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# Ethereum 2.0 Phase 0 -- The Beacon Chain
**NOTICE**: This document is a work in progress for researchers and implementers.
## Table of contents
<!-- TOC -->
- [Ethereum 2.0 Phase 0 -- The Beacon Chain](#ethereum-20-phase-0----the-beacon-chain)
- [Table of contents](#table-of-contents)
- [Introduction](#introduction)
- [Notation](#notation)
- [Terminology](#terminology)
- [Constants](#constants)
- [Misc](#misc)
- [Deposit contract](#deposit-contract)
- [Gwei values](#gwei-values)
- [Initial values](#initial-values)
- [Time parameters](#time-parameters)
- [State list lengths](#state-list-lengths)
- [Reward and penalty quotients](#reward-and-penalty-quotients)
- [Max operations per block](#max-operations-per-block)
- [Signature domains](#signature-domains)
- [Data structures](#data-structures)
- [Misc dependencies](#misc-dependencies)
- [`Fork`](#fork)
- [`Crosslink`](#crosslink)
- [`Eth1Data`](#eth1data)
- [`AttestationData`](#attestationdata)
- [`AttestationDataAndCustodyBit`](#attestationdataandcustodybit)
- [`IndexedAttestation`](#indexedattestation)
- [`DepositData`](#depositdata)
- [`BeaconBlockHeader`](#beaconblockheader)
- [`Validator`](#validator)
- [`PendingAttestation`](#pendingattestation)
- [`HistoricalBatch`](#historicalbatch)
- [Beacon operations](#beacon-operations)
- [`ProposerSlashing`](#proposerslashing)
- [`AttesterSlashing`](#attesterslashing)
- [`Attestation`](#attestation)
- [`Deposit`](#deposit)
- [`VoluntaryExit`](#voluntaryexit)
- [`Transfer`](#transfer)
- [Beacon blocks](#beacon-blocks)
- [`BeaconBlockBody`](#beaconblockbody)
- [`BeaconBlock`](#beaconblock)
- [Beacon state](#beacon-state)
- [`BeaconState`](#beaconstate)
- [Custom Types](#custom-types)
- [Helper functions](#helper-functions)
- [`xor`](#xor)
- [`hash`](#hash)
- [`hash_tree_root`](#hash_tree_root)
- [`signing_root`](#signing_root)
- [`slot_to_epoch`](#slot_to_epoch)
- [`get_previous_epoch`](#get_previous_epoch)
- [`get_current_epoch`](#get_current_epoch)
- [`get_epoch_start_slot`](#get_epoch_start_slot)
- [`is_active_validator`](#is_active_validator)
- [`is_slashable_validator`](#is_slashable_validator)
- [`get_active_validator_indices`](#get_active_validator_indices)
- [`increase_balance`](#increase_balance)
- [`decrease_balance`](#decrease_balance)
- [`get_permuted_index`](#get_permuted_index)
- [`get_split_offset`](#get_split_offset)
- [`get_epoch_committee_count`](#get_epoch_committee_count)
- [`get_shard_delta`](#get_shard_delta)
- [`compute_committee`](#compute_committee)
- [`get_crosslink_committees_at_slot`](#get_crosslink_committees_at_slot)
- [`get_block_root_at_slot`](#get_block_root_at_slot)
- [`get_block_root`](#get_block_root)
- [`get_randao_mix`](#get_randao_mix)
- [`get_active_index_root`](#get_active_index_root)
- [`generate_seed`](#generate_seed)
- [`get_beacon_proposer_index`](#get_beacon_proposer_index)
- [`verify_merkle_branch`](#verify_merkle_branch)
- [`get_attesting_indices`](#get_attesting_indices)
- [`int_to_bytes1`, `int_to_bytes2`, ...](#int_to_bytes1-int_to_bytes2-)
- [`bytes_to_int`](#bytes_to_int)
- [`get_total_balance`](#get_total_balance)
- [`get_domain`](#get_domain)
- [`get_bitfield_bit`](#get_bitfield_bit)
- [`verify_bitfield`](#verify_bitfield)
- [`convert_to_indexed`](#convert_to_indexed)
- [`verify_indexed_attestation`](#verify_indexed_attestation)
- [`is_double_vote`](#is_double_vote)
- [`is_surround_vote`](#is_surround_vote)
- [`integer_squareroot`](#integer_squareroot)
- [`get_delayed_activation_exit_epoch`](#get_delayed_activation_exit_epoch)
- [`get_churn_limit`](#get_churn_limit)
- [`bls_verify`](#bls_verify)
- [`bls_verify_multiple`](#bls_verify_multiple)
- [`bls_aggregate_pubkeys`](#bls_aggregate_pubkeys)
- [Routines for updating validator status](#routines-for-updating-validator-status)
- [`initiate_validator_exit`](#initiate_validator_exit)
- [`slash_validator`](#slash_validator)
- [On genesis](#on-genesis)
- [Beacon chain state transition function](#beacon-chain-state-transition-function)
- [State caching](#state-caching)
- [Per-epoch processing](#per-epoch-processing)
- [Helper functions](#helper-functions-1)
- [Justification and finalization](#justification-and-finalization)
- [Crosslinks](#crosslinks)
- [Rewards and penalties](#rewards-and-penalties)
- [Registry updates](#registry-updates)
- [Slashings](#slashings)
- [Final updates](#final-updates)
- [Per-slot processing](#per-slot-processing)
- [Per-block processing](#per-block-processing)
- [Block header](#block-header)
- [RANDAO](#randao)
- [Eth1 data](#eth1-data)
- [Operations](#operations)
- [Proposer slashings](#proposer-slashings)
- [Attester slashings](#attester-slashings)
- [Attestations](#attestations)
- [Deposits](#deposits)
- [Voluntary exits](#voluntary-exits)
- [Transfers](#transfers)
- [State root verification](#state-root-verification)
<!-- /TOC -->
## Introduction
This document represents the specification for Phase 0 of Ethereum 2.0 -- The Beacon Chain.
At the core of Ethereum 2.0 is a system chain called the "beacon chain". The beacon chain stores and manages the registry of [validators](#dfn-validator). In the initial deployment phases of Ethereum 2.0, the only mechanism to become a [validator](#dfn-validator) is to make a one-way ETH transaction to a deposit contract on Ethereum 1.0. Activation as a [validator](#dfn-validator) happens when Ethereum 1.0 deposit receipts are processed by the beacon chain, the activation balance is reached, and a queuing process is completed. Exit is either voluntary or done forcibly as a penalty for misbehavior.
The primary source of load on the beacon chain is "attestations". Attestations are simultaneously availability votes for a shard block and proof-of-stake votes for a beacon block. A sufficient number of attestations for the same shard block create a "crosslink", confirming the shard segment up to that shard block into the beacon chain. Crosslinks also serve as infrastructure for asynchronous cross-shard communication.
## Notation
Code snippets appearing in `this style` are to be interpreted as Python code.
## Terminology
* **Validator** <a id="dfn-validator"></a> - a registered participant in the beacon chain. You can become one by sending ether into the Ethereum 1.0 deposit contract.
* **Active validator** <a id="dfn-active-validator"></a> - an active participant in the Ethereum 2.0 consensus invited to, among other things, propose and attest to blocks and vote for crosslinks.
* **Committee** - a (pseudo-) randomly sampled subset of [active validators](#dfn-active-validator). When a committee is referred to collectively, as in "this committee attests to X", this is assumed to mean "some subset of that committee that contains enough [validators](#dfn-validator) that the protocol recognizes it as representing the committee".
* **Proposer** - the [validator](#dfn-validator) that creates a beacon chain block.
* **Attester** - a [validator](#dfn-validator) that is part of a committee that needs to sign off on a beacon chain block while simultaneously creating a link (crosslink) to a recent shard block on a particular shard chain.
* **Beacon chain** - the central PoS chain that is the base of the sharding system.
* **Shard chain** - one of the chains on which user transactions take place and account data is stored.
* **Block root** - a 32-byte Merkle root of a beacon chain block or shard chain block. Previously called "block hash".
* **Crosslink** - a set of signatures from a committee attesting to a block in a shard chain that can be included into the beacon chain. Crosslinks are the main means by which the beacon chain "learns about" the updated state of shard chains.
* **Slot** - a period during which one proposer has the ability to create a beacon chain block and some attesters have the ability to make attestations.
* **Epoch** - an aligned span of slots during which all [validators](#dfn-validator) get exactly one chance to make an attestation.
* **Finalized**, **justified** - see the [Casper FFG paper](https://arxiv.org/abs/1710.09437).
* **Withdrawal period** - the number of slots between a [validator](#dfn-validator) exit and the [validator](#dfn-validator) balance being withdrawable.
* **Genesis time** - the Unix time of the genesis beacon chain block at slot 0.
## Constants
Note: the default mainnet values for the constants are included here for spec-design purposes.
The different configurations for mainnet, testnets, and yaml-based testing can be found in the `configs/constant_presets/` directory.
These configurations are updated for releases, but may be out of sync during `dev` changes.
### Misc
| Name | Value |
| - | - |
| `SHARD_COUNT` | `2**10` (= 1,024) |
| `TARGET_COMMITTEE_SIZE` | `2**7` (= 128) |
| `MAX_INDICES_PER_ATTESTATION` | `2**12` (= 4,096) |
| `MIN_PER_EPOCH_CHURN_LIMIT` | `2**2` (= 4) |
| `CHURN_LIMIT_QUOTIENT` | `2**16` (= 65,536) |
| `BASE_REWARDS_PER_EPOCH` | `5` |
| `SHUFFLE_ROUND_COUNT` | 90 |
* For the safety of crosslinks `TARGET_COMMITTEE_SIZE` exceeds [the recommended minimum committee size of 111](https://vitalik.ca/files/Ithaca201807_Sharding.pdf); with sufficient active validators (at least `SLOTS_PER_EPOCH * TARGET_COMMITTEE_SIZE`), the shuffling algorithm ensures committee sizes of at least `TARGET_COMMITTEE_SIZE`. (Unbiasable randomness with a Verifiable Delay Function (VDF) will improve committee robustness and lower the safe minimum committee size.)
### Deposit contract
| Name | Value |
| - | - |
| `DEPOSIT_CONTRACT_TREE_DEPTH` | `2**5` (= 32) |
### Gwei values
| Name | Value | Unit |
| - | - | :-: |
| `MIN_DEPOSIT_AMOUNT` | `2**0 * 10**9` (= 1,000,000,000) | Gwei |
| `MAX_EFFECTIVE_BALANCE` | `2**5 * 10**9` (= 32,000,000,000) | Gwei |
| `EJECTION_BALANCE` | `2**4 * 10**9` (= 16,000,000,000) | Gwei |
| `EFFECTIVE_BALANCE_INCREMENT` | `2**0 * 10**9` (= 1,000,000,000) | Gwei |
### Initial values
| Name | Value |
| - | - |
| `GENESIS_SLOT` | `0` |
| `GENESIS_EPOCH` | `0` |
| `FAR_FUTURE_EPOCH` | `2**64 - 1` |
| `ZERO_HASH` | `int_to_bytes32(0)` |
| `BLS_WITHDRAWAL_PREFIX_BYTE` | `int_to_bytes1(0)` |
### Time parameters
| Name | Value | Unit | Duration |
| - | - | :-: | :-: |
| `MIN_ATTESTATION_INCLUSION_DELAY` | `2**2` (= 4) | slots | 24 seconds |
| `SLOTS_PER_EPOCH` | `2**6` (= 64) | slots | 6.4 minutes |
| `MIN_SEED_LOOKAHEAD` | `2**0` (= 1) | epochs | 6.4 minutes |
| `ACTIVATION_EXIT_DELAY` | `2**2` (= 4) | epochs | 25.6 minutes |
| `SLOTS_PER_ETH1_VOTING_PERIOD` | `2**10` (= 1,024) | slots | ~1.7 hours |
| `SLOTS_PER_HISTORICAL_ROOT` | `2**13` (= 8,192) | slots | ~13 hours |
| `MIN_VALIDATOR_WITHDRAWABILITY_DELAY` | `2**8` (= 256) | epochs | ~27 hours |
| `PERSISTENT_COMMITTEE_PERIOD` | `2**11` (= 2,048) | epochs | 9 days |
| `MAX_CROSSLINK_EPOCHS` | `2**6` (= 64) | epochs | ~7 hours |
| `MIN_EPOCHS_TO_INACTIVITY_PENALTY` | `2**2` (= 4) | epochs | 25.6 minutes |
* `MAX_CROSSLINK_EPOCHS` should be a small constant times `SHARD_COUNT // SLOTS_PER_EPOCH`
### State list lengths
| Name | Value | Unit | Duration |
| - | - | :-: | :-: |
| `LATEST_RANDAO_MIXES_LENGTH` | `2**13` (= 8,192) | epochs | ~36 days |
| `LATEST_ACTIVE_INDEX_ROOTS_LENGTH` | `2**13` (= 8,192) | epochs | ~36 days |
| `LATEST_SLASHED_EXIT_LENGTH` | `2**13` (= 8,192) | epochs | ~36 days |
### Reward and penalty quotients
| Name | Value |
| - | - |
| `BASE_REWARD_QUOTIENT` | `2**5` (= 32) |
| `WHISTLEBLOWING_REWARD_QUOTIENT` | `2**9` (= 512) |
| `PROPOSER_REWARD_QUOTIENT` | `2**3` (= 8) |
| `INACTIVITY_PENALTY_QUOTIENT` | `2**25` (= 33,554,432) |
| `MIN_SLASHING_PENALTY_QUOTIENT` | `2**5` (= 32) |
* **The `BASE_REWARD_QUOTIENT` is NOT final. Once all other protocol details are finalized it will be adjusted, to target a theoretical maximum total issuance of `2**21` ETH per year if `2**27` ETH is validating (and therefore `2**20` per year if `2**25` ETH is validating, etc etc)**
* The `INACTIVITY_PENALTY_QUOTIENT` equals `INVERSE_SQRT_E_DROP_TIME**2` where `INVERSE_SQRT_E_DROP_TIME := 2**12 epochs` (~18 days) is the time it takes the inactivity penalty to reduce the balance of non-participating [validators](#dfn-validator) to about `1/sqrt(e) ~= 60.6%`. Indeed, the balance retained by offline [validators](#dfn-validator) after `n` epochs is about `(1 - 1/INACTIVITY_PENALTY_QUOTIENT)**(n**2/2)` so after `INVERSE_SQRT_E_DROP_TIME` epochs it is roughly `(1 - 1/INACTIVITY_PENALTY_QUOTIENT)**(INACTIVITY_PENALTY_QUOTIENT/2) ~= 1/sqrt(e)`.
### Max operations per block
| Name | Value |
| - | - |
| `MAX_PROPOSER_SLASHINGS` | `2**4` (= 16) |
| `MAX_ATTESTER_SLASHINGS` | `2**0` (= 1) |
| `MAX_ATTESTATIONS` | `2**7` (= 128) |
| `MAX_DEPOSITS` | `2**4` (= 16) |
| `MAX_VOLUNTARY_EXITS` | `2**4` (= 16) |
| `MAX_TRANSFERS` | `0` |
### Signature domains
| Name | Value |
| - | - |
| `DOMAIN_BEACON_PROPOSER` | `0` |
| `DOMAIN_RANDAO` | `1` |
| `DOMAIN_ATTESTATION` | `2` |
| `DOMAIN_DEPOSIT` | `3` |
| `DOMAIN_VOLUNTARY_EXIT` | `4` |
| `DOMAIN_TRANSFER` | `5` |
## Data structures
The following data structures are defined as [SimpleSerialize (SSZ)](../simple-serialize.md) objects.
The types are defined topologically to aid in facilitating an executable version of the spec.
### Misc dependencies
#### `Fork`
```python
{
# Previous fork version
'previous_version': 'bytes4',
# Current fork version
'current_version': 'bytes4',
# Fork epoch number
'epoch': 'uint64',
}
```
#### `Crosslink`
```python
{
# Epoch number
'epoch': 'uint64',
# Root of the previous crosslink
'previous_crosslink_root': 'bytes32',
# Root of the crosslinked shard data since the previous crosslink
'crosslink_data_root': 'bytes32',
}
```
#### `Eth1Data`
```python
{
# Root of the deposit tree
'deposit_root': 'bytes32',
# Total number of deposits
'deposit_count': 'uint64',
# Block hash
'block_hash': 'bytes32',
}
```
#### `AttestationData`
```python
{
# LMD GHOST vote
'slot': 'uint64',
'beacon_block_root': 'bytes32',
# FFG vote
'source_epoch': 'uint64',
'source_root': 'bytes32',
'target_root': 'bytes32',
# Crosslink vote
'shard': 'uint64',
'previous_crosslink_root': 'bytes32',
'crosslink_data_root': 'bytes32',
}
```
#### `AttestationDataAndCustodyBit`
```python
{
# Attestation data
'data': AttestationData,
# Custody bit
'custody_bit': 'bool',
}
```
#### `IndexedAttestation`
```python
{
# Validator indices
'custody_bit_0_indices': ['uint64'],
'custody_bit_1_indices': ['uint64'],
# Attestation data
'data': AttestationData,
# Aggregate signature
'signature': 'bytes96',
}
```
#### `DepositData`
```python
{
# BLS pubkey
'pubkey': 'bytes48',
# Withdrawal credentials
'withdrawal_credentials': 'bytes32',
# Amount in Gwei
'amount': 'uint64',
# Container self-signature
'signature': 'bytes96',
}
```
#### `BeaconBlockHeader`
```python
{
'slot': 'uint64',
'previous_block_root': 'bytes32',
'state_root': 'bytes32',
'block_body_root': 'bytes32',
'signature': 'bytes96',
}
```
#### `Validator`
```python
{
# BLS public key
'pubkey': 'bytes48',
# Withdrawal credentials
'withdrawal_credentials': 'bytes32',
# Epoch when became eligible for activation
'activation_eligibility_epoch': 'uint64',
# Epoch when validator activated
'activation_epoch': 'uint64',
# Epoch when validator exited
'exit_epoch': 'uint64',
# Epoch when validator is eligible to withdraw
'withdrawable_epoch': 'uint64',
# Was the validator slashed
'slashed': 'bool',
# Effective balance
'effective_balance': 'uint64',
}
```
#### `PendingAttestation`
```python
{
# Attester aggregation bitfield
'aggregation_bitfield': 'bytes',
# Attestation data
'data': AttestationData,
# Inclusion slot
'inclusion_slot': 'uint64',
# Proposer index
'proposer_index': 'uint64',
}
```
#### `HistoricalBatch`
```python
{
# Block roots
'block_roots': ['bytes32', SLOTS_PER_HISTORICAL_ROOT],
# State roots
'state_roots': ['bytes32', SLOTS_PER_HISTORICAL_ROOT],
}
```
### Beacon operations
#### `ProposerSlashing`
```python
{
# Proposer index
'proposer_index': 'uint64',
# First block header
'header_1': BeaconBlockHeader,
# Second block header
'header_2': BeaconBlockHeader,
}
```
#### `AttesterSlashing`
```python
{
# First attestation
'attestation_1': IndexedAttestation,
# Second attestation
'attestation_2': IndexedAttestation,
}
```
#### `Attestation`
```python
{
# Attester aggregation bitfield
'aggregation_bitfield': 'bytes',
# Attestation data
'data': AttestationData,
# Custody bitfield
'custody_bitfield': 'bytes',
# BLS aggregate signature
'signature': 'bytes96',
}
```
#### `Deposit`
```python
{
# Branch in the deposit tree
'proof': ['bytes32', DEPOSIT_CONTRACT_TREE_DEPTH],
# Index in the deposit tree
'index': 'uint64',
# Data
'data': DepositData,
}
```
#### `VoluntaryExit`
```python
{
# Minimum epoch for processing exit
'epoch': 'uint64',
# Index of the exiting validator
'validator_index': 'uint64',
# Validator signature
'signature': 'bytes96',
}
```
#### `Transfer`
```python
{
# Sender index
'sender': 'uint64',
# Recipient index
'recipient': 'uint64',
# Amount in Gwei
'amount': 'uint64',
# Fee in Gwei for block proposer
'fee': 'uint64',
# Inclusion slot
'slot': 'uint64',
# Sender withdrawal pubkey
'pubkey': 'bytes48',
# Sender signature
'signature': 'bytes96',
}
```
### Beacon blocks
#### `BeaconBlockBody`
```python
{
'randao_reveal': 'bytes96',
'eth1_data': Eth1Data,
'proposer_slashings': [ProposerSlashing],
'attester_slashings': [AttesterSlashing],
'attestations': [Attestation],
'deposits': [Deposit],
'voluntary_exits': [VoluntaryExit],
'transfers': [Transfer],
}
```
#### `BeaconBlock`
```python
{
# Header
'slot': 'uint64',
'previous_block_root': 'bytes32',
'state_root': 'bytes32',
'body': BeaconBlockBody,
'signature': 'bytes96',
}
```
### Beacon state
#### `BeaconState`
```python
{
# Misc
'slot': 'uint64',
'genesis_time': 'uint64',
'fork': Fork, # For versioning hard forks
# Validator registry
'validator_registry': [Validator],
'balances': ['uint64'],
# Randomness and committees
'latest_randao_mixes': ['bytes32', LATEST_RANDAO_MIXES_LENGTH],
'latest_start_shard': 'uint64',
# Finality
'previous_epoch_attestations': [PendingAttestation],
'current_epoch_attestations': [PendingAttestation],
'previous_justified_epoch': 'uint64',
'current_justified_epoch': 'uint64',
'previous_justified_root': 'bytes32',
'current_justified_root': 'bytes32',
'justification_bitfield': 'uint64',
'finalized_epoch': 'uint64',
'finalized_root': 'bytes32',
# Recent state
'current_crosslinks': [Crosslink, SHARD_COUNT],
'previous_crosslinks': [Crosslink, SHARD_COUNT],
'latest_block_roots': ['bytes32', SLOTS_PER_HISTORICAL_ROOT],
'latest_state_roots': ['bytes32', SLOTS_PER_HISTORICAL_ROOT],
'latest_active_index_roots': ['bytes32', LATEST_ACTIVE_INDEX_ROOTS_LENGTH],
'latest_slashed_balances': ['uint64', LATEST_SLASHED_EXIT_LENGTH], # Balances slashed at every withdrawal period
'latest_block_header': BeaconBlockHeader, # `latest_block_header.state_root == ZERO_HASH` temporarily
'historical_roots': ['bytes32'],
# Ethereum 1.0 chain data
'latest_eth1_data': Eth1Data,
'eth1_data_votes': [Eth1Data],
'deposit_index': 'uint64',
}
```
## Custom Types
We define the following Python custom types for type hinting and readability:
| Name | SSZ equivalent | Description |
| - | - | - |
| `Slot` | `uint64` | a slot number |
| `Epoch` | `uint64` | an epoch number |
| `Shard` | `uint64` | a shard number |
| `ValidatorIndex` | `uint64` | a validator registry index |
| `Gwei` | `uint64` | an amount in Gwei |
| `Bytes32` | `bytes32` | 32 bytes of binary data |
| `BLSPubkey` | `bytes48` | a BLS12-381 public key |
| `BLSSignature` | `bytes96` | a BLS12-381 signature |
## Helper functions
Note: The definitions below are for specification purposes and are not necessarily optimal implementations.
### `xor`
```python
def xor(bytes1: Bytes32, bytes2: Bytes32) -> Bytes32:
return bytes(a ^ b for a, b in zip(bytes1, bytes2))
```
### `hash`
The `hash` function is SHA256.
Note: We aim to migrate to a S[T/N]ARK-friendly hash function in a future Ethereum 2.0 deployment phase.
### `hash_tree_root`
`def hash_tree_root(object: SSZSerializable) -> Bytes32` is a function for hashing objects into a single root utilizing a hash tree structure. `hash_tree_root` is defined in the [SimpleSerialize spec](../simple-serialize.md#merkleization).
### `signing_root`
`def signing_root(object: SSZContainer) -> Bytes32` is a function defined in the [SimpleSerialize spec](../simple-serialize.md#self-signed-containers) to compute signing messages.
### `slot_to_epoch`
```python
def slot_to_epoch(slot: Slot) -> Epoch:
"""
Return the epoch number of the given ``slot``.
"""
return slot // SLOTS_PER_EPOCH
```
### `get_previous_epoch`
```python
def get_previous_epoch(state: BeaconState) -> Epoch:
"""`
Return the previous epoch of the given ``state``.
Return the current epoch if it's genesis epoch.
"""
current_epoch = get_current_epoch(state)
return (current_epoch - 1) if current_epoch > GENESIS_EPOCH else current_epoch
```
### `get_current_epoch`
```python
def get_current_epoch(state: BeaconState) -> Epoch:
"""
Return the current epoch of the given ``state``.
"""
return slot_to_epoch(state.slot)
```
### `get_epoch_start_slot`
```python
def get_epoch_start_slot(epoch: Epoch) -> Slot:
"""
Return the starting slot of the given ``epoch``.
"""
return epoch * SLOTS_PER_EPOCH
```
### `is_active_validator`
```python
def is_active_validator(validator: Validator, epoch: Epoch) -> bool:
"""
Check if ``validator`` is active.
"""
return validator.activation_epoch <= epoch < validator.exit_epoch
```
### `is_slashable_validator`
```python
def is_slashable_validator(validator: Validator, epoch: Epoch) -> bool:
"""
Check if ``validator`` is slashable.
"""
return validator.slashed is False and (validator.activation_epoch <= epoch < validator.withdrawable_epoch)
```
### `get_active_validator_indices`
```python
def get_active_validator_indices(state: BeaconState, epoch: Epoch) -> List[ValidatorIndex]:
"""
Get active validator indices at ``epoch``.
"""
return [i for i, v in enumerate(state.validator_registry) if is_active_validator(v, epoch)]
```
### `increase_balance`
```python
def increase_balance(state: BeaconState, index: ValidatorIndex, delta: Gwei) -> None:
"""
Increase validator balance by ``delta``.
"""
state.balances[index] += delta
```
### `decrease_balance`
```python
def decrease_balance(state: BeaconState, index: ValidatorIndex, delta: Gwei) -> None:
"""
Decrease validator balance by ``delta`` with underflow protection.
"""
state.balances[index] = 0 if delta > state.balances[index] else state.balances[index] - delta
```
### `get_permuted_index`
```python
def get_permuted_index(index: int, list_size: int, seed: Bytes32) -> int:
"""
Return `p(index)` in a pseudorandom permutation `p` of `0...list_size - 1` with ``seed`` as entropy.
Utilizes 'swap or not' shuffling found in
https://link.springer.com/content/pdf/10.1007%2F978-3-642-32009-5_1.pdf
See the 'generalized domain' algorithm on page 3.
"""
assert index < list_size
assert list_size <= 2**40
for round in range(SHUFFLE_ROUND_COUNT):
pivot = bytes_to_int(hash(seed + int_to_bytes1(round))[0:8]) % list_size
flip = (pivot - index) % list_size
position = max(index, flip)
source = hash(seed + int_to_bytes1(round) + int_to_bytes4(position // 256))
byte = source[(position % 256) // 8]
bit = (byte >> (position % 8)) % 2
index = flip if bit else index
return index
```
### `get_split_offset`
```python
def get_split_offset(list_size: int, chunks: int, index: int) -> int:
"""
Returns a value such that for a list L, chunk count k and index i,
split(L, k)[i] == L[get_split_offset(len(L), k, i): get_split_offset(len(L), k, i+1)]
"""
return (list_size * index) // chunks
```
### `get_epoch_committee_count`
```python
def get_epoch_committee_count(state: BeaconState, epoch: Epoch) -> int:
"""
Return the number of committees at ``epoch``.
"""
active_validator_indices = get_active_validator_indices(state, epoch)
return max(
1,
min(
SHARD_COUNT // SLOTS_PER_EPOCH,
len(active_validator_indices) // SLOTS_PER_EPOCH // TARGET_COMMITTEE_SIZE,
)
) * SLOTS_PER_EPOCH
```
### `get_shard_delta`
```python
def get_shard_delta(state: BeaconState, epoch: Epoch) -> int:
"""
Return the number of shards to increment ``state.latest_start_shard`` during ``epoch``.
"""
return min(get_epoch_committee_count(state, epoch), SHARD_COUNT - SHARD_COUNT // SLOTS_PER_EPOCH)
```
### `compute_committee`
```python
def compute_committee(validator_indices: List[ValidatorIndex],
seed: Bytes32,
index: int,
total_committees: int) -> List[ValidatorIndex]:
"""
Return the ``index``'th shuffled committee out of a total ``total_committees``
using ``validator_indices`` and ``seed``.
"""
start_offset = get_split_offset(len(validator_indices), total_committees, index)
end_offset = get_split_offset(len(validator_indices), total_committees, index + 1)
return [
validator_indices[get_permuted_index(i, len(validator_indices), seed)]
for i in range(start_offset, end_offset)
]
```
Note: this definition and the next few definitions are highly inefficient as algorithms, as they re-calculate many sub-expressions. Production implementations are expected to appropriately use caching/memoization to avoid redoing work.
### `get_crosslink_committees_at_slot`
```python
def get_crosslink_committees_at_slot(state: BeaconState,
slot: Slot) -> List[Tuple[List[ValidatorIndex], Shard]]:
"""
Return the list of ``(committee, shard)`` tuples for the ``slot``.
"""
epoch = slot_to_epoch(slot)
current_epoch = get_current_epoch(state)
previous_epoch = get_previous_epoch(state)
next_epoch = current_epoch + 1
assert previous_epoch <= epoch <= next_epoch
indices = get_active_validator_indices(state, epoch)
if epoch == current_epoch:
start_shard = state.latest_start_shard
elif epoch == previous_epoch:
previous_shard_delta = get_shard_delta(state, previous_epoch)
start_shard = (state.latest_start_shard - previous_shard_delta) % SHARD_COUNT
elif epoch == next_epoch:
current_shard_delta = get_shard_delta(state, current_epoch)
start_shard = (state.latest_start_shard + current_shard_delta) % SHARD_COUNT
committees_per_epoch = get_epoch_committee_count(state, epoch)
committees_per_slot = committees_per_epoch // SLOTS_PER_EPOCH
offset = slot % SLOTS_PER_EPOCH
slot_start_shard = (start_shard + committees_per_slot * offset) % SHARD_COUNT
seed = generate_seed(state, epoch)
return [
(
compute_committee(indices, seed, committees_per_slot * offset + i, committees_per_epoch),
(slot_start_shard + i) % SHARD_COUNT,
)
for i in range(committees_per_slot)
]
```
### `get_block_root_at_slot`
```python
def get_block_root_at_slot(state: BeaconState,
slot: Slot) -> Bytes32:
"""
Return the block root at a recent ``slot``.
"""
assert slot < state.slot <= slot + SLOTS_PER_HISTORICAL_ROOT
return state.latest_block_roots[slot % SLOTS_PER_HISTORICAL_ROOT]
```
### `get_block_root`
```python
def get_block_root(state: BeaconState,
epoch: Epoch) -> Bytes32:
"""
Return the block root at a recent ``epoch``.
"""
return get_block_root_at_slot(state, get_epoch_start_slot(epoch))
```
### `get_randao_mix`
```python
def get_randao_mix(state: BeaconState,
epoch: Epoch) -> Bytes32:
"""
Return the randao mix at a recent ``epoch``.
"""
assert get_current_epoch(state) - LATEST_RANDAO_MIXES_LENGTH < epoch <= get_current_epoch(state)
return state.latest_randao_mixes[epoch % LATEST_RANDAO_MIXES_LENGTH]
```
### `get_active_index_root`
```python
def get_active_index_root(state: BeaconState,
epoch: Epoch) -> Bytes32:
"""
Return the index root at a recent ``epoch``.
"""
assert get_current_epoch(state) - LATEST_ACTIVE_INDEX_ROOTS_LENGTH + ACTIVATION_EXIT_DELAY < epoch <= get_current_epoch(state) + ACTIVATION_EXIT_DELAY
return state.latest_active_index_roots[epoch % LATEST_ACTIVE_INDEX_ROOTS_LENGTH]
```
### `generate_seed`
```python
def generate_seed(state: BeaconState,
epoch: Epoch) -> Bytes32:
"""
Generate a seed for the given ``epoch``.
"""
return hash(
get_randao_mix(state, epoch - MIN_SEED_LOOKAHEAD) +
get_active_index_root(state, epoch) +
int_to_bytes32(epoch)
)
```
### `get_beacon_proposer_index`
```python
def get_beacon_proposer_index(state: BeaconState) -> ValidatorIndex:
"""
Return the beacon proposer index at ``state.slot``.
"""
current_epoch = get_current_epoch(state)
first_committee, _ = get_crosslink_committees_at_slot(state, state.slot)[0]
MAX_RANDOM_BYTE = 2**8 - 1
i = 0
while True:
candidate_index = first_committee[(current_epoch + i) % len(first_committee)]
random_byte = hash(generate_seed(state, current_epoch) + int_to_bytes8(i // 32))[i % 32]
effective_balance = state.validator_registry[candidate_index].effective_balance
if effective_balance * MAX_RANDOM_BYTE >= MAX_EFFECTIVE_BALANCE * random_byte:
return candidate_index
i += 1
```
### `verify_merkle_branch`
```python
def verify_merkle_branch(leaf: Bytes32, proof: List[Bytes32], depth: int, index: int, root: Bytes32) -> bool:
"""
Verify that the given ``leaf`` is on the merkle branch ``proof``
starting with the given ``root``.
"""
value = leaf
for i in range(depth):
if index // (2**i) % 2:
value = hash(proof[i] + value)
else:
value = hash(value + proof[i])
return value == root
```
### `get_attesting_indices`
```python
def get_attesting_indices(state: BeaconState,
attestation_data: AttestationData,
bitfield: bytes) -> List[ValidatorIndex]:
"""
Return the sorted attesting indices corresponding to ``attestation_data`` and ``bitfield``.
"""
crosslink_committees = get_crosslink_committees_at_slot(state, attestation_data.slot)
crosslink_committee = [committee for committee, shard in crosslink_committees if shard == attestation_data.shard][0]
assert verify_bitfield(bitfield, len(crosslink_committee))
return sorted([index for i, index in enumerate(crosslink_committee) if get_bitfield_bit(bitfield, i) == 0b1])
```
### `int_to_bytes1`, `int_to_bytes2`, ...
`int_to_bytes1(x): return x.to_bytes(1, 'little')`, `int_to_bytes2(x): return x.to_bytes(2, 'little')`, and so on for all integers, particularly 1, 2, 3, 4, 8, 32, 48, 96.
### `bytes_to_int`
```python
def bytes_to_int(data: bytes) -> int:
return int.from_bytes(data, 'little')
```
### `get_total_balance`
```python
def get_total_balance(state: BeaconState, indices: List[ValidatorIndex]) -> Gwei:
"""
Return the combined effective balance of an array of ``validators``.
"""
return sum([state.validator_registry[index].effective_balance for index in indices])
```
### `get_domain`
```python
def get_domain(state: BeaconState,
domain_type: int,
message_epoch: int=None) -> int:
"""
Return the signature domain (fork version concatenated with domain type) of a message.
"""
epoch = get_current_epoch(state) if message_epoch is None else message_epoch
fork_version = state.fork.previous_version if epoch < state.fork.epoch else state.fork.current_version
return bytes_to_int(fork_version + int_to_bytes4(domain_type))
```
### `get_bitfield_bit`
```python
def get_bitfield_bit(bitfield: bytes, i: int) -> int:
"""
Extract the bit in ``bitfield`` at position ``i``.
"""
return (bitfield[i // 8] >> (i % 8)) % 2
```
### `verify_bitfield`
```python
def verify_bitfield(bitfield: bytes, committee_size: int) -> bool:
"""
Verify ``bitfield`` against the ``committee_size``.
"""
if len(bitfield) != (committee_size + 7) // 8:
return False
# Check `bitfield` is padded with zero bits only
for i in range(committee_size, len(bitfield) * 8):
if get_bitfield_bit(bitfield, i) == 0b1:
return False
return True
```
### `convert_to_indexed`
```python
def convert_to_indexed(state: BeaconState, attestation: Attestation) -> IndexedAttestation:
"""
Convert ``attestation`` to (almost) indexed-verifiable form.
"""
attesting_indices = get_attesting_indices(state, attestation.data, attestation.aggregation_bitfield)
custody_bit_1_indices = get_attesting_indices(state, attestation.data, attestation.custody_bitfield)
custody_bit_0_indices = [index for index in attesting_indices if index not in custody_bit_1_indices]
return IndexedAttestation(
custody_bit_0_indices=custody_bit_0_indices,
custody_bit_1_indices=custody_bit_1_indices,
data=attestation.data,
signature=attestation.signature,
)
```
### `verify_indexed_attestation`
```python
def verify_indexed_attestation(state: BeaconState, indexed_attestation: IndexedAttestation) -> bool:
"""
Verify validity of ``indexed_attestation`` fields.
"""
custody_bit_0_indices = indexed_attestation.custody_bit_0_indices
custody_bit_1_indices = indexed_attestation.custody_bit_1_indices
# Ensure no duplicate indices across custody bits
assert len(set(custody_bit_0_indices).intersection(set(custody_bit_1_indices))) == 0
if len(custody_bit_1_indices) > 0: # [TO BE REMOVED IN PHASE 1]
return False
if not (1 <= len(custody_bit_0_indices) + len(custody_bit_1_indices) <= MAX_INDICES_PER_ATTESTATION):
return False
if custody_bit_0_indices != sorted(custody_bit_0_indices):
return False
if custody_bit_1_indices != sorted(custody_bit_1_indices):
return False
return bls_verify_multiple(
pubkeys=[
bls_aggregate_pubkeys([state.validator_registry[i].pubkey for i in custody_bit_0_indices]),
bls_aggregate_pubkeys([state.validator_registry[i].pubkey for i in custody_bit_1_indices]),
],
message_hashes=[
hash_tree_root(AttestationDataAndCustodyBit(data=indexed_attestation.data, custody_bit=0b0)),
hash_tree_root(AttestationDataAndCustodyBit(data=indexed_attestation.data, custody_bit=0b1)),
],
signature=indexed_attestation.signature,
domain=get_domain(state, DOMAIN_ATTESTATION, slot_to_epoch(indexed_attestation.data.slot)),
)
```
### `is_double_vote`
```python
def is_double_vote(attestation_data_1: AttestationData,
attestation_data_2: AttestationData) -> bool:
"""
Check if ``attestation_data_1`` and ``attestation_data_2`` have the same target.
"""
target_epoch_1 = slot_to_epoch(attestation_data_1.slot)
target_epoch_2 = slot_to_epoch(attestation_data_2.slot)
return target_epoch_1 == target_epoch_2
```
### `is_surround_vote`
```python
def is_surround_vote(attestation_data_1: AttestationData,
attestation_data_2: AttestationData) -> bool:
"""
Check if ``attestation_data_1`` surrounds ``attestation_data_2``.
"""
source_epoch_1 = attestation_data_1.source_epoch
source_epoch_2 = attestation_data_2.source_epoch
target_epoch_1 = slot_to_epoch(attestation_data_1.slot)
target_epoch_2 = slot_to_epoch(attestation_data_2.slot)
return source_epoch_1 < source_epoch_2 and target_epoch_2 < target_epoch_1
```
### `integer_squareroot`
```python
def integer_squareroot(n: int) -> int:
"""
The largest integer ``x`` such that ``x**2`` is less than or equal to ``n``.
"""
assert n >= 0
x = n
y = (x + 1) // 2
while y < x:
x = y
y = (x + n // x) // 2
return x
```
### `get_delayed_activation_exit_epoch`
```python
def get_delayed_activation_exit_epoch(epoch: Epoch) -> Epoch:
"""
Return the epoch at which an activation or exit triggered in ``epoch`` takes effect.
"""
return epoch + 1 + ACTIVATION_EXIT_DELAY
```
### `get_churn_limit`
```python
def get_churn_limit(state: BeaconState) -> int:
return max(
MIN_PER_EPOCH_CHURN_LIMIT,
len(get_active_validator_indices(state, get_current_epoch(state))) // CHURN_LIMIT_QUOTIENT
)
```
### `bls_verify`
`bls_verify` is a function for verifying a BLS signature, defined in the [BLS Signature spec](../bls_signature.md#bls_verify).
### `bls_verify_multiple`
`bls_verify_multiple` is a function for verifying a BLS signature constructed from multiple messages, defined in the [BLS Signature spec](../bls_signature.md#bls_verify_multiple).
### `bls_aggregate_pubkeys`
`bls_aggregate_pubkeys` is a function for aggregating multiple BLS public keys into a single aggregate key, defined in the [BLS Signature spec](../bls_signature.md#bls_aggregate_pubkeys).
### Routines for updating validator status
Note: All functions in this section mutate `state`.
#### `initiate_validator_exit`
```python
def initiate_validator_exit(state: BeaconState, index: ValidatorIndex) -> None:
"""
Initiate the validator of the given ``index``.
Note that this function mutates ``state``.
"""
# Return if validator already initiated exit
validator = state.validator_registry[index]
if validator.exit_epoch != FAR_FUTURE_EPOCH:
return
# Compute exit queue epoch
exit_epochs = [v.exit_epoch for v in state.validator_registry if v.exit_epoch != FAR_FUTURE_EPOCH]
exit_queue_epoch = max(exit_epochs + [get_delayed_activation_exit_epoch(get_current_epoch(state))])
exit_queue_churn = len([v for v in state.validator_registry if v.exit_epoch == exit_queue_epoch])
if exit_queue_churn >= get_churn_limit(state):
exit_queue_epoch += 1
# Set validator exit epoch and withdrawable epoch
validator.exit_epoch = exit_queue_epoch
validator.withdrawable_epoch = validator.exit_epoch + MIN_VALIDATOR_WITHDRAWABILITY_DELAY
```
#### `slash_validator`
```python
def slash_validator(state: BeaconState, slashed_index: ValidatorIndex, whistleblower_index: ValidatorIndex=None) -> None:
"""
Slash the validator with index ``slashed_index``.
Note that this function mutates ``state``.
"""
current_epoch = get_current_epoch(state)
initiate_validator_exit(state, slashed_index)
state.validator_registry[slashed_index].slashed = True
state.validator_registry[slashed_index].withdrawable_epoch = current_epoch + LATEST_SLASHED_EXIT_LENGTH
slashed_balance = state.validator_registry[slashed_index].effective_balance
state.latest_slashed_balances[current_epoch % LATEST_SLASHED_EXIT_LENGTH] += slashed_balance
proposer_index = get_beacon_proposer_index(state)
if whistleblower_index is None:
whistleblower_index = proposer_index
whistleblowing_reward = slashed_balance // WHISTLEBLOWING_REWARD_QUOTIENT
proposer_reward = whistleblowing_reward // PROPOSER_REWARD_QUOTIENT
increase_balance(state, proposer_index, proposer_reward)
increase_balance(state, whistleblower_index, whistleblowing_reward - proposer_reward)
decrease_balance(state, slashed_index, whistleblowing_reward)
```
## On genesis
When enough full deposits have been made to the deposit contract, an `Eth2Genesis` log is emitted. Construct a corresponding `genesis_state` and `genesis_block` as follows:
* Let `genesis_validator_deposits` be the list of deposits, ordered chronologically, up to and including the deposit that triggered the `Eth2Genesis` log.
* Let `genesis_time` be the timestamp specified in the `Eth2Genesis` log.
* Let `genesis_eth1_data` be the `Eth1Data` object where:
* `genesis_eth1_data.deposit_root` is the `deposit_root` contained in the `Eth2Genesis` log.
* `genesis_eth1_data.deposit_count` is the `deposit_count` contained in the `Eth2Genesis` log.
* `genesis_eth1_data.block_hash` is the hash of the Ethereum 1.0 block that emitted the `Eth2Genesis` log.
* Let `genesis_state = get_genesis_beacon_state(genesis_validator_deposits, genesis_time, genesis_eth1_data)`.
* Let `genesis_block = BeaconBlock(state_root=hash_tree_root(genesis_state))`.
```python
def get_genesis_beacon_state(genesis_validator_deposits: List[Deposit],
genesis_time: int,
genesis_eth1_data: Eth1Data) -> BeaconState:
"""
Get the genesis ``BeaconState``.
"""
state = BeaconState(genesis_time=genesis_time, latest_eth1_data=genesis_eth1_data)
# Process genesis deposits
for deposit in genesis_validator_deposits:
process_deposit(state, deposit)
# Process genesis activations
for index, validator in enumerate(state.validator_registry):
if validator.effective_balance >= MAX_EFFECTIVE_BALANCE:
validator.activation_eligibility_epoch = GENESIS_EPOCH
validator.activation_epoch = GENESIS_EPOCH
genesis_active_index_root = hash_tree_root(get_active_validator_indices(state, GENESIS_EPOCH))
for index in range(LATEST_ACTIVE_INDEX_ROOTS_LENGTH):
state.latest_active_index_roots[index] = genesis_active_index_root
return state
```
## Beacon chain state transition function
We now define the state transition function. At a high level, the state transition is made up of four parts:
1. State caching, which happens at the start of every slot.
2. The per-epoch transitions, which happens at the start of the first slot of every epoch.
3. The per-slot transitions, which happens at every slot.
4. The per-block transitions, which happens at every block.
Transition section notes:
* The state caching caches the state root of the previous slot and updates block and state roots records.
* The per-epoch transitions focus on the [validator](#dfn-validator) registry, including adjusting balances and activating and exiting [validators](#dfn-validator), as well as processing crosslinks and managing block justification/finalization.
* The per-slot transitions focus on the slot counter.
* The per-block transitions generally focus on verifying aggregate signatures and saving temporary records relating to the per-block activity in the `BeaconState`.
Beacon blocks that trigger unhandled Python exceptions (e.g. out-of-range list accesses) and failed `assert`s during the state transition are considered invalid.
Note: If there are skipped slots between a block and its parent block, run the steps in the [state-root](#state-caching), [per-epoch](#per-epoch-processing), and [per-slot](#per-slot-processing) sections once for each skipped slot and then once for the slot containing the new block.
### State caching
At every `slot > GENESIS_SLOT` run the following function:
```python
def cache_state(state: BeaconState) -> None:
# Cache latest known state root (for previous slot)
latest_state_root = hash_tree_root(state)
state.latest_state_roots[state.slot % SLOTS_PER_HISTORICAL_ROOT] = latest_state_root
# Store latest known state root (for previous slot) in latest_block_header if it is empty
if state.latest_block_header.state_root == ZERO_HASH:
state.latest_block_header.state_root = latest_state_root
# Cache latest known block root (for previous slot)
latest_block_root = signing_root(state.latest_block_header)
state.latest_block_roots[state.slot % SLOTS_PER_HISTORICAL_ROOT] = latest_block_root
```
### Per-epoch processing
The steps below happen when `state.slot > GENESIS_SLOT and (state.slot + 1) % SLOTS_PER_EPOCH == 0`.
#### Helper functions
We define epoch transition helper functions:
```python
def get_total_active_balance(state: BeaconState) -> Gwei:
return get_total_balance(state, get_active_validator_indices(state, get_current_epoch(state)))
```
```python
def get_matching_source_attestations(state: BeaconState, epoch: Epoch) -> List[PendingAttestation]:
assert epoch in (get_current_epoch(state), get_previous_epoch(state))
return state.current_epoch_attestations if epoch == get_current_epoch(state) else state.previous_epoch_attestations
```
```python
def get_matching_target_attestations(state: BeaconState, epoch: Epoch) -> List[PendingAttestation]:
return [
a for a in get_matching_source_attestations(state, epoch)
if a.data.target_root == get_block_root(state, epoch)
]
```
```python
def get_matching_head_attestations(state: BeaconState, epoch: Epoch) -> List[PendingAttestation]:
return [
a for a in get_matching_source_attestations(state, epoch)
if a.data.beacon_block_root == get_block_root_at_slot(state, a.data.slot)
]
```
```python
def get_unslashed_attesting_indices(state: BeaconState, attestations: List[PendingAttestation]) -> List[ValidatorIndex]:
output = set()
for a in attestations:
output = output.union(get_attesting_indices(state, a.data, a.aggregation_bitfield))
return sorted(filter(lambda index: not state.validator_registry[index].slashed, list(output)))
```
```python
def get_attesting_balance(state: BeaconState, attestations: List[PendingAttestation]) -> Gwei:
return get_total_balance(state, get_unslashed_attesting_indices(state, attestations))
```
```python
def get_crosslink_from_attestation_data(state: BeaconState, data: AttestationData) -> Crosslink:
return Crosslink(
epoch=min(slot_to_epoch(data.slot), state.current_crosslinks[data.shard].epoch + MAX_CROSSLINK_EPOCHS),
previous_crosslink_root=data.previous_crosslink_root,
crosslink_data_root=data.crosslink_data_root,
)
```
```python
def get_winning_crosslink_and_attesting_indices(state: BeaconState, shard: Shard, epoch: Epoch) -> Tuple[Crosslink, List[ValidatorIndex]]:
shard_attestations = [a for a in get_matching_source_attestations(state, epoch) if a.data.shard == shard]
shard_crosslinks = [get_crosslink_from_attestation_data(state, a.data) for a in shard_attestations]
candidate_crosslinks = [
c for c in shard_crosslinks
if hash_tree_root(state.current_crosslinks[shard]) in (c.previous_crosslink_root, hash_tree_root(c))
]
if len(candidate_crosslinks) == 0:
return Crosslink(epoch=GENESIS_EPOCH, previous_crosslink_root=ZERO_HASH, crosslink_data_root=ZERO_HASH), []
def get_attestations_for(crosslink: Crosslink) -> List[PendingAttestation]:
return [a for a in shard_attestations if get_crosslink_from_attestation_data(state, a.data) == crosslink]
# Winning crosslink has the crosslink data root with the most balance voting for it (ties broken lexicographically)
winning_crosslink = max(candidate_crosslinks, key=lambda crosslink: (
get_attesting_balance(state, get_attestations_for(crosslink)), crosslink.crosslink_data_root
))
return winning_crosslink, get_unslashed_attesting_indices(state, get_attestations_for(winning_crosslink))
```
```python
def get_earliest_attestation(state: BeaconState, attestations: List[PendingAttestation], index: ValidatorIndex) -> PendingAttestation:
return min([
a for a in attestations if index in get_attesting_indices(state, a.data, a.aggregation_bitfield)
], key=lambda a: a.inclusion_slot)
```
#### Justification and finalization
Run the following function:
```python
def process_justification_and_finalization(state: BeaconState) -> None:
if get_current_epoch(state) <= GENESIS_EPOCH + 1:
return
previous_epoch = get_previous_epoch(state)
current_epoch = get_current_epoch(state)
old_previous_justified_epoch = state.previous_justified_epoch
old_current_justified_epoch = state.current_justified_epoch
# Process justifications
state.previous_justified_epoch = state.current_justified_epoch
state.previous_justified_root = state.current_justified_root
state.justification_bitfield = (state.justification_bitfield << 1) % 2**64
previous_epoch_matching_target_balance = get_attesting_balance(state, get_matching_target_attestations(state, previous_epoch))
if previous_epoch_matching_target_balance * 3 >= get_total_active_balance(state) * 2:
state.current_justified_epoch = previous_epoch
state.current_justified_root = get_block_root(state, state.current_justified_epoch)
state.justification_bitfield |= (1 << 1)
current_epoch_matching_target_balance = get_attesting_balance(state, get_matching_target_attestations(state, current_epoch))
if current_epoch_matching_target_balance * 3 >= get_total_active_balance(state) * 2:
state.current_justified_epoch = current_epoch
state.current_justified_root = get_block_root(state, state.current_justified_epoch)
state.justification_bitfield |= (1 << 0)
# Process finalizations
bitfield = state.justification_bitfield
# The 2nd/3rd/4th most recent epochs are justified, the 2nd using the 4th as source
if (bitfield >> 1) % 8 == 0b111 and old_previous_justified_epoch == current_epoch - 3:
state.finalized_epoch = old_previous_justified_epoch
state.finalized_root = get_block_root(state, state.finalized_epoch)
# The 2nd/3rd most recent epochs are justified, the 2nd using the 3rd as source
if (bitfield >> 1) % 4 == 0b11 and old_previous_justified_epoch == current_epoch - 2:
state.finalized_epoch = old_previous_justified_epoch
state.finalized_root = get_block_root(state, state.finalized_epoch)
# The 1st/2nd/3rd most recent epochs are justified, the 1st using the 3rd as source
if (bitfield >> 0) % 8 == 0b111 and old_current_justified_epoch == current_epoch - 2:
state.finalized_epoch = old_current_justified_epoch
state.finalized_root = get_block_root(state, state.finalized_epoch)
# The 1st/2nd most recent epochs are justified, the 1st using the 2nd as source
if (bitfield >> 0) % 4 == 0b11 and old_current_justified_epoch == current_epoch - 1:
state.finalized_epoch = old_current_justified_epoch
state.finalized_root = get_block_root(state, state.finalized_epoch)
```
#### Crosslinks
Run the following function:
```python
def process_crosslinks(state: BeaconState) -> None:
state.previous_crosslinks = [c for c in state.current_crosslinks]
previous_epoch = get_previous_epoch(state)
next_epoch = get_current_epoch(state) + 1
for slot in range(get_epoch_start_slot(previous_epoch), get_epoch_start_slot(next_epoch)):
epoch = slot_to_epoch(slot)
for crosslink_committee, shard in get_crosslink_committees_at_slot(state, slot):
winning_crosslink, attesting_indices = get_winning_crosslink_and_attesting_indices(state, shard, epoch)
if 3 * get_total_balance(state, attesting_indices) >= 2 * get_total_balance(state, crosslink_committee):
state.current_crosslinks[shard] = winning_crosslink
```
#### Rewards and penalties
First, we define additional helpers:
```python
def get_base_reward(state: BeaconState, index: ValidatorIndex) -> Gwei:
adjusted_quotient = integer_squareroot(get_total_active_balance(state)) // BASE_REWARD_QUOTIENT
if adjusted_quotient == 0:
return 0
return state.validator_registry[index].effective_balance // adjusted_quotient // BASE_REWARDS_PER_EPOCH
```
```python
def get_attestation_deltas(state: BeaconState) -> Tuple[List[Gwei], List[Gwei]]:
previous_epoch = get_previous_epoch(state)
total_balance = get_total_active_balance(state)
rewards = [0 for index in range(len(state.validator_registry))]
penalties = [0 for index in range(len(state.validator_registry))]
eligible_validator_indices = [
index for index, v in enumerate(state.validator_registry)
if is_active_validator(v, previous_epoch) or (v.slashed and previous_epoch + 1 < v.withdrawable_epoch)
]
# Micro-incentives for matching FFG source, FFG target, and head
matching_source_attestations = get_matching_source_attestations(state, previous_epoch)
matching_target_attestations = get_matching_target_attestations(state, previous_epoch)
matching_head_attestations = get_matching_head_attestations(state, previous_epoch)
for attestations in (matching_source_attestations, matching_target_attestations, matching_head_attestations):
unslashed_attesting_indices = get_unslashed_attesting_indices(state, attestations)
attesting_balance = get_attesting_balance(state, attestations)
for index in eligible_validator_indices:
if index in unslashed_attesting_indices:
rewards[index] += get_base_reward(state, index) * attesting_balance // total_balance
else:
penalties[index] += get_base_reward(state, index)
# Proposer and inclusion delay micro-rewards
for index in get_unslashed_attesting_indices(state, matching_source_attestations):
earliest_attestation = get_earliest_attestation(state, matching_source_attestations, index)
rewards[earliest_attestation.proposer_index] += get_base_reward(state, index) // PROPOSER_REWARD_QUOTIENT
inclusion_delay = earliest_attestation.inclusion_slot - earliest_attestation.data.slot
rewards[index] += get_base_reward(state, index) * MIN_ATTESTATION_INCLUSION_DELAY // inclusion_delay
# Inactivity penalty
finality_delay = previous_epoch - state.finalized_epoch
if finality_delay > MIN_EPOCHS_TO_INACTIVITY_PENALTY:
matching_target_attesting_indices = get_unslashed_attesting_indices(state, matching_target_attestations)
for index in eligible_validator_indices:
penalties[index] += BASE_REWARDS_PER_EPOCH * get_base_reward(state, index)
if index not in matching_target_attesting_indices:
penalties[index] += state.validator_registry[index].effective_balance * finality_delay // INACTIVITY_PENALTY_QUOTIENT
return [rewards, penalties]
```
```python
def get_crosslink_deltas(state: BeaconState) -> Tuple[List[Gwei], List[Gwei]]:
rewards = [0 for index in range(len(state.validator_registry))]
penalties = [0 for index in range(len(state.validator_registry))]
for slot in range(get_epoch_start_slot(get_previous_epoch(state)), get_epoch_start_slot(get_current_epoch(state))):
epoch = slot_to_epoch(slot)
for crosslink_committee, shard in get_crosslink_committees_at_slot(state, slot):
winning_crosslink, attesting_indices = get_winning_crosslink_and_attesting_indices(state, shard, epoch)
attesting_balance = get_total_balance(state, attesting_indices)
committee_balance = get_total_balance(state, crosslink_committee)
for index in crosslink_committee:
base_reward = get_base_reward(state, index)
if index in attesting_indices:
rewards[index] += base_reward * attesting_balance // committee_balance
else:
penalties[index] += base_reward
return [rewards, penalties]
```
Run the following function:
```python
def process_rewards_and_penalties(state: BeaconState) -> None:
if get_current_epoch(state) == GENESIS_EPOCH:
return
rewards1, penalties1 = get_attestation_deltas(state)
rewards2, penalties2 = get_crosslink_deltas(state)
for i in range(len(state.validator_registry)):
increase_balance(state, i, rewards1[i] + rewards2[i])
decrease_balance(state, i, penalties1[i] + penalties2[i])
```
#### Registry updates
Run the following function:
```python
def process_registry_updates(state: BeaconState) -> None:
# Process activation eligibility and ejections
for index, validator in enumerate(state.validator_registry):
if validator.activation_eligibility_epoch == FAR_FUTURE_EPOCH and validator.effective_balance >= MAX_EFFECTIVE_BALANCE:
validator.activation_eligibility_epoch = get_current_epoch(state)
if is_active_validator(validator, get_current_epoch(state)) and validator.effective_balance <= EJECTION_BALANCE:
initiate_validator_exit(state, index)
# Queue validators eligible for activation and not dequeued for activation prior to finalized epoch
activation_queue = sorted([
index for index, validator in enumerate(state.validator_registry) if
validator.activation_eligibility_epoch != FAR_FUTURE_EPOCH and
validator.activation_epoch >= get_delayed_activation_exit_epoch(state.finalized_epoch)
], key=lambda index: state.validator_registry[index].activation_eligibility_epoch)
# Dequeued validators for activation up to churn limit (without resetting activation epoch)
for index in activation_queue[:get_churn_limit(state)]:
if validator.activation_epoch == FAR_FUTURE_EPOCH:
validator.activation_epoch = get_delayed_activation_exit_epoch(get_current_epoch(state))
```
#### Slashings
Run the following function:
```python
def process_slashings(state: BeaconState) -> None:
current_epoch = get_current_epoch(state)
active_validator_indices = get_active_validator_indices(state, current_epoch)
total_balance = get_total_balance(state, active_validator_indices)
# Compute `total_penalties`
total_at_start = state.latest_slashed_balances[(current_epoch + 1) % LATEST_SLASHED_EXIT_LENGTH]
total_at_end = state.latest_slashed_balances[current_epoch % LATEST_SLASHED_EXIT_LENGTH]
total_penalties = total_at_end - total_at_start
for index, validator in enumerate(state.validator_registry):
if validator.slashed and current_epoch == validator.withdrawable_epoch - LATEST_SLASHED_EXIT_LENGTH // 2:
penalty = max(
validator.effective_balance * min(total_penalties * 3, total_balance) // total_balance,
validator.effective_balance // MIN_SLASHING_PENALTY_QUOTIENT
)
decrease_balance(state, index, penalty)
```
#### Final updates
Run the following function:
```python
def process_final_updates(state: BeaconState) -> None:
current_epoch = get_current_epoch(state)
next_epoch = current_epoch + 1
# Reset eth1 data votes
if (state.slot + 1) % SLOTS_PER_ETH1_VOTING_PERIOD == 0:
state.eth1_data_votes = []
# Update effective balances with hysteresis
for index, validator in enumerate(state.validator_registry):
balance = min(state.balances[index], MAX_EFFECTIVE_BALANCE)
HALF_INCREMENT = EFFECTIVE_BALANCE_INCREMENT // 2
if balance < validator.effective_balance or validator.effective_balance + 3 * HALF_INCREMENT < balance:
validator.effective_balance = balance - balance % EFFECTIVE_BALANCE_INCREMENT
# Update start shard
state.latest_start_shard = (state.latest_start_shard + get_shard_delta(state, current_epoch)) % SHARD_COUNT
# Set active index root
index_root_position = (next_epoch + ACTIVATION_EXIT_DELAY) % LATEST_ACTIVE_INDEX_ROOTS_LENGTH
state.latest_active_index_roots[index_root_position] = hash_tree_root(
get_active_validator_indices(state, next_epoch + ACTIVATION_EXIT_DELAY)
)
# Set total slashed balances
state.latest_slashed_balances[next_epoch % LATEST_SLASHED_EXIT_LENGTH] = (
state.latest_slashed_balances[current_epoch % LATEST_SLASHED_EXIT_LENGTH]
)
# Set randao mix
state.latest_randao_mixes[next_epoch % LATEST_RANDAO_MIXES_LENGTH] = get_randao_mix(state, current_epoch)
# Set historical root accumulator
if next_epoch % (SLOTS_PER_HISTORICAL_ROOT // SLOTS_PER_EPOCH) == 0:
historical_batch = HistoricalBatch(
block_roots=state.latest_block_roots,
state_roots=state.latest_state_roots,
)
state.historical_roots.append(hash_tree_root(historical_batch))
# Rotate current/previous epoch attestations
state.previous_epoch_attestations = state.current_epoch_attestations
state.current_epoch_attestations = []
```
### Per-slot processing
At every `slot > GENESIS_SLOT` run the following function:
```python
def advance_slot(state: BeaconState) -> None:
state.slot += 1
```
### Per-block processing
For every `block` except the genesis block, run `process_block_header(state, block)`, `process_randao(state, block)` and `process_eth1_data(state, block)`.
#### Block header
```python
def process_block_header(state: BeaconState, block: BeaconBlock) -> None:
# Verify that the slots match
assert block.slot == state.slot
# Verify that the parent matches
assert block.previous_block_root == signing_root(state.latest_block_header)
# Save current block as the new latest block
state.latest_block_header = BeaconBlockHeader(
slot=block.slot,
previous_block_root=block.previous_block_root,
block_body_root=hash_tree_root(block.body),
)
# Verify proposer is not slashed
proposer = state.validator_registry[get_beacon_proposer_index(state)]
assert not proposer.slashed
# Verify proposer signature
assert bls_verify(proposer.pubkey, signing_root(block), block.signature, get_domain(state, DOMAIN_BEACON_PROPOSER))
```
#### RANDAO
```python
def process_randao(state: BeaconState, block: BeaconBlock) -> None:
proposer = state.validator_registry[get_beacon_proposer_index(state)]
# Verify that the provided randao value is valid
assert bls_verify(proposer.pubkey, hash_tree_root(get_current_epoch(state)), block.body.randao_reveal, get_domain(state, DOMAIN_RANDAO))
# Mix it in
state.latest_randao_mixes[get_current_epoch(state) % LATEST_RANDAO_MIXES_LENGTH] = (
xor(get_randao_mix(state, get_current_epoch(state)),
hash(block.body.randao_reveal))
)
```
#### Eth1 data
```python
def process_eth1_data(state: BeaconState, block: BeaconBlock) -> None:
state.eth1_data_votes.append(block.body.eth1_data)
if state.eth1_data_votes.count(block.body.eth1_data) * 2 > SLOTS_PER_ETH1_VOTING_PERIOD:
state.latest_eth1_data = block.body.eth1_data
```
#### Operations
##### Proposer slashings
Verify that `len(block.body.proposer_slashings) <= MAX_PROPOSER_SLASHINGS`.
For each `proposer_slashing` in `block.body.proposer_slashings`, run the following function:
```python
def process_proposer_slashing(state: BeaconState,
proposer_slashing: ProposerSlashing) -> None:
"""
Process ``ProposerSlashing`` operation.
Note that this function mutates ``state``.
"""
proposer = state.validator_registry[proposer_slashing.proposer_index]
# Verify that the epoch is the same
assert slot_to_epoch(proposer_slashing.header_1.slot) == slot_to_epoch(proposer_slashing.header_2.slot)
# But the headers are different
assert proposer_slashing.header_1 != proposer_slashing.header_2
# Check proposer is slashable
assert is_slashable_validator(proposer, get_current_epoch(state))
# Signatures are valid
for header in (proposer_slashing.header_1, proposer_slashing.header_2):
domain = get_domain(state, DOMAIN_BEACON_PROPOSER, slot_to_epoch(header.slot))
assert bls_verify(proposer.pubkey, signing_root(header), header.signature, domain)
slash_validator(state, proposer_slashing.proposer_index)
```
##### Attester slashings
Verify that `len(block.body.attester_slashings) <= MAX_ATTESTER_SLASHINGS`.
For each `attester_slashing` in `block.body.attester_slashings`, run the following function:
```python
def process_attester_slashing(state: BeaconState,
attester_slashing: AttesterSlashing) -> None:
"""
Process ``AttesterSlashing`` operation.
Note that this function mutates ``state``.
"""
attestation1 = attester_slashing.attestation_1
attestation2 = attester_slashing.attestation_2
# Check that the attestations are conflicting
assert attestation1.data != attestation2.data
assert (
is_double_vote(attestation1.data, attestation2.data) or
is_surround_vote(attestation1.data, attestation2.data)
)
assert verify_indexed_attestation(state, attestation1)
assert verify_indexed_attestation(state, attestation2)
attesting_indices_1 = attestation1.custody_bit_0_indices + attestation1.custody_bit_1_indices
attesting_indices_2 = attestation2.custody_bit_0_indices + attestation2.custody_bit_1_indices
slashable_indices = [
index for index in attesting_indices_1
if (
index in attesting_indices_2 and
is_slashable_validator(state.validator_registry[index], get_current_epoch(state))
)
]
assert len(slashable_indices) >= 1
for index in slashable_indices:
slash_validator(state, index)
```
##### Attestations
Verify that `len(block.body.attestations) <= MAX_ATTESTATIONS`.
For each `attestation` in `block.body.attestations`, run the following function:
```python
def process_attestation(state: BeaconState, attestation: Attestation) -> None:
"""
Process ``Attestation`` operation.
Note that this function mutates ``state``.
"""
data = attestation.data
min_slot = state.slot - SLOTS_PER_EPOCH if get_current_epoch(state) > GENESIS_EPOCH else GENESIS_SLOT
assert min_slot <= data.slot <= state.slot - MIN_ATTESTATION_INCLUSION_DELAY
# Check target epoch, source epoch, source root, and source crosslink
target_epoch = slot_to_epoch(data.slot)
assert (target_epoch, data.source_epoch, data.source_root, data.previous_crosslink_root) in {
(get_current_epoch(state), state.current_justified_epoch, state.current_justified_root, hash_tree_root(state.current_crosslinks[data.shard])),
(get_previous_epoch(state), state.previous_justified_epoch, state.previous_justified_root, hash_tree_root(state.previous_crosslinks[data.shard])),
}
# Check crosslink data root
assert data.crosslink_data_root == ZERO_HASH # [to be removed in phase 1]
# Check signature and bitfields
assert verify_indexed_attestation(state, convert_to_indexed(state, attestation))
# Cache pending attestation
pending_attestation = PendingAttestation(
data=data,
aggregation_bitfield=attestation.aggregation_bitfield,
inclusion_slot=state.slot,
proposer_index=get_beacon_proposer_index(state),
)
if target_epoch == get_current_epoch(state):
state.current_epoch_attestations.append(pending_attestation)
else:
state.previous_epoch_attestations.append(pending_attestation)
```
##### Deposits
Verify that `len(block.body.deposits) == min(MAX_DEPOSITS, state.latest_eth1_data.deposit_count - state.deposit_index)`.
For each `deposit` in `block.body.deposits`, run the following function:
```python
def process_deposit(state: BeaconState, deposit: Deposit) -> None:
"""
Process an Eth1 deposit, registering a validator or increasing its balance.
Note that this function mutates ``state``.
"""
# Verify the Merkle branch
assert verify_merkle_branch(
leaf=hash_tree_root(deposit.data),
proof=deposit.proof,
depth=DEPOSIT_CONTRACT_TREE_DEPTH,
index=deposit.index,
root=state.latest_eth1_data.deposit_root,
)
# Deposits must be processed in order
assert deposit.index == state.deposit_index
state.deposit_index += 1
pubkey = deposit.data.pubkey
amount = deposit.data.amount
validator_pubkeys = [v.pubkey for v in state.validator_registry]
if pubkey not in validator_pubkeys:
# Verify the deposit signature (proof of possession)
if not bls_verify(pubkey, signing_root(deposit.data), deposit.data.signature, get_domain(state, DOMAIN_DEPOSIT)):
return
# Add validator and balance entries
state.validator_registry.append(Validator(
pubkey=pubkey,
withdrawal_credentials=deposit.data.withdrawal_credentials,
activation_eligibility_epoch=FAR_FUTURE_EPOCH,
activation_epoch=FAR_FUTURE_EPOCH,
exit_epoch=FAR_FUTURE_EPOCH,
withdrawable_epoch=FAR_FUTURE_EPOCH,
effective_balance=amount - amount % EFFECTIVE_BALANCE_INCREMENT
))
state.balances.append(amount)
else:
# Increase balance by deposit amount
index = validator_pubkeys.index(pubkey)
increase_balance(state, index, amount)
```
##### Voluntary exits
Verify that `len(block.body.voluntary_exits) <= MAX_VOLUNTARY_EXITS`.
For each `exit` in `block.body.voluntary_exits`, run the following function:
```python
def process_voluntary_exit(state: BeaconState, exit: VoluntaryExit) -> None:
"""
Process ``VoluntaryExit`` operation.
Note that this function mutates ``state``.
"""
validator = state.validator_registry[exit.validator_index]
# Verify the validator is active
assert is_active_validator(validator, get_current_epoch(state))
# Verify the validator has not yet exited
assert validator.exit_epoch == FAR_FUTURE_EPOCH
# Exits must specify an epoch when they become valid; they are not valid before then
assert get_current_epoch(state) >= exit.epoch
# Verify the validator has been active long enough
assert get_current_epoch(state) - validator.activation_epoch >= PERSISTENT_COMMITTEE_PERIOD
# Verify signature
domain = get_domain(state, DOMAIN_VOLUNTARY_EXIT, exit.epoch)
assert bls_verify(validator.pubkey, signing_root(exit), exit.signature, domain)
# Initiate exit
initiate_validator_exit(state, exit.validator_index)
```
##### Transfers
Verify that `len(block.body.transfers) <= MAX_TRANSFERS` and that all transfers are distinct.
For each `transfer` in `block.body.transfers`, run the following function:
```python
def process_transfer(state: BeaconState, transfer: Transfer) -> None:
"""
Process ``Transfer`` operation.
Note that this function mutates ``state``.
"""
# Verify the amount and fee are not individually too big (for anti-overflow purposes)
assert state.balances[transfer.sender] >= max(transfer.amount, transfer.fee)
# A transfer is valid in only one slot
assert state.slot == transfer.slot
# Sender must be not yet eligible for activation, withdrawn, or transfer balance over MAX_EFFECTIVE_BALANCE
assert (
state.validator_registry[transfer.sender].activation_eligibility_epoch == FAR_FUTURE_EPOCH or
get_current_epoch(state) >= state.validator_registry[transfer.sender].withdrawable_epoch or
transfer.amount + transfer.fee + MAX_EFFECTIVE_BALANCE <= state.balances[transfer.sender]
)
# Verify that the pubkey is valid
assert (
state.validator_registry[transfer.sender].withdrawal_credentials ==
BLS_WITHDRAWAL_PREFIX_BYTE + hash(transfer.pubkey)[1:]
)
# Verify that the signature is valid
assert bls_verify(transfer.pubkey, signing_root(transfer), transfer.signature, get_domain(state, DOMAIN_TRANSFER))
# Process the transfer
decrease_balance(state, transfer.sender, transfer.amount + transfer.fee)
increase_balance(state, transfer.recipient, transfer.amount)
increase_balance(state, get_beacon_proposer_index(state), transfer.fee)
# Verify balances are not dust
assert not (0 < state.balances[transfer.sender] < MIN_DEPOSIT_AMOUNT)
assert not (0 < state.balances[transfer.recipient] < MIN_DEPOSIT_AMOUNT)
```
#### State root verification
Verify the block's `state_root` by running the following function:
```python
def verify_block_state_root(state: BeaconState, block: BeaconBlock) -> None:
assert block.state_root == hash_tree_root(state)
```
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