feat(avm): merkle tree gadget (#9205)

Resolves #9458

barretenberg/cpp/pil/avm/gadgets/merkle_tree.pil

@@ -0,0 +1,57 @@
+include "./poseidon2_full.pil";
+
+// Handles membership and insertion into a merkle tree
+namespace merkle_tree(256);
+
+    pol commit sel_merkle_tree;
+    sel_merkle_tree * (1 - sel_merkle_tree) = 0;
+    // Gotta stop using clk for things that are more like foreign keys
+    pol commit clk;
+    // Inputs to the gadget
+    pol commit leaf_value;
+    pol commit leaf_index;
+    pol commit path_len;
+    pol commit expected_tree_root;
+    // These are all hinted
+    pol commit sibling_value;
+
+    // If we are not done, the path_len decrements by 1
+    sel_merkle_tree * (1 - latch) * (path_len' - path_len + 1) = 0;
+
+    pol commit latch;
+    latch * (1 - latch) = 0;
+    pol commit path_len_inv;
+    // latch == 1 when the path_len == 0
+    sel_merkle_tree * (path_len * (latch * (1 - path_len_inv) + path_len_inv) - 1 + latch) = 0;
+
+    // TODO: Constrain that this is indeed even.
+    // Note this might be covered by the halving of the leaf index if the original index is constrained.
+    pol commit leaf_index_is_even;
+    leaf_index_is_even * (1 - leaf_index_is_even) = 0;
+    pol LEAF_INDEX_IS_ODD = (1 - leaf_index_is_even);
+    // If we are not done, the next leaf index is half the current leaf index;
+    // We don't need to worry about underflowing the field since (leaf_index - LEAF_INDEX_IS_ODD)
+    // wil be even (over the integers) and as the field is not of characteristic 2, leaf_index' == leaf_index / 2 over the integers
+    sel_merkle_tree * (1 - latch) * (leaf_index' * 2 + LEAF_INDEX_IS_ODD - leaf_index) = 0;
+
+    // These are what are sent to poseidon2
+    // These arrange the leaf_value and sibling_value in the correct order
+    pol commit left_hash;
+    pol commit right_hash;
+    // I dont think these can be safely combined
+    // if the leaf index is even, the leaf value is the left hash and the sibling value is the right hash
+    // vice-versa
+    sel_merkle_tree * leaf_index_is_even * (left_hash - right_hash) + right_hash - leaf_value = 0;
+    sel_merkle_tree * leaf_index_is_even * (right_hash - left_hash) + left_hash - sibling_value = 0;
+    pol commit output_hash;
+
+    // If we are not done, the output hash is the next value in
+    sel_merkle_tree * (1 - latch) * (leaf_value' - output_hash) = 0;
+
+    pol LAST_COMPUTE = sel_merkle_tree * latch;
+    // LAST_COMPUTE will be used in permutations to other traces
+
+    // Permutation to the full poseidon2 gadget
+    #[PERM_MERKLE_POSEIDON2]
+    sel_merkle_tree { clk, left_hash, right_hash, output_hash } is
+    poseidon2_full.sel_merkle_tree {poseidon2_full.clk, poseidon2_full.input_0, poseidon2_full.input_1, poseidon2_full.output };

barretenberg/cpp/pil/avm/gadgets/poseidon2_full.pil

@@ -87,3 +87,5 @@ namespace poseidon2_full(256);
         { poseidon2.clk, poseidon2.a_0, poseidon2.a_1, poseidon2.a_2, poseidon2.a_3,
           poseidon2.b_0, poseidon2.b_1, poseidon2.b_2, poseidon2.b_3 };
 
+    // ======== Merkle Tree Selector ======================
+    pol commit sel_merkle_tree;

barretenberg/cpp/pil/avm/main.pil

@@ -13,6 +13,7 @@ include "gadgets/poseidon2.pil";
 include "gadgets/poseidon2_full.pil";
 include "gadgets/keccakf1600.pil";
 include "gadgets/mem_slice.pil";
+include "gadgets/merkle_tree.pil";
 
 namespace main(256);
     //===== CONSTANT POLYNOMIALS ==================================================

barretenberg/cpp/src/barretenberg/vm/avm/generated/circuit_builder.cpp

@@ -423,6 +423,19 @@ AvmCircuitBuilder::ProverPolynomials AvmCircuitBuilder::compute_polynomials() co
                 polys.mem_tsp.set_if_valid_index(i, rows[i].mem_tsp);
                 polys.mem_val.set_if_valid_index(i, rows[i].mem_val);
                 polys.mem_w_in_tag.set_if_valid_index(i, rows[i].mem_w_in_tag);
+                polys.merkle_tree_clk.set_if_valid_index(i, rows[i].merkle_tree_clk);
+                polys.merkle_tree_expected_tree_root.set_if_valid_index(i, rows[i].merkle_tree_expected_tree_root);
+                polys.merkle_tree_latch.set_if_valid_index(i, rows[i].merkle_tree_latch);
+                polys.merkle_tree_leaf_index.set_if_valid_index(i, rows[i].merkle_tree_leaf_index);
+                polys.merkle_tree_leaf_index_is_even.set_if_valid_index(i, rows[i].merkle_tree_leaf_index_is_even);
+                polys.merkle_tree_leaf_value.set_if_valid_index(i, rows[i].merkle_tree_leaf_value);
+                polys.merkle_tree_left_hash.set_if_valid_index(i, rows[i].merkle_tree_left_hash);
+                polys.merkle_tree_output_hash.set_if_valid_index(i, rows[i].merkle_tree_output_hash);
+                polys.merkle_tree_path_len.set_if_valid_index(i, rows[i].merkle_tree_path_len);
+                polys.merkle_tree_path_len_inv.set_if_valid_index(i, rows[i].merkle_tree_path_len_inv);
+                polys.merkle_tree_right_hash.set_if_valid_index(i, rows[i].merkle_tree_right_hash);
+                polys.merkle_tree_sel_merkle_tree.set_if_valid_index(i, rows[i].merkle_tree_sel_merkle_tree);
+                polys.merkle_tree_sibling_value.set_if_valid_index(i, rows[i].merkle_tree_sibling_value);
                 polys.poseidon2_B_10_0.set_if_valid_index(i, rows[i].poseidon2_B_10_0);
                 polys.poseidon2_B_10_1.set_if_valid_index(i, rows[i].poseidon2_B_10_1);
                 polys.poseidon2_B_10_2.set_if_valid_index(i, rows[i].poseidon2_B_10_2);
@@ -714,6 +727,7 @@ AvmCircuitBuilder::ProverPolynomials AvmCircuitBuilder::compute_polynomials() co
                     i, rows[i].poseidon2_full_num_perm_rounds_rem_inv);
                 polys.poseidon2_full_output.set_if_valid_index(i, rows[i].poseidon2_full_output);
                 polys.poseidon2_full_padding.set_if_valid_index(i, rows[i].poseidon2_full_padding);
+                polys.poseidon2_full_sel_merkle_tree.set_if_valid_index(i, rows[i].poseidon2_full_sel_merkle_tree);
                 polys.poseidon2_full_sel_poseidon.set_if_valid_index(i, rows[i].poseidon2_full_sel_poseidon);
                 polys.poseidon2_full_start_poseidon.set_if_valid_index(i, rows[i].poseidon2_full_start_poseidon);
                 polys.poseidon2_input_addr.set_if_valid_index(i, rows[i].poseidon2_input_addr);