chore: add example for recursion on the CLI (#6389)

This PR adds an example script which does recursive verification using
`nargo` and the `bb` binary as based on @jzaki's javascript example
noir-lang/noir#4969.

noir/noir-repo/examples/recursion/.gitignore

@@ -0,0 +1,2 @@
+recurse_leaf/Prover.toml
+recurse_node/Prover.toml

noir/noir-repo/examples/recursion/Nargo.toml

@@ -0,0 +1,2 @@
+[workspace]
+members = ["recurse_leaf", "recurse_node", "sum"]

noir/noir-repo/examples/recursion/generate_recursive_proof.sh

@@ -0,0 +1,61 @@
+#!/bin/bash
+set -eu
+
+BACKEND=${BACKEND:-bb}
+
+nargo execute sum_witness --package sum
+$BACKEND prove -b ./target/sum.json -w ./target/sum_witness.gz -o ./target/sum_proof
+
+# Once we have generated our inner proof, we must use this to generate inputs to `recurse_leaf``
+
+$BACKEND write_vk -b ./target/sum.json -o ./target/sum_key
+$BACKEND vk_as_fields -k ./target/sum_key -o ./target/sum_vk_as_fields
+VK_HASH=$(jq -r '.[0]' ./target/sum_vk_as_fields)
+VK_AS_FIELDS=$(jq -r '.[1:]' ./target/sum_vk_as_fields) 
+
+FULL_PROOF_AS_FIELDS="$($BACKEND proof_as_fields -p ./target/sum_proof -k ./target/sum_key -o -)"
+# sum has 3 public inputs
+PUBLIC_INPUTS=$(echo $FULL_PROOF_AS_FIELDS | jq -r '.[:3]')
+PROOF_AS_FIELDS=$(echo $FULL_PROOF_AS_FIELDS | jq -r '.[3:]')
+
+RECURSE_LEAF_PROVER_TOML=./recurse_leaf/Prover.toml
+echo "num = 2" > $RECURSE_LEAF_PROVER_TOML
+echo "key_hash = \"$VK_HASH\"" >> $RECURSE_LEAF_PROVER_TOML
+echo "verification_key = $VK_AS_FIELDS"  >> $RECURSE_LEAF_PROVER_TOML
+echo "proof = $PROOF_AS_FIELDS" >> $RECURSE_LEAF_PROVER_TOML
+echo "public_inputs = $PUBLIC_INPUTS" >> $RECURSE_LEAF_PROVER_TOML
+
+# We can now execute and prove `recurse_leaf`
+
+nargo execute recurse_leaf_witness --package recurse_leaf
+$BACKEND prove -b ./target/recurse_leaf.json -w ./target/recurse_leaf_witness.gz -o ./target/recurse_leaf_proof
+
+# Let's do a sanity check that the proof we've generated so far is valid.
+$BACKEND write_vk -b ./target/recurse_leaf.json -o ./target/recurse_leaf_key
+$BACKEND verify -p ./target/recurse_leaf_proof -k ./target/recurse_leaf_key
+
+# Now we generate the final `recurse_node` proof similarly to how we did for `recurse_leaf`.
+
+$BACKEND vk_as_fields -k ./target/recurse_leaf_key -o ./target/recurse_leaf_vk_as_fields
+VK_HASH=$(jq -r '.[0]' ./target/recurse_leaf_vk_as_fields)
+VK_AS_FIELDS=$(jq -r '.[1:]' ./target/recurse_leaf_vk_as_fields) 
+
+FULL_PROOF_AS_FIELDS="$($BACKEND proof_as_fields -p ./target/recurse_leaf_proof -k ./target/recurse_leaf_key -o -)"
+# recurse_leaf has 4 public inputs (excluding aggregation object)
+PUBLIC_INPUTS=$(echo $FULL_PROOF_AS_FIELDS | jq -r '.[:4]')
+PROOF_AS_FIELDS=$(echo $FULL_PROOF_AS_FIELDS | jq -r '.[4:]')
+
+RECURSE_NODE_PROVER_TOML=./recurse_node/Prover.toml
+echo "key_hash = \"$VK_HASH\"" > $RECURSE_NODE_PROVER_TOML
+echo "verification_key = $VK_AS_FIELDS"  >> $RECURSE_NODE_PROVER_TOML
+echo "proof = $PROOF_AS_FIELDS" >> $RECURSE_NODE_PROVER_TOML
+echo "public_inputs = $PUBLIC_INPUTS" >> $RECURSE_NODE_PROVER_TOML
+
+# We can now execute and prove `recurse_node`
+
+nargo execute recurse_node_witness --package recurse_node
+$BACKEND prove -b ./target/recurse_node.json -w ./target/recurse_node_witness.gz -o ./target/recurse_node_proof
+
+# We finally verify that the generated recursive proof is valid.
+$BACKEND write_vk -b ./target/recurse_node.json -o ./target/recurse_node_key
+$BACKEND verify -p ./target/recurse_node_proof -k ./target/recurse_node_key

noir/noir-repo/examples/recursion/recurse_leaf/Nargo.toml

@@ -0,0 +1,7 @@
+[package]
+name = "recurse_leaf"
+type = "bin"
+authors = [""]
+compiler_version = ">=0.26.0"
+
+[dependencies]

noir/noir-repo/examples/recursion/recurse_leaf/src/main.nr

@@ -0,0 +1,20 @@
+use dep::std;
+
+#[recursive]
+fn main(
+    verification_key: [Field; 114],
+    public_inputs: pub [Field; 3],
+    key_hash: Field,
+    proof: [Field; 93],
+    num: u64
+) -> pub u64 {
+    // verify sum so far was computed correctly
+    std::verify_proof(
+        verification_key.as_slice(),
+        proof.as_slice(),
+        public_inputs.as_slice(),
+        key_hash
+    );
+    // Take output of previous proof and add another number to it.
+    public_inputs[2] as u64 + num
+}

noir/noir-repo/examples/recursion/recurse_node/Nargo.toml

@@ -0,0 +1,7 @@
+[package]
+name = "recurse_node"
+type = "bin"
+authors = [""]
+compiler_version = ">=0.26.0"
+
+[dependencies]

noir/noir-repo/examples/recursion/recurse_node/src/main.nr

@@ -0,0 +1,17 @@
+use dep::std;
+
+fn main(
+    verification_key: [Field; 114],
+    public_inputs: pub [Field; 4],
+    key_hash: Field,
+    proof: [Field; 109]
+) -> pub u64 {
+    // verify sum was computed correctly
+    std::verify_proof(
+        verification_key.as_slice(),
+        proof.as_slice(),
+        public_inputs.as_slice(),
+        key_hash
+    );
+    public_inputs[3] as u64
+}

noir/noir-repo/examples/recursion/sum/Nargo.toml

@@ -0,0 +1,7 @@
+[package]
+name = "sum"
+type = "bin"
+authors = [""]
+compiler_version = ">=0.26.0"
+
+[dependencies]

noir/noir-repo/examples/recursion/sum/Prover.toml

@@ -0,0 +1,2 @@
+a = 1
+b = 2

noir/noir-repo/examples/recursion/sum/src/main.nr

@@ -0,0 +1,4 @@
+#[recursive]
+fn main(a: pub u64, b: pub u64) -> pub u64 {
+    a + b
+}

noir/noir-repo/examples/recursion/test.sh

@@ -0,0 +1,8 @@
+#!/bin/bash
+set -eu
+
+# This file is used for Noir CI and is not required.
+
+BACKEND=${BACKEND:-bb}
+
+./generate_recursive_proof.sh