# Binary trie format EIP draft

###### tags: binary EIP

eip:
title: Binary trie structure
author: Guillaume Ballet (@gballet), Vitalik Buterin (@vbuterin)
discussions-to: https://ethresear.ch/t/binary-trie-format/7621
status: Draft
type: Standards Track
category: Core
created: 2020-09-01
requires: None

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## Abstract

This proposal presents a binary structure and merkelization rule for the account and storage tries, which are merged into a single "state" trie. RLP and most of the MPT's optimizations are dropped to simplify the design.

## Motivation

The current design of the Merkle Patricia Trie (MPT) uses an hexary trie. Hexary Merkle trees are more shallow than their binary counterparts, which means less hashing.
Over the course of the 5 years of Ethereum's existence, it has become apparent that disk accesses are a greater bottleneck than hashing. Clients are therefore moving away from a storage model in which all internal nodes are stored, in favor of a flat (key, value) storage model first used by turbo-geth, in which the intermediate nodes are recalculated only when needed.

There is a push for making Ethereum easier to use in a stateless fashion. Binary tries make for smaller (~4x) proofs than hexary tries, making it the design of choice for a stateless-friendly Ethereum.

For that same reason, the account and storage tries are merged in order to have a single proof for all changes.

The MPT design is also rife with uncanny optimizations for size, that have a limited effect - at the cost of prohibitive complexity. For example, nesting for children whose RLP is less than 32 bytes saves an estimated 1MB of disk space. A paltry compared to the 300GB required by a fast sync at the time of this writing. These optimizations are a significant source of errors, and therefore a consensus-breaking risk.
The reliance on RLP has also been criticized for its complexity, while the overhead of a general-purpose encoding scheme isn't warranted for the rigid structure of a Merkle trie.

The desire to switch the storage model from an hexary trie to a binary trie provides an opportunity to adopt a simpler trie architecture that pushes optimizations from the protocol level down to that of the client implementors.

## Specification

### Conventions

As defined in the yellow paper, $\mathbb{B}$ is the set of all possible byte sequences, and $\mathbb{B}_{32}$ is the set of all byte sequences of length 32. Accordingly, $\mathbb{B}_{l}$ is the set of all byte sequences of length $l$.

|Code|Symbolic|Description|
|:-:|:-:|-|
|`u256(x)`|$\texttt{U256}(x)$|Big endian, 32-byte representation of number _x_|
|`||`| $\|$ |Byte-wise concatenation operator|
|`++`| $\texttt{++}$ | Bit-wise concatenation operator|
|`0b0101`|$0101_2$|The binary string `0101`|
|`hash()`|$\texttt{H}()$|The usual hashing function|
|`empty_hash`|$H_{\varnothing}$|The empty hash: `hash("")`|
|`length(x)`|$\lVert x \rVert$|The byte length of object `x`|
|`d[a..b]`|$d_{a..b}$|The big-endian bit sequence taken from $d$, starting at bit index $a$, up to and including the bit at index $b$.|

### Notable changes from the hexary structure

* Account and storage tries are merged, with key length between 32 and 64 bytes;
* RLP is no longer used;
* The "leaf marker" bit used in the hex prefix is also dropped. Leaves are identified as nodes with no children;
* Serialized nodes are hashed, no matter how small the byte length of the serialized nodes.

### The trie

#### Structure

The trie structure is made up of _nodes_. A node $N \equiv \left(N_{l},N_{r},N_{p},N_{v}\right)$ has the following 4 components:

* $N_{l}$ is the hash to the node's _left child_. If the node does not have a left child, then $N_{l}$ is the empty hash $H_{\varnothing}$;
* $N_{r}$ is the hash to the node's _right child_. If the node does not have a right child, then $N_{r}$ is the empty hash $H_{\varnothing}$;
* the optional $N_{p}$ is the node's _prefix_ : every key into the subtrie rooted at $N$ is prefixed by this bit string;
* $N_{v}$ is the _value_ stored at this node. The value is **only present in leaf nodes**.

Nodes with $H_\varnothing$ as both children are called _leaf nodes_, and the remaining nodes are known as _internal nodes_.

#### Accessing account's balance, nonce, code, storage root and storage slots

Assuming an account $A \equiv \left(A_b, A_n, A_c, A_s\right)$ at address $A_a$, the following elements can be found at the following addresses:

* The account balance $A_b \in \mathbb{B}_{32}$ can be found at key $\texttt{H}(A_a)_{0..253} \space\texttt{++}\space 00_2$;
* The account nonce $A_n \in \mathbb{B}_8$ can be found at key $\texttt{H}(A_a)_{0..253} \space\texttt{++}\space 01_2$;
* The code $A_c \in \mathbb{B}$ can be found at key $\texttt{H}(A_a)_{0..253} \space\texttt{++}\space 10_2$;
* The root of the storage trie $A_s$ can be found at key $\texttt{H}(A_a)_{0..253} \space\texttt{++}\space 11_2$
* The storage slot number $k \in \mathbb{B}_{32}$ can be found at key $\texttt{H}(A_a)_{0..253} \space\texttt{++}\space 11_2 \space\texttt{++}\space \texttt{H}(k)$;

After [EIP-2926](https://eips.ethereum.org/EIPS/eip-2926) has been rolled out, $A_c$ will represent the root of the code merkelization tree. The key accessing code chunk $c$ is $\texttt{H}(A_a)_{0..253} \space\texttt{++}\space 10_2 \space\texttt{++}\space \texttt{U256}(c)$.

### Node merkelization rule

Leaves and nodes that have no prefix are hashed according to the rule below:

internal_hash = hash(left_child_hash || right_child_hash)
leaf_hash = hash(hash(key) || hash(leaf_value))

If a prefix is present, the prefix length (as a 32-byte integer) is further concatenated with the output of the prefix-less rule, and hashed again:

internal_hash_with_prefix = hash(u256(prefix_length) || internal_hash)
leaf_hash_with_prefix = hash(u256(prefix_length_u256 - 1) || leaf_hash)

## Rationale

### Merging of the account and storage tries

The trend in clients is to store the keys and values in a "flat" database. Having the key of any storage slot prefixed with the address key of the account it belongs to helps grouping all of an account's data on disk, as well as simplifying the witness structure.

### Prefixes and extension nodes

The removal of extension nodes has been considered as a way to greatly reduce code complexity. The removal induces really prohibitive hashing costs (on the order of 10 seconds for a trie with only 1k leaves) and as a result they have been kept.

An attempt to keep extension nodes for witness and not the merkelization rule can be found [here](https://notes.ethereum.org/m5VMkX8FRvi0Q_OOR7TF4A).

It has been deemed a good trade-off to keep extension nodes, and to only get rid of complex methods like the hex prefix and children nesting.

### 2x32-byte inputs

It has been requested to keep each node hash calculation as a function that takes two 256-bit integer as an input and outputs one 256-bit integer. This property is expected to play nice with circuit constructions and is therefore expected to greatly help with future zero-knowledge applications.

### Binary tries

Binary tries have been chosen primarily because they reduce the witness size. In general, in an `N`-element tree with each element having `k` children, the average length of a branch is roughly `32 * (k-1) * log(N) / log(k)` plus a few percent for overhead. 32 is the length of a hash; the `k-1` refers to the fact that a Merkle proof needs to provide all `k-1` sister nodes at each level, and `log(N) / log(k)` is an approximation of the number of levels in the tree (eg. a tree where each node has 5 children, with 625 nodes total, would have depth 4, as `625 = 5**4` and so `log(625) / log(5) = 4`).

For any `N`, the expression is minimized at `k = 2`. Here's a table of branch lengths for different `k` values assuming `N = 2**24`:

| `k` (children per node) | Branch length (chunks) | Branch length (bytes) |
| - | - | - |
| 2 | 1 * 24 = 24 | 768 |
| 4 | 3 * 12 = 36 | 1152 |
| 8 | 7 * 8 = 56 | 1792 |
| 16 | 15 * 6 = 90 | 2880 |

Actual branch lengths will be slightly larger than this due to uneven distribution and overhead, but the pattern of `k=2` being by far the best remains.

The ethereum tree was originally hexary because this would reduce the number of database reads (eg. 6 instead of 24 in the above example). It is now understood that this reasoning was flawed, because nodes can still "pretend" that a binary tree is a hexary (or even 256-ary) tree at the database layer (eg. see https://ethresear.ch/t/optimizing-sparse-merkle-trees/3751), and thereby get the best-of-both-worlds of having the low proof sizes of the tree being binary from a hash-structure perspective and at the same time a much more efficient representation in the database.

Additionally, binary trees are expected to be widely used in Eth2, so this path improves forward-compatibility and reduces long-run total complexity for the protocol.

### Key length instead of bit prefix

In order to remove the complexity associated with byte manipulation, only the bit-length of the extension is used to merkelize a node with a prefix.

## Backwards compatibility

A hard fork is required in order for blocks to have a trie root using a different structure.

## Test Cases

This section is still incomplete and will be fleshed out as clients start implementing the support for binary tries.

### Hard fork transition test

Input: an hexary trie with N accounts and associated storages.
Output: a binary trie with the same N accounts and associated storages, as well as an extra account for the miner.

## Implementation

This section is still incomplete and will be fleshed out as clients start implementing the support for binary tries.

* As of [commit 0db87e187dc0bfb96046a47e3d6768c93a2e3331](https://github.com/gballet/multiproof-rs/commit/6d22b1aef9548581826b3c04b3e00d6cc709388c), [multiproof-rs](https://github.com/gballet/multiproof-rs) implements this merkelization rule in the `hash_m5()` function, found in file `src/binary_tree.rs`.
* An implementation of this structure for go-ethereum is available [in this branch](https://github.com/gballet/go-ethereum/tree/binary-trie-m5).

## Security considerations

Security issues could arise when performing the transition. In particular, a heavy conversion process could incentivize clients to wait the transition out. This could lead to a lowered network security at the time of the transition. A transition process has been proposed with [EIP-2584](https://github.com/ethereum/EIPs/blob/master/EIPS/eip-2584.md).

## Copyright
Copyright and related rights waived via [CC0](https://creativecommons.org/publicdomain/zero/1.0/).