.com>

arithmetic/src/lib.rs

@@ -12,4 +12,6 @@ pub use multilinear_polynomial::{
 };
 pub use univariate_polynomial::{build_l, get_uni_domain};
 pub use util::{bit_decompose, gen_eval_point, get_batched_nv, get_index};
-pub use virtual_polynomial::{build_eq_x_r, build_eq_x_r_vec, VPAuxInfo, VirtualPolynomial};
+pub use virtual_polynomial::{
+    build_eq_x_r, build_eq_x_r_vec, eq_eval, VPAuxInfo, VirtualPolynomial,
+};

arithmetic/src/virtual_polynomial.rs

@@ -325,6 +325,23 @@ impl<F: PrimeField> VirtualPolynomial<F> {
     }
 }
 
+/// Evaluate eq polynomial.
+pub fn eq_eval<F: PrimeField>(x: &[F], y: &[F]) -> Result<F, ArithErrors> {
+    if x.len() != y.len() {
+        return Err(ArithErrors::InvalidParameters(
+            "x and y have different length".to_string(),
+        ));
+    }
+    let start = start_timer!(|| "eq_eval");
+    let mut res = F::one();
+    for (&xi, &yi) in x.iter().zip(y.iter()) {
+        let xi_yi = xi * yi;
+        res *= xi_yi + xi_yi - xi - yi + F::one();
+    }
+    end_timer!(start);
+    Ok(res)
+}
+
 /// This function build the eq(x, r) polynomial for any given r.
 ///
 /// Evaluate