Introduce SmtProof (#270)

* add conversion for `SmtLeaf`

* introduce `SmtProof` scaffolding

* implement `verify_membership()`

* SmtLeaf: knows its index

* `SmtLeaf::index`

* `SmtLeaf::get_value()` returns an Option

* fix `verify_membership()`

* impl `SmtProof::get`

* impl `into_parts()`

* `SmtProof::compute_root`

* use `SmtProof` in `Smt::open`

* `SmtLeaf` constructors

* Vec

* impl `Error` for `SmtLeafError`

* fix std Error

* move Word/Digest conversions to LeafIndex

* `SmtProof::new()` returns an error

* `SparseMerkleTree::path_and_leaf_to_opening`

* `SmtLeaf`: serializable/deserializable

* `SmtProof`: serializable/deserializable

* add tests for SmtLeaf serialization

* move `SmtLeaf` to submodule

* use constructors internally

* fix docs

* Add `Vec`

* add `Vec` to tests

* no_std use statements

* fmt

* `Errors`: make heading

* use `SMT_DEPTH`

* SmtLeaf single case: check leaf index

* Multiple case: check consistency with leaf index

* use `pub(super)` instead of `pub(crate)`

* use `pub(super)`

* `SmtLeaf`: add `num_entries()` accessor

* Fix `SmtLeaf` serialization

* improve leaf serialization tests

src/merkle/error.rs

@@ -4,6 +4,8 @@ use crate::{
 };
 use core::fmt;
 
+use super::smt::SmtLeafError;
+
 #[derive(Clone, Debug, PartialEq, Eq)]
 pub enum MerkleError {
     ConflictingRoots(Vec<RpoDigest>),
@@ -20,6 +22,7 @@ pub enum MerkleError {
     NodeNotInStore(RpoDigest, NodeIndex),
     NumLeavesNotPowerOfTwo(usize),
     RootNotInStore(RpoDigest),
+    SmtLeaf(SmtLeafError),
 }
 
 impl fmt::Display for MerkleError {
@@ -50,9 +53,16 @@ impl fmt::Display for MerkleError {
                 write!(f, "the leaves count {leaves} is not a power of 2")
             }
             RootNotInStore(root) => write!(f, "the root {:?} is not in the store", root),
+            SmtLeaf(smt_leaf_error) => write!(f, "smt leaf error: {smt_leaf_error}"),
         }
     }
 }
 
 #[cfg(feature = "std")]
 impl std::error::Error for MerkleError {}
+
+impl From<SmtLeafError> for MerkleError {
+    fn from(value: SmtLeafError) -> Self {
+        Self::SmtLeaf(value)
+    }
+}

src/merkle/smt/full/error.rs

@@ -0,0 +1,86 @@
+use core::fmt;
+
+use crate::{
+    hash::rpo::RpoDigest,
+    merkle::{LeafIndex, SMT_DEPTH},
+    utils::collections::Vec,
+    Word,
+};
+
+// SMT LEAF ERROR
+// =================================================================================================
+
+#[derive(Clone, Debug, PartialEq, Eq)]
+pub enum SmtLeafError {
+    InconsistentKeys {
+        entries: Vec<(RpoDigest, Word)>,
+        key_1: RpoDigest,
+        key_2: RpoDigest,
+    },
+    InvalidNumEntriesForMultiple(usize),
+    SingleKeyInconsistentWithLeafIndex {
+        key: RpoDigest,
+        leaf_index: LeafIndex<SMT_DEPTH>,
+    },
+    MultipleKeysInconsistentWithLeafIndex {
+        leaf_index_from_keys: LeafIndex<SMT_DEPTH>,
+        leaf_index_supplied: LeafIndex<SMT_DEPTH>,
+    },
+}
+
+#[cfg(feature = "std")]
+impl std::error::Error for SmtLeafError {}
+
+impl fmt::Display for SmtLeafError {
+    fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result {
+        use SmtLeafError::*;
+        match self {
+            InvalidNumEntriesForMultiple(num_entries) => {
+                write!(f, "Multiple leaf requires 2 or more entries. Got: {num_entries}")
+            }
+            InconsistentKeys { entries, key_1, key_2 } => {
+                write!(f, "Multiple leaf requires all keys to map to the same leaf index. Offending keys: {key_1} and {key_2}. Entries: {entries:?}.")
+            }
+            SingleKeyInconsistentWithLeafIndex { key, leaf_index } => {
+                write!(
+                    f,
+                    "Single key in leaf inconsistent with leaf index. Key: {key}, leaf index: {}",
+                    leaf_index.value()
+                )
+            }
+            MultipleKeysInconsistentWithLeafIndex {
+                leaf_index_from_keys,
+                leaf_index_supplied,
+            } => {
+                write!(
+                    f,
+                    "Keys in entries map to leaf index {}, but leaf index {} was supplied",
+                    leaf_index_from_keys.value(),
+                    leaf_index_supplied.value()
+                )
+            }
+        }
+    }
+}
+
+// SMT PROOF ERROR
+// =================================================================================================
+
+#[derive(Clone, Debug, PartialEq, Eq)]
+pub enum SmtProofError {
+    InvalidPathLength(usize),
+}
+
+#[cfg(feature = "std")]
+impl std::error::Error for SmtProofError {}
+
+impl fmt::Display for SmtProofError {
+    fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result {
+        use SmtProofError::*;
+        match self {
+            InvalidPathLength(path_length) => {
+                write!(f, "Invalid Merkle path length. Expected {SMT_DEPTH}, got {path_length}")
+            }
+        }
+    }
+}

src/merkle/smt/full/leaf.rs

@@ -0,0 +1,373 @@
+use core::cmp::Ordering;
+
+use crate::utils::{collections::Vec, string::ToString, vec};
+use winter_math::StarkField;
+use winter_utils::{ByteReader, ByteWriter, Deserializable, DeserializationError, Serializable};
+
+use super::{Felt, LeafIndex, Rpo256, RpoDigest, SmtLeafError, Word, EMPTY_WORD, SMT_DEPTH};
+
+#[derive(Clone, Debug, PartialEq, Eq)]
+#[cfg_attr(feature = "serde", derive(serde::Deserialize, serde::Serialize))]
+pub enum SmtLeaf {
+    Empty(LeafIndex<SMT_DEPTH>),
+    Single((RpoDigest, Word)),
+    Multiple(Vec<(RpoDigest, Word)>),
+}
+
+impl SmtLeaf {
+    // CONSTRUCTORS
+    // ---------------------------------------------------------------------------------------------
+
+    /// Returns a new leaf with the specified entries
+    ///
+    /// # Errors
+    ///   - Returns an error if 2 keys in `entries` map to a different leaf index
+    ///   - Returns an error if 1 or more keys in `entries` map to a leaf index
+    ///     different from `leaf_index`
+    pub fn new(
+        entries: Vec<(RpoDigest, Word)>,
+        leaf_index: LeafIndex<SMT_DEPTH>,
+    ) -> Result<Self, SmtLeafError> {
+        match entries.len() {
+            0 => Ok(Self::new_empty(leaf_index)),
+            1 => {
+                let (key, value) = entries[0];
+
+                if LeafIndex::<SMT_DEPTH>::from(key) != leaf_index {
+                    return Err(SmtLeafError::SingleKeyInconsistentWithLeafIndex {
+                        key,
+                        leaf_index,
+                    });
+                }
+
+                Ok(Self::new_single(key, value))
+            }
+            _ => {
+                let leaf = Self::new_multiple(entries)?;
+
+                // `new_multiple()` checked that all keys map to the same leaf index. We still need
+                // to ensure that that leaf index is `leaf_index`.
+                if leaf.index() != leaf_index {
+                    Err(SmtLeafError::MultipleKeysInconsistentWithLeafIndex {
+                        leaf_index_from_keys: leaf.index(),
+                        leaf_index_supplied: leaf_index,
+                    })
+                } else {
+                    Ok(leaf)
+                }
+            }
+        }
+    }
+
+    /// Returns a new empty leaf with the specified leaf index
+    pub fn new_empty(leaf_index: LeafIndex<SMT_DEPTH>) -> Self {
+        Self::Empty(leaf_index)
+    }
+
+    /// Returns a new single leaf with the specified entry. The leaf index is derived from the
+    /// entry's key.
+    pub fn new_single(key: RpoDigest, value: Word) -> Self {
+        Self::Single((key, value))
+    }
+
+    /// Returns a new single leaf with the specified entry. The leaf index is derived from the
+    /// entries' keys.
+    ///
+    /// # Errors
+    ///   - Returns an error if 2 keys in `entries` map to a different leaf index
+    pub fn new_multiple(entries: Vec<(RpoDigest, Word)>) -> Result<Self, SmtLeafError> {
+        if entries.len() < 2 {
+            return Err(SmtLeafError::InvalidNumEntriesForMultiple(entries.len()));
+        }
+
+        // Check that all keys map to the same leaf index
+        {
+            let mut keys = entries.iter().map(|(key, _)| key);
+
+            let first_key = *keys.next().expect("ensured at least 2 entries");
+            let first_leaf_index: LeafIndex<SMT_DEPTH> = first_key.into();
+
+            for &next_key in keys {
+                let next_leaf_index: LeafIndex<SMT_DEPTH> = next_key.into();
+
+                if next_leaf_index != first_leaf_index {
+                    return Err(SmtLeafError::InconsistentKeys {
+                        entries,
+                        key_1: first_key,
+                        key_2: next_key,
+                    });
+                }
+            }
+        }
+
+        Ok(Self::Multiple(entries))
+    }
+
+    // PUBLIC ACCESSORS
+    // ---------------------------------------------------------------------------------------------
+
+    /// Returns true if the leaf is empty
+    pub fn is_empty(&self) -> bool {
+        matches!(self, Self::Empty(_))
+    }
+
+    /// Returns the leaf's index in the [`super::Smt`]
+    pub fn index(&self) -> LeafIndex<SMT_DEPTH> {
+        match self {
+            SmtLeaf::Empty(leaf_index) => *leaf_index,
+            SmtLeaf::Single((key, _)) => key.into(),
+            SmtLeaf::Multiple(entries) => {
+                // Note: All keys are guaranteed to have the same leaf index
+                let (first_key, _) = entries[0];
+                first_key.into()
+            }
+        }
+    }
+
+    /// Returns the number of entries stored in the leaf
+    pub fn num_entries(&self) -> u64 {
+        match self {
+            SmtLeaf::Empty(_) => 0,
+            SmtLeaf::Single(_) => 1,
+            SmtLeaf::Multiple(entries) => {
+                entries.len().try_into().expect("shouldn't have more than 2^64 entries")
+            }
+        }
+    }
+
+    /// Computes the hash of the leaf
+    pub fn hash(&self) -> RpoDigest {
+        match self {
+            SmtLeaf::Empty(_) => EMPTY_WORD.into(),
+            SmtLeaf::Single((key, value)) => Rpo256::merge(&[*key, value.into()]),
+            SmtLeaf::Multiple(kvs) => {
+                let elements: Vec<Felt> = kvs.iter().copied().flat_map(kv_to_elements).collect();
+                Rpo256::hash_elements(&elements)
+            }
+        }
+    }
+
+    // ITERATORS
+    // ---------------------------------------------------------------------------------------------
+
+    /// Returns the key-value pairs in the leaf
+    pub fn entries(&self) -> Vec<&(RpoDigest, Word)> {
+        match self {
+            SmtLeaf::Empty(_) => Vec::new(),
+            SmtLeaf::Single(kv_pair) => vec![kv_pair],
+            SmtLeaf::Multiple(kv_pairs) => kv_pairs.iter().collect(),
+        }
+    }
+
+    // CONVERSIONS
+    // ---------------------------------------------------------------------------------------------
+
+    /// Converts a leaf to a list of field elements
+    pub fn to_elements(&self) -> Vec<Felt> {
+        self.clone().into_elements()
+    }
+
+    /// Converts a leaf to a list of field elements
+    pub fn into_elements(self) -> Vec<Felt> {
+        self.into_entries().into_iter().flat_map(kv_to_elements).collect()
+    }
+
+    /// Converts a leaf the key-value pairs in the leaf
+    pub fn into_entries(self) -> Vec<(RpoDigest, Word)> {
+        match self {
+            SmtLeaf::Empty(_) => Vec::new(),
+            SmtLeaf::Single(kv_pair) => vec![kv_pair],
+            SmtLeaf::Multiple(kv_pairs) => kv_pairs,
+        }
+    }
+
+    // HELPERS
+    // ---------------------------------------------------------------------------------------------
+
+    /// Returns the value associated with `key` in the leaf, or `None` if `key` maps to another leaf.
+    pub(super) fn get_value(&self, key: &RpoDigest) -> Option<Word> {
+        // Ensure that `key` maps to this leaf
+        if self.index() != key.into() {
+            return None;
+        }
+
+        match self {
+            SmtLeaf::Empty(_) => Some(EMPTY_WORD),
+            SmtLeaf::Single((key_in_leaf, value_in_leaf)) => {
+                if key == key_in_leaf {
+                    Some(*value_in_leaf)
+                } else {
+                    Some(EMPTY_WORD)
+                }
+            }
+            SmtLeaf::Multiple(kv_pairs) => {
+                for (key_in_leaf, value_in_leaf) in kv_pairs {
+                    if key == key_in_leaf {
+                        return Some(*value_in_leaf);
+                    }
+                }
+
+                Some(EMPTY_WORD)
+            }
+        }
+    }
+
+    /// Inserts key-value pair into the leaf; returns the previous value associated with `key`, if
+    /// any.
+    ///
+    /// The caller needs to ensure that `key` has the same leaf index as all other keys in the leaf
+    pub(super) fn insert(&mut self, key: RpoDigest, value: Word) -> Option<Word> {
+        match self {
+            SmtLeaf::Empty(_) => {
+                *self = SmtLeaf::new_single(key, value);
+                None
+            }
+            SmtLeaf::Single(kv_pair) => {
+                if kv_pair.0 == key {
+                    // the key is already in this leaf. Update the value and return the previous
+                    // value
+                    let old_value = kv_pair.1;
+                    kv_pair.1 = value;
+                    Some(old_value)
+                } else {
+                    // Another entry is present in this leaf. Transform the entry into a list
+                    // entry, and make sure the key-value pairs are sorted by key
+                    let mut pairs = vec![*kv_pair, (key, value)];
+                    pairs.sort_by(|(key_1, _), (key_2, _)| cmp_keys(*key_1, *key_2));
+
+                    *self = SmtLeaf::Multiple(pairs);
+
+                    None
+                }
+            }
+            SmtLeaf::Multiple(kv_pairs) => {
+                match kv_pairs.binary_search_by(|kv_pair| cmp_keys(kv_pair.0, key)) {
+                    Ok(pos) => {
+                        let old_value = kv_pairs[pos].1;
+                        kv_pairs[pos].1 = value;
+
+                        Some(old_value)
+                    }
+                    Err(pos) => {
+                        kv_pairs.insert(pos, (key, value));
+
+                        None
+                    }
+                }
+            }
+        }
+    }
+
+    /// Removes key-value pair from the leaf stored at key; returns the previous value associated
+    /// with `key`, if any. Also returns an `is_empty` flag, indicating whether the leaf became
+    /// empty, and must be removed from the data structure it is contained in.
+    pub(super) fn remove(&mut self, key: RpoDigest) -> (Option<Word>, bool) {
+        match self {
+            SmtLeaf::Empty(_) => (None, false),
+            SmtLeaf::Single((key_at_leaf, value_at_leaf)) => {
+                if *key_at_leaf == key {
+                    // our key was indeed stored in the leaf, so we return the value that was stored
+                    // in it, and indicate that the leaf should be removed
+                    let old_value = *value_at_leaf;
+
+                    // Note: this is not strictly needed, since the caller is expected to drop this
+                    // `SmtLeaf` object.
+                    *self = SmtLeaf::new_empty(key.into());
+
+                    (Some(old_value), true)
+                } else {
+                    // another key is stored at leaf; nothing to update
+                    (None, false)
+                }
+            }
+            SmtLeaf::Multiple(kv_pairs) => {
+                match kv_pairs.binary_search_by(|kv_pair| cmp_keys(kv_pair.0, key)) {
+                    Ok(pos) => {
+                        let old_value = kv_pairs[pos].1;
+
+                        kv_pairs.remove(pos);
+                        debug_assert!(!kv_pairs.is_empty());
+
+                        if kv_pairs.len() == 1 {
+                            // convert the leaf into `Single`
+                            *self = SmtLeaf::Single(kv_pairs[0]);
+                        }
+
+                        (Some(old_value), false)
+                    }
+                    Err(_) => {
+                        // other keys are stored at leaf; nothing to update
+                        (None, false)
+                    }
+                }
+            }
+        }
+    }
+}
+
+impl Serializable for SmtLeaf {
+    fn write_into<W: ByteWriter>(&self, target: &mut W) {
+        // Write: num entries
+        self.num_entries().write_into(target);
+
+        // Write: leaf index
+        let leaf_index: u64 = self.index().value();
+        leaf_index.write_into(target);
+
+        // Write: entries
+        for (key, value) in self.entries() {
+            key.write_into(target);
+            value.write_into(target);
+        }
+    }
+}
+
+impl Deserializable for SmtLeaf {
+    fn read_from<R: ByteReader>(source: &mut R) -> Result<Self, DeserializationError> {
+        // Read: num entries
+        let num_entries = source.read_u64()?;
+
+        // Read: leaf index
+        let leaf_index: LeafIndex<SMT_DEPTH> = {
+            let value = source.read_u64()?;
+            LeafIndex::new_max_depth(value)
+        };
+
+        // Read: entries
+        let mut entries: Vec<(RpoDigest, Word)> = Vec::new();
+        for _ in 0..num_entries {
+            let key: RpoDigest = source.read()?;
+            let value: Word = source.read()?;
+
+            entries.push((key, value));
+        }
+
+        Self::new(entries, leaf_index)
+            .map_err(|err| DeserializationError::InvalidValue(err.to_string()))
+    }
+}
+
+// HELPER FUNCTIONS
+// ================================================================================================
+
+/// Converts a key-value tuple to an iterator of `Felt`s
+fn kv_to_elements((key, value): (RpoDigest, Word)) -> impl Iterator<Item = Felt> {
+    let key_elements = key.into_iter();
+    let value_elements = value.into_iter();
+
+    key_elements.chain(value_elements)
+}
+
+/// Compares two keys, compared element-by-element using their integer representations starting with
+/// the most significant element.
+fn cmp_keys(key_1: RpoDigest, key_2: RpoDigest) -> Ordering {
+    for (v1, v2) in key_1.iter().zip(key_2.iter()).rev() {
+        let v1 = v1.as_int();
+        let v2 = v2.as_int();
+        if v1 != v2 {
+            return v1.cmp(&v2);
+        }
+    }
+
+    Ordering::Equal
+}

src/merkle/smt/full/mod.rs

@@ -1,19 +1,23 @@
-use core::cmp::Ordering;
-
 use winter_math::StarkField;
 
 use crate::hash::rpo::Rpo256;
 use crate::merkle::{EmptySubtreeRoots, InnerNodeInfo};
-use crate::utils::{
-    collections::{BTreeMap, BTreeSet, Vec},
-    vec,
-};
+use crate::utils::collections::{BTreeMap, BTreeSet};
 use crate::{Felt, EMPTY_WORD};
 
 use super::{
     InnerNode, LeafIndex, MerkleError, MerklePath, NodeIndex, RpoDigest, SparseMerkleTree, Word,
 };
 
+mod error;
+pub use error::{SmtLeafError, SmtProofError};
+
+mod leaf;
+pub use leaf::SmtLeaf;
+
+mod proof;
+pub use proof::SmtProof;
+
 #[cfg(test)]
 mod tests;
 
@@ -119,14 +123,14 @@ impl Smt {
         let leaf_pos = LeafIndex::<SMT_DEPTH>::from(*key).value();
 
         match self.leaves.get(&leaf_pos) {
-            Some(leaf) => leaf.get_value(key),
+            Some(leaf) => leaf.get_value(key).unwrap_or_default(),
             None => EMPTY_WORD,
         }
     }
 
     /// Returns an opening of the leaf associated with `key`. Conceptually, an opening is a Merkle
     /// path to the leaf, as well as the leaf itself.
-    pub fn open(&self, key: &RpoDigest) -> (MerklePath, SmtLeaf) {
+    pub fn open(&self, key: &RpoDigest) -> SmtProof {
         <Self as SparseMerkleTree<SMT_DEPTH>>::open(self, key)
     }
 
@@ -208,7 +212,7 @@ impl SparseMerkleTree<SMT_DEPTH> for Smt {
     type Key = RpoDigest;
     type Value = Word;
     type Leaf = SmtLeaf;
-    type Opening = (MerklePath, SmtLeaf);
+    type Opening = SmtProof;
 
     const EMPTY_VALUE: Self::Value = EMPTY_WORD;
 
@@ -250,7 +254,7 @@ impl SparseMerkleTree<SMT_DEPTH> for Smt {
 
         match self.leaves.get(&leaf_pos) {
             Some(leaf) => leaf.clone(),
-            None => SmtLeaf::Empty,
+            None => SmtLeaf::new_empty(key.into()),
         }
     }
 
@@ -262,6 +266,10 @@ impl SparseMerkleTree<SMT_DEPTH> for Smt {
         let most_significant_felt = key[3];
         LeafIndex::new_max_depth(most_significant_felt.as_int())
     }
+
+    fn path_and_leaf_to_opening(path: MerklePath, leaf: SmtLeaf) -> SmtProof {
+        SmtProof::new_unchecked(path, leaf)
+    }
 }
 
 impl Default for Smt {
@@ -270,196 +278,24 @@ impl Default for Smt {
     }
 }
 
-// LEAF
+// CONVERSIONS
 // ================================================================================================
 
-#[derive(Clone, Debug, PartialEq, Eq)]
-#[cfg_attr(feature = "serde", derive(serde::Deserialize, serde::Serialize))]
-pub enum SmtLeaf {
-    Empty,
-    Single((RpoDigest, Word)),
-    Multiple(Vec<(RpoDigest, Word)>),
-}
-
-impl SmtLeaf {
-    /// Converts a leaf to a list of field elements
-    pub fn to_elements(&self) -> Vec<Felt> {
-        self.clone().into_elements()
-    }
-
-    /// Converts a leaf to a list of field elements
-    pub fn into_elements(self) -> Vec<Felt> {
-        self.into_entries().into_iter().flat_map(kv_to_elements).collect()
-    }
-
-    /// Returns the key-value pairs in the leaf
-    pub fn entries(&self) -> Vec<&(RpoDigest, Word)> {
-        match self {
-            SmtLeaf::Empty => Vec::new(),
-            SmtLeaf::Single(kv_pair) => vec![kv_pair],
-            SmtLeaf::Multiple(kv_pairs) => kv_pairs.iter().collect(),
+impl From<Word> for LeafIndex<SMT_DEPTH> {
+    fn from(value: Word) -> Self {
+        // We use the most significant `Felt` of a `Word` as the leaf index.
+        Self::new_max_depth(value[3].as_int())
     }
 }
 
-    /// Converts a leaf the key-value pairs in the leaf
-    pub fn into_entries(self) -> Vec<(RpoDigest, Word)> {
-        match self {
-            SmtLeaf::Empty => Vec::new(),
-            SmtLeaf::Single(kv_pair) => vec![kv_pair],
-            SmtLeaf::Multiple(kv_pairs) => kv_pairs,
+impl From<RpoDigest> for LeafIndex<SMT_DEPTH> {
+    fn from(value: RpoDigest) -> Self {
+        Word::from(value).into()
     }
 }
 
-    /// Computes the hash of the leaf
-    pub fn hash(&self) -> RpoDigest {
-        match self {
-            SmtLeaf::Empty => EMPTY_WORD.into(),
-            SmtLeaf::Single((key, value)) => Rpo256::merge(&[*key, value.into()]),
-            SmtLeaf::Multiple(kvs) => {
-                let elements: Vec<Felt> = kvs.iter().copied().flat_map(kv_to_elements).collect();
-                Rpo256::hash_elements(&elements)
+impl From<&RpoDigest> for LeafIndex<SMT_DEPTH> {
+    fn from(value: &RpoDigest) -> Self {
+        Word::from(value).into()
     }
 }
-    }
-
-    // HELPERS
-    // ---------------------------------------------------------------------------------------------
-
-    /// Returns the value associated with `key` in the leaf
-    fn get_value(&self, key: &RpoDigest) -> Word {
-        match self {
-            SmtLeaf::Empty => EMPTY_WORD,
-            SmtLeaf::Single((key_in_leaf, value_in_leaf)) => {
-                if key == key_in_leaf {
-                    *value_in_leaf
-                } else {
-                    EMPTY_WORD
-                }
-            }
-            SmtLeaf::Multiple(kv_pairs) => {
-                for (key_in_leaf, value_in_leaf) in kv_pairs {
-                    if key == key_in_leaf {
-                        return *value_in_leaf;
-                    }
-                }
-
-                EMPTY_WORD
-            }
-        }
-    }
-
-    /// Inserts key-value pair into the leaf; returns the previous value associated with `key`, if
-    /// any.
-    fn insert(&mut self, key: RpoDigest, value: Word) -> Option<Word> {
-        match self {
-            SmtLeaf::Empty => {
-                *self = SmtLeaf::Single((key, value));
-                None
-            }
-            SmtLeaf::Single(kv_pair) => {
-                if kv_pair.0 == key {
-                    // the key is already in this leaf. Update the value and return the previous
-                    // value
-                    let old_value = kv_pair.1;
-                    kv_pair.1 = value;
-                    Some(old_value)
-                } else {
-                    // Another entry is present in this leaf. Transform the entry into a list
-                    // entry, and make sure the key-value pairs are sorted by key
-                    let mut pairs = vec![*kv_pair, (key, value)];
-                    pairs.sort_by(|(key_1, _), (key_2, _)| cmp_keys(*key_1, *key_2));
-
-                    *self = SmtLeaf::Multiple(pairs);
-
-                    None
-                }
-            }
-            SmtLeaf::Multiple(kv_pairs) => {
-                match kv_pairs.binary_search_by(|kv_pair| cmp_keys(kv_pair.0, key)) {
-                    Ok(pos) => {
-                        let old_value = kv_pairs[pos].1;
-                        kv_pairs[pos].1 = value;
-
-                        Some(old_value)
-                    }
-                    Err(pos) => {
-                        kv_pairs.insert(pos, (key, value));
-
-                        None
-                    }
-                }
-            }
-        }
-    }
-
-    /// Removes key-value pair from the leaf stored at key; returns the previous value associated
-    /// with `key`, if any. Also returns an `is_empty` flag, indicating whether the leaf became
-    /// empty, and must be removed from the data structure it is contained in.
-    fn remove(&mut self, key: RpoDigest) -> (Option<Word>, bool) {
-        match self {
-            SmtLeaf::Empty => (None, false),
-            SmtLeaf::Single((key_at_leaf, value_at_leaf)) => {
-                if *key_at_leaf == key {
-                    // our key was indeed stored in the leaf, so we return the value that was stored
-                    // in it, and indicate that the leaf should be removed
-                    let old_value = *value_at_leaf;
-
-                    // Note: this is not strictly needed, since the caller is expected to drop this
-                    // `SmtLeaf` object.
-                    *self = SmtLeaf::Empty;
-
-                    (Some(old_value), true)
-                } else {
-                    // another key is stored at leaf; nothing to update
-                    (None, false)
-                }
-            }
-            SmtLeaf::Multiple(kv_pairs) => {
-                match kv_pairs.binary_search_by(|kv_pair| cmp_keys(kv_pair.0, key)) {
-                    Ok(pos) => {
-                        let old_value = kv_pairs[pos].1;
-
-                        kv_pairs.remove(pos);
-                        debug_assert!(!kv_pairs.is_empty());
-
-                        if kv_pairs.len() == 1 {
-                            // convert the leaf into `Single`
-                            *self = SmtLeaf::Single(kv_pairs[0]);
-                        }
-
-                        (Some(old_value), false)
-                    }
-                    Err(_) => {
-                        // other keys are stored at leaf; nothing to update
-                        (None, false)
-                    }
-                }
-            }
-        }
-    }
-}
-
-// HELPER FUNCTIONS
-// ================================================================================================
-
-/// Converts a key-value tuple to an iterator of `Felt`s
-fn kv_to_elements((key, value): (RpoDigest, Word)) -> impl Iterator<Item = Felt> {
-    let key_elements = key.into_iter();
-    let value_elements = value.into_iter();
-
-    key_elements.chain(value_elements)
-}
-
-/// Compares two keys, compared element-by-element using their integer representations starting with
-/// the most significant element.
-fn cmp_keys(key_1: RpoDigest, key_2: RpoDigest) -> Ordering {
-    for (v1, v2) in key_1.iter().zip(key_2.iter()).rev() {
-        let v1 = v1.as_int();
-        let v2 = v2.as_int();
-        if v1 != v2 {
-            return v1.cmp(&v2);
-        }
-    }
-
-    Ordering::Equal
-}

src/merkle/smt/full/proof.rs

@@ -0,0 +1,104 @@
+use crate::utils::string::ToString;
+use winter_utils::{ByteReader, ByteWriter, Deserializable, DeserializationError, Serializable};
+
+use super::{MerklePath, RpoDigest, SmtLeaf, SmtProofError, Word, SMT_DEPTH};
+
+/// A proof which can be used to assert membership (or non-membership) of key-value pairs in a
+/// [`super::Smt`].
+///
+/// The proof consists of a Merkle path and leaf which describes the node located at the base of the
+/// path.
+#[derive(PartialEq, Eq, Debug)]
+pub struct SmtProof {
+    path: MerklePath,
+    leaf: SmtLeaf,
+}
+
+impl SmtProof {
+    // CONSTRUCTOR
+    // --------------------------------------------------------------------------------------------
+
+    /// Returns a new instance of [`SmtProof`] instantiated from the specified path and leaf.
+    ///
+    /// # Errors
+    /// Returns an error if the path length is not [`SMT_DEPTH`].
+    pub fn new(path: MerklePath, leaf: SmtLeaf) -> Result<Self, SmtProofError> {
+        if path.len() != SMT_DEPTH.into() {
+            return Err(SmtProofError::InvalidPathLength(path.len()));
+        }
+
+        Ok(Self { path, leaf })
+    }
+
+    /// Returns a new instance of [`SmtProof`] instantiated from the specified path and leaf.
+    ///
+    /// The length of the path is not checked. Reserved for internal use.
+    pub(super) fn new_unchecked(path: MerklePath, leaf: SmtLeaf) -> Self {
+        Self { path, leaf }
+    }
+
+    // PROOF VERIFIER
+    // --------------------------------------------------------------------------------------------
+
+    /// Returns true if a [`super::Smt`] with the specified root contains the provided
+    /// key-value pair.
+    ///
+    /// Note: this method cannot be used to assert non-membership. That is, if false is returned,
+    /// it does not mean that the provided key-value pair is not in the tree.
+    pub fn verify_membership(&self, key: &RpoDigest, value: &Word, root: &RpoDigest) -> bool {
+        let maybe_value_in_leaf = self.leaf.get_value(key);
+
+        match maybe_value_in_leaf {
+            Some(value_in_leaf) => {
+                // The value must match for the proof to be valid
+                if value_in_leaf != *value {
+                    return false;
+                }
+
+                // make sure the Merkle path resolves to the correct root
+                self.compute_root() == *root
+            }
+            // If the key maps to a different leaf, the proof cannot verify membership of `value`
+            None => false,
+        }
+    }
+
+    // PUBLIC ACCESSORS
+    // --------------------------------------------------------------------------------------------
+
+    /// Returns the value associated with the specific key according to this proof, or None if
+    /// this proof does not contain a value for the specified key.
+    ///
+    /// A key-value pair generated by using this method should pass the `verify_membership()` check.
+    pub fn get(&self, key: &RpoDigest) -> Option<Word> {
+        self.leaf.get_value(key)
+    }
+
+    /// Computes the root of a [`super::Smt`] to which this proof resolves.
+    pub fn compute_root(&self) -> RpoDigest {
+        self.path
+            .compute_root(self.leaf.index().value(), self.leaf.hash())
+            .expect("failed to compute Merkle path root")
+    }
+
+    /// Consume the proof and returns its parts.
+    pub fn into_parts(self) -> (MerklePath, SmtLeaf) {
+        (self.path, self.leaf)
+    }
+}
+
+impl Serializable for SmtProof {
+    fn write_into<W: ByteWriter>(&self, target: &mut W) {
+        self.path.write_into(target);
+        self.leaf.write_into(target);
+    }
+}
+
+impl Deserializable for SmtProof {
+    fn read_from<R: ByteReader>(source: &mut R) -> Result<Self, DeserializationError> {
+        let path = MerklePath::read_from(source)?;
+        let leaf = SmtLeaf::read_from(source)?;
+
+        Self::new(path, leaf).map_err(|err| DeserializationError::InvalidValue(err.to_string()))
+    }
+}

src/merkle/smt/full/tests.rs

@@ -1,9 +1,15 @@
+use winter_utils::{Deserializable, Serializable};
+
 use super::*;
 use crate::{
     merkle::{EmptySubtreeRoots, MerkleStore},
+    utils::collections::Vec,
     ONE, WORD_SIZE,
 };
 
+// SMT
+// --------------------------------------------------------------------------------------------
+
 /// This test checks that inserting twice at the same key functions as expected. The test covers
 /// only the case where the key is alone in its leaf
 #[test]
@@ -129,7 +135,7 @@ fn test_smt_insert_and_remove_multiple_values() {
             assert_eq!(smt.root(), tree_root);
 
             let expected_path = store.get_path(tree_root, key_index).unwrap();
-            assert_eq!(smt.open(&key).0, expected_path.path);
+            assert_eq!(smt.open(&key).into_parts().0, expected_path.path);
         }
     }
     let mut smt = Smt::default();
@@ -248,7 +254,7 @@ fn test_smt_removal() {
         let old_value_1 = smt.insert(key_1, EMPTY_WORD);
         assert_eq!(old_value_1, value_1);
 
-        assert_eq!(smt.get_leaf(&key_1), SmtLeaf::Empty);
+        assert_eq!(smt.get_leaf(&key_1), SmtLeaf::new_empty(key_1.into()));
     }
 }
 
@@ -323,12 +329,65 @@ fn test_smt_entries() {
     assert_eq!(&(key_2, value_2), entries.next().unwrap());
     assert!(entries.next().is_none());
 }
+
+// SMT LEAF
+// --------------------------------------------------------------------------------------------
+
+#[test]
+fn test_empty_smt_leaf_serialization() {
+    let empty_leaf = SmtLeaf::new_empty(LeafIndex::new_max_depth(42));
+
+    let mut serialized = empty_leaf.to_bytes();
+    // extend buffer with random bytes
+    serialized.extend([1, 2, 3, 4, 5]);
+    let deserialized = SmtLeaf::read_from_bytes(&serialized).unwrap();
+
+    assert_eq!(empty_leaf, deserialized);
+}
+
+#[test]
+fn test_single_smt_leaf_serialization() {
+    let single_leaf = SmtLeaf::new_single(
+        RpoDigest::from([10_u64.into(), 11_u64.into(), 12_u64.into(), 13_u64.into()]),
+        [1_u64.into(), 2_u64.into(), 3_u64.into(), 4_u64.into()],
+    );
+
+    let mut serialized = single_leaf.to_bytes();
+    // extend buffer with random bytes
+    serialized.extend([1, 2, 3, 4, 5]);
+    let deserialized = SmtLeaf::read_from_bytes(&serialized).unwrap();
+
+    assert_eq!(single_leaf, deserialized);
+}
+
+#[test]
+fn test_multiple_smt_leaf_serialization_success() {
+    let multiple_leaf = SmtLeaf::new_multiple(vec![
+        (
+            RpoDigest::from([10_u64.into(), 11_u64.into(), 12_u64.into(), 13_u64.into()]),
+            [1_u64.into(), 2_u64.into(), 3_u64.into(), 4_u64.into()],
+        ),
+        (
+            RpoDigest::from([100_u64.into(), 101_u64.into(), 102_u64.into(), 13_u64.into()]),
+            [11_u64.into(), 12_u64.into(), 13_u64.into(), 14_u64.into()],
+        ),
+    ])
+    .unwrap();
+
+    let mut serialized = multiple_leaf.to_bytes();
+    // extend buffer with random bytes
+    serialized.extend([1, 2, 3, 4, 5]);
+    let deserialized = SmtLeaf::read_from_bytes(&serialized).unwrap();
+
+    assert_eq!(multiple_leaf, deserialized);
+}
+
 // HELPERS
 // --------------------------------------------------------------------------------------------
 
 fn build_empty_or_single_leaf_node(key: RpoDigest, value: Word) -> RpoDigest {
     if value == EMPTY_WORD {
-        SmtLeaf::Empty.hash()
+        SmtLeaf::new_empty(key.into()).hash()
     } else {
         SmtLeaf::Single((key, value)).hash()
     }

src/merkle/smt/mod.rs

@@ -1,5 +1,3 @@
-use winter_math::StarkField;
-
 use crate::{
     hash::rpo::{Rpo256, RpoDigest},
     Word,
@@ -8,7 +6,7 @@ use crate::{
 use super::{EmptySubtreeRoots, MerkleError, MerklePath, NodeIndex, Vec};
 
 mod full;
-pub use full::{Smt, SmtLeaf, SMT_DEPTH};
+pub use full::{Smt, SmtLeaf, SmtLeafError, SMT_DEPTH};
 
 mod simple;
 pub use simple::SimpleSmt;
@@ -52,7 +50,7 @@ pub(crate) trait SparseMerkleTree<const DEPTH: u8> {
     /// The type for a leaf
     type Leaf;
     /// The type for an opening (i.e. a "proof") of a leaf
-    type Opening: From<(MerklePath, Self::Leaf)>;
+    type Opening;
 
     /// The default value used to compute the hash of empty leaves
     const EMPTY_VALUE: Self::Value;
@@ -83,7 +81,7 @@ pub(crate) trait SparseMerkleTree<const DEPTH: u8> {
             MerklePath::new(path)
         };
 
-        (merkle_path, leaf).into()
+        Self::path_and_leaf_to_opening(merkle_path, leaf)
     }
 
     /// Inserts a value at the specified key, returning the previous value associated with that key.
@@ -170,6 +168,11 @@ pub(crate) trait SparseMerkleTree<const DEPTH: u8> {
 
     /// Maps a key to a leaf index
     fn key_to_leaf_index(key: &Self::Key) -> LeafIndex<DEPTH>;
+
+    /// Maps a (MerklePath, Self::Leaf) to an opening.
+    ///
+    /// The length `path` is guaranteed to be equal to `DEPTH`
+    fn path_and_leaf_to_opening(path: MerklePath, leaf: Self::Leaf) -> Self::Opening;
 }
 
 // INNER NODE
@@ -240,16 +243,3 @@ impl<const DEPTH: u8> TryFrom<NodeIndex> for LeafIndex<DEPTH> {
         Self::new(node_index.value())
     }
 }
-
-impl From<Word> for LeafIndex<SMT_MAX_DEPTH> {
-    fn from(value: Word) -> Self {
-        // We use the most significant `Felt` of a `Word` as the leaf index.
-        Self::new_max_depth(value[3].as_int())
-    }
-}
-
-impl From<RpoDigest> for LeafIndex<SMT_MAX_DEPTH> {
-    fn from(value: RpoDigest) -> Self {
-        Word::from(value).into()
-    }
-}

src/merkle/smt/simple/mod.rs

@@ -1,5 +1,5 @@
 use crate::{
-    merkle::{EmptySubtreeRoots, InnerNodeInfo, ValuePath},
+    merkle::{EmptySubtreeRoots, InnerNodeInfo, MerklePath, ValuePath},
     EMPTY_WORD,
 };
 
@@ -302,4 +302,8 @@ impl<const DEPTH: u8> SparseMerkleTree<DEPTH> for SimpleSmt<DEPTH> {
     fn key_to_leaf_index(key: &LeafIndex<DEPTH>) -> LeafIndex<DEPTH> {
         *key
     }
+
+    fn path_and_leaf_to_opening(path: MerklePath, leaf: Word) -> ValuePath {
+        (path, leaf).into()
+    }
 }