merkletree: reduce gate amount (-23%) by custom poseidon to use flag as initial state (#472)

* merkletree: custom poseidon to use flag as initial state.

This allows to do the merkletree related hashing in 1 gate instead of 2,
reducing ~23% of gates per merkle proof.

| tree levels   | 10 | 16 | 32  | 40  | 64  | 128 | 130 | 250  | 256  |
|---------------|----|----|-----|-----|-----|-----|-----|------|------|
| old num gates | 50 | 76 | 144 | 178 | 280 | 554 | 564 | 1076 | 1102 |
| new num gates | 39 | 59 | 111 | 137 | 215 | 425 | 433 | 825  | 845  |

* update docs with new tree hashing approach

* add inline comment stating clear how the flag is used in the state permutation

book/src/merkletree.md

@@ -37,10 +37,12 @@ A Merkle tree with no entry at all is represented by the hash value
 (With the Plonky2 backend, the hash function ```hash``` will output a 4-tuple of field elements.)
 
 A Merkle tree with a single entry ```(key, value)``` is called a "leaf".  It is represented by the hash value
-```root = hash((key, value, 1)).```
+```root = hash(1, (key, value))```, where `1` is a flag indicating that it is a leaf, and it's used as the initial state of the hash (Poseidon) permutation.
 
 A Merkle tree ```tree``` with more than one entry is required to have two subtrees, ```left``` and ```right```.  It is then represented by the hash value
-```root = hash((left_root, right_root, 2)).```
+```root = hash(2, (left_root, right_root))```, where `2` is a flag indicating that it is an intermediate node, and it's used as the initial state of the hash (Poseidon) permutation.
+
+The flags are used as the initial state of the Poseidon permutation so that they don't account for extra inputs in the Poseidon gadget, needing only 1 gate for each node/leaf hash.
 
 (The role of the constants 1 and 2 is to prevent collisions between leaves and non-leaf Merkle roots.  If the constants were omitted, a large Merkle tree could be dishonestly interpreted as a leaf, leading to security vulnerabilities.)
 

src/backends/plonky2/primitives/merkletree/circuit.rs

@@ -14,14 +14,15 @@ use itertools::zip_eq;
 use plonky2::{
     field::types::Field,
     hash::{
-        hash_types::{HashOut, HashOutTarget},
+        hash_types::{HashOut, HashOutTarget, NUM_HASH_OUT_ELTS},
+        hashing::PlonkyPermutation,
         poseidon::PoseidonHash,
     },
     iop::{
         target::{BoolTarget, Target},
         witness::{PartialWitness, WitnessWrite},
     },
-    plonk::circuit_builder::CircuitBuilder,
+    plonk::{circuit_builder::CircuitBuilder, config::AlgebraicHasher},
 };
 use serde::{Deserialize, Serialize};
 
@@ -349,7 +350,7 @@ fn compute_root_from_leaf(
             .map(|j| builder.select(path[i], h.elements[j], sibling.elements[j]))
             .collect();
         let new_h =
-            builder.hash_n_to_hash_no_pad::<PoseidonHash>([input_1, input_2, vec![two]].concat());
+            hash_with_flag_target::<PoseidonHash>(builder, two, [input_1, input_2].concat());
 
         let h_targ: Vec<Target> = (0..HASH_SIZE)
             .map(|j| builder.select(*selector, new_h.elements[j], h.elements[j]))
@@ -401,9 +402,51 @@ fn kv_hash_target(
         .iter()
         .chain(value.elements.iter())
         .cloned()
-        .chain(iter::once(builder.one()))
         .collect();
-    builder.hash_n_to_hash_no_pad::<PoseidonHash>(inputs)
+    let one = builder.one();
+    hash_with_flag_target::<PoseidonHash>(builder, one, inputs)
+}
+
+/// Circuit matching the `merkletree::hash_with_flag` function. Variation of
+/// Poseidon hash which takes as input 1 Goldilock element as a flag, and 8
+/// Goldilocks elements as inputs to the hash. Performs the hashing in a single
+/// gate.
+/// The function is a fork of
+/// [hash_n_to_m_no_pad](https://github.com/0xPolygonZero/plonky2/tree/5d9da5a65bbcba2c66eb29c035090eb2e9ccb05f/plonky2/src/hash/hashing.rs#L118)
+/// from plonky2.
+fn hash_with_flag_target<H: AlgebraicHasher<F>>(
+    builder: &mut CircuitBuilder<F, D>,
+    flag: Target,
+    inputs: Vec<Target>,
+) -> HashOutTarget {
+    assert_eq!(inputs.len(), H::AlgebraicPermutation::RATE);
+
+    // here we set `state` to a  `SPONGE_RATE+SPONGE_CAPACITY` (8+4=12) in our
+    // case to a vector of repeated `flag` value. Later at the absorption step,
+    // it will fit the inputs values at positions 0-8, keeping the flag values
+    // at positions 8-12.
+    let mut state = H::AlgebraicPermutation::new(core::iter::repeat(flag));
+
+    // Absorb all input chunks.
+    for input_chunk in inputs.chunks(H::AlgebraicPermutation::RATE) {
+        // Overwrite the first r elements with the inputs. This differs from a standard sponge,
+        // where we would xor or add in the inputs. This is a well-known variant, though,
+        // sometimes called "overwrite mode".
+        state.set_from_slice(input_chunk, 0);
+        state = builder.permute::<H>(state);
+    }
+
+    // Squeeze until we have the desired number of outputs.
+    let mut outputs = Vec::with_capacity(NUM_HASH_OUT_ELTS);
+    loop {
+        for &s in state.squeeze() {
+            outputs.push(s);
+            if outputs.len() == NUM_HASH_OUT_ELTS {
+                return HashOutTarget::from_vec(outputs);
+            }
+        }
+        state = builder.permute::<H>(state);
+    }
 }
 
 /// Verifies that the merkletree state transition (from old_root to new_root)

src/backends/plonky2/primitives/merkletree/mod.rs

@@ -3,10 +3,15 @@
 use std::{collections::HashMap, fmt, iter::IntoIterator};
 
 use itertools::zip_eq;
-use plonky2::field::types::Field;
+use plonky2::{
+    field::types::Field,
+    hash::{
+        hash_types::NUM_HASH_OUT_ELTS, hashing::PlonkyPermutation, poseidon::PoseidonPermutation,
+    },
+};
 use serde::{Deserialize, Serialize};
 
-use crate::middleware::{hash_fields, Hash, RawValue, EMPTY_HASH, EMPTY_VALUE, F};
+use crate::middleware::{Hash, RawValue, EMPTY_HASH, EMPTY_VALUE, F};
 
 pub mod circuit;
 pub use circuit::*;
@@ -376,10 +381,47 @@ impl MerkleTree {
 /// mitigate fake proofs.
 pub fn kv_hash(key: &RawValue, value: Option<RawValue>) -> Hash {
     value
-        .map(|v| hash_fields(&[key.0.to_vec(), v.0.to_vec(), vec![F::ONE]].concat()))
+        .map(|v| hash_with_flag(F::ONE, &[key.0.to_vec(), v.0.to_vec()].concat()))
         .unwrap_or(EMPTY_HASH)
 }
 
+/// Variation of Poseidon hash which takes as input 1 Goldilock element as a
+/// flag, and 8 Goldilocks elements as inputs to the hash. Performs the hashing
+/// in a single gate.
+/// The function is a fork of
+/// [hash_n_to_m_no_pad](https://github.com/0xPolygonZero/plonky2/tree/5d9da5a65bbcba2c66eb29c035090eb2e9ccb05f/plonky2/src/hash/hashing.rs#L30)
+/// from plonky2.
+fn hash_with_flag(flag: F, inputs: &[F]) -> Hash {
+    assert_eq!(
+        inputs.len(),
+        <PoseidonPermutation<F> as PlonkyPermutation<F>>::RATE
+    );
+
+    // this will set `perm` to a  `SPONGE_RATE+SPONGE_CAPACITY` (8+4=12) in our
+    // case to a vector of repeated `flag` value. Later at the absorption step,
+    // it will fit the inputs values at positions 0-8, keeping the flag values
+    // at positions 8-12.
+    let mut perm = <PoseidonPermutation<F> as PlonkyPermutation<F>>::new(core::iter::repeat(flag));
+
+    // Absorb all input chunks.
+    for input_chunk in inputs.chunks(<PoseidonPermutation<F> as PlonkyPermutation<F>>::RATE) {
+        perm.set_from_slice(input_chunk, 0);
+        perm.permute();
+    }
+
+    // Squeeze until we have the desired number of outputs.
+    let mut outputs = Vec::new();
+    loop {
+        for &item in perm.squeeze() {
+            outputs.push(item);
+            if outputs.len() == NUM_HASH_OUT_ELTS {
+                return Hash(crate::middleware::HashOut::from_vec(outputs).elements);
+            }
+        }
+        perm.permute();
+    }
+}
+
 impl<'a> IntoIterator for &'a MerkleTree {
     type Item = (&'a RawValue, &'a RawValue);
     type IntoIter = Iter<'a>;
@@ -437,13 +479,12 @@ impl MerkleProof {
     fn compute_root_from_node(&self, node_hash: &Hash, path: Vec<bool>) -> TreeResult<Hash> {
         let mut h = *node_hash;
         for (i, sibling) in self.siblings.iter().enumerate().rev() {
-            let mut input: Vec<F> = if path[i] {
+            let input: Vec<F> = if path[i] {
                 [sibling.0, h.0].concat()
             } else {
                 [h.0, sibling.0].concat()
             };
-            input.push(F::TWO);
+            h = hash_with_flag(F::TWO, &input);
-            h = hash_fields(&input);
         }
         Ok(h)
     }
@@ -876,8 +917,8 @@ impl Intermediate {
         }
         let l_hash = self.left.compute_hash();
         let r_hash = self.right.compute_hash();
-        let input: Vec<F> = [l_hash.0.to_vec(), r_hash.0.to_vec(), vec![F::TWO]].concat();
+        let input: Vec<F> = [l_hash.0.to_vec(), r_hash.0.to_vec()].concat();
-        let h = hash_fields(&input);
+        let h = hash_with_flag(F::TWO, &input);
         self.hash = Some(h);
         h
     }