chore: move shared pcs functionality to internal library in solidity …

…and small refactorings in sumcheck (#11230)

* functionality that is shared in PCS between the ZK and non-ZK contract
has been moved to a separate internal library.
* simplified ZK sumcheck and pcs logic

barretenberg/cpp/src/barretenberg/commitment_schemes/gemini/gemini_impl.hpp

@@ -53,8 +53,8 @@ std::vector<typename GeminiProver_<Curve>::Claim> GeminiProver_<Curve>::prove(
     bool has_zk)
 
 {
-    size_t log_n = numeric::get_msb(static_cast<uint32_t>(circuit_size));
-    size_t n = 1 << log_n;
+    const size_t log_n = numeric::get_msb(static_cast<uint32_t>(circuit_size));
+    const size_t n = 1 << log_n;
 
     // Compute batched polynomials
     Polynomial batched_unshifted(n);
@@ -66,10 +66,8 @@ std::vector<typename GeminiProver_<Curve>::Claim> GeminiProver_<Curve>::prove(
         transcript->send_to_verifier("Gemini:masking_poly_comm", commitment_key->commit(batched_unshifted));
         // In the provers, the size of multilinear_challenge is CONST_PROOF_SIZE_LOG_N, but we need to evaluate the
         // hiding polynomial as multilinear in log_n variables
-        std::vector<Fr> multilinear_challenge_resized(multilinear_challenge.begin(), multilinear_challenge.end());
-        multilinear_challenge_resized.resize(log_n);
         transcript->send_to_verifier("Gemini:masking_poly_eval",
-                                     batched_unshifted.evaluate_mle(multilinear_challenge_resized));
+                                     batched_unshifted.evaluate_mle(multilinear_challenge.subspan(0, log_n)));
     }
 
     // Get the batching challenge

barretenberg/cpp/src/barretenberg/commitment_schemes/shplonk/shplemini.hpp

@@ -169,7 +169,7 @@ template <typename Curve> class ShpleminiVerifier_ {
         if (has_zk) {
             hiding_polynomial_commitment =
                 transcript->template receive_from_prover<Commitment>("Gemini:masking_poly_comm");
-            batched_evaluation += transcript->template receive_from_prover<Fr>("Gemini:masking_poly_eval");
+            batched_evaluation = transcript->template receive_from_prover<Fr>("Gemini:masking_poly_eval");
         }
 
         // Get the challenge ρ to batch commitments to multilinear polynomials and their shifts
@@ -399,7 +399,7 @@ template <typename Curve> class ShpleminiVerifier_ {
         Fr current_batching_challenge = Fr(1);
 
         if (has_zk) {
-            // ρ⁰ is used to batch the hiding polynomial
+            // ρ⁰ is used to batch the hiding polynomial which has already been added to the commitments vector
             current_batching_challenge *= multivariate_batching_challenge;
         }
 
@@ -635,12 +635,12 @@ template <typename Curve> class ShpleminiVerifier_ {
         // need to keep track of the contribution to the constant term
         Fr& constant_term = scalars.back();
 
-        // add Libra commitments to the vector of commitments; compute corresponding scalars and the correction to
-        // the constant term
+        // add Libra commitments to the vector of commitments
         for (size_t idx = 0; idx < libra_commitments.size(); idx++) {
             commitments.push_back(libra_commitments[idx]);
         }
 
+        // compute corresponding scalars and the correction to the constant term
         std::array<Fr, NUM_LIBRA_EVALUATIONS> denominators;
         std::array<Fr, NUM_LIBRA_EVALUATIONS> batching_scalars;
         // compute Shplonk denominators and invert them