# Ethereum 2.0 Phase 0 -- The Beacon Chain
**Notice**: This document is a work-in-progress for researchers and implementers.
## Table of contents
- [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)
- [Custom types](#custom-types)
- [Constants](#constants)
- [Configuration](#configuration)
- [Misc](#misc)
- [Gwei values](#gwei-values)
- [Initial values](#initial-values)
- [Time parameters](#time-parameters)
- [State list lengths](#state-list-lengths)
- [Rewards and penalties](#rewards-and-penalties)
- [Max operations per block](#max-operations-per-block)
- [Signature domains](#signature-domains)
- [Containers](#containers)
- [Misc dependencies](#misc-dependencies)
- [`Fork`](#fork)
- [`Validator`](#validator)
- [`Crosslink`](#crosslink)
- [`AttestationData`](#attestationdata)
- [`AttestationDataAndCustodyBit`](#attestationdataandcustodybit)
- [`IndexedAttestation`](#indexedattestation)
- [`PendingAttestation`](#pendingattestation)
- [`Eth1Data`](#eth1data)
- [`HistoricalBatch`](#historicalbatch)
- [`DepositData`](#depositdata)
- [`BeaconBlockHeader`](#beaconblockheader)
- [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)
- [Helper functions](#helper-functions)
- [`xor`](#xor)
- [`hash`](#hash)
- [`hash_tree_root`](#hash_tree_root)
- [`signing_root`](#signing_root)
- [`calc_merkle_tree_from_leaves`](#calc_merkle_tree_from_leaves)
- [`get_merkle_root`](#get_merkle_root)
- [`bls_domain`](#bls_domain)
- [`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_epoch_committee_count`](#get_epoch_committee_count)
- [`get_shard_delta`](#get_shard_delta)
- [`get_epoch_start_shard`](#get_epoch_start_shard)
- [`get_attestation_data_slot`](#get_attestation_data_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_shuffled_index`](#get_shuffled_index)
- [`compute_committee`](#compute_committee)
- [`get_crosslink_committee`](#get_crosslink_committee)
- [`get_attesting_indices`](#get_attesting_indices)
- [`int_to_bytes`](#int_to_bytes)
- [`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)
- [`validate_indexed_attestation`](#validate_indexed_attestation)
- [`is_slashable_attestation_data`](#is_slashable_attestation_data)
- [`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)
- [Genesis](#genesis)
- [Genesis trigger](#genesis-trigger)
- [Genesis state](#genesis-state)
- [Genesis block](#genesis-block)
- [Beacon chain state transition function](#beacon-chain-state-transition-function)
- [Epoch processing](#epoch-processing)
- [Helper functions](#helper-functions-1)
- [Justification and finalization](#justification-and-finalization)
- [Crosslinks](#crosslinks)
- [Rewards and penalties](#rewards-and-penalties-1)
- [Registry updates](#registry-updates)
- [Slashings](#slashings)
- [Final updates](#final-updates)
- [Block processing](#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)
## 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 Eth 1.0. Activation as a [validator](#dfn-validator) happens when Eth 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 registered participant in the beacon chain. You can become one by sending ether into the Eth 1.0 deposit contract.
* **Active validator**—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 proof-of-stake 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.
## 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 |
| `Version` | `Bytes4` | a fork version number |
| `Hash` | `Bytes32` | a hashed result |
| `BLSPubkey` | `Bytes48` | a BLS12-381 public key |
| `BLSSignature` | `Bytes96` | a BLS12-381 signature |
## Constants
The following values are (non-configurable) constants used throughout the specification.
| Name | Value |
| - | - |
| `FAR_FUTURE_EPOCH` | `Epoch(2**64 - 1)` |
| `ZERO_HASH` | `Hash(b'\x00' * 32)` |
| `BASE_REWARDS_PER_EPOCH` | `5` |
| `DEPOSIT_CONTRACT_TREE_DEPTH` | `2**5` (= 32) |
## Configuration
*Note*: The default mainnet configuration values 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`](../../configs/constant_presets) directory. These configurations are updated for releases and 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) |
| `SHUFFLE_ROUND_COUNT` | `90` |
| `GENESIS_ACTIVE_VALIDATOR_COUNT` | `2**16` (= 65,536) |
| `MIN_GENESIS_TIME` | `1578009600` (Jan 3, 2020) |
* 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.)
### Gwei values
| Name | Value |
| - | - |
| `MIN_DEPOSIT_AMOUNT` | `Gwei(2**0 * 10**9)` (= 1,000,000,000) |
| `MAX_EFFECTIVE_BALANCE` | `Gwei(2**5 * 10**9)` (= 32,000,000,000) |
| `EJECTION_BALANCE` | `Gwei(2**4 * 10**9)` (= 16,000,000,000) |
| `EFFECTIVE_BALANCE_INCREMENT` | `Gwei(2**0 * 10**9)` (= 1,000,000,000) |
### Initial values
| Name | Value |
| - | - |
| `GENESIS_SLOT` | `Slot(0)` |
| `GENESIS_EPOCH` | `Epoch(0)` |
| `BLS_WITHDRAWAL_PREFIX` | `0` |
### Time parameters
| Name | Value | Unit | Duration |
| - | - | :-: | :-: |
| `MIN_ATTESTATION_INCLUSION_DELAY` | `2**0` (= 1) | slots | 6 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_EPOCHS_PER_CROSSLINK` | `2**6` (= 64) | epochs | ~7 hours |
| `MIN_EPOCHS_TO_INACTIVITY_PENALTY` | `2**2` (= 4) | epochs | 25.6 minutes |
* `MAX_EPOCHS_PER_CROSSLINK` should be a small constant times `SHARD_COUNT // SLOTS_PER_EPOCH`.
### State list lengths
| Name | Value | Unit | Duration |
| - | - | :-: | :-: |
| `EPOCHS_PER_HISTORICAL_VECTOR` | `2**16` (= 65,536) | epochs | ~0.8 years |
| `EPOCHS_PER_SLASHED_BALANCES_VECTOR` | `2**13` (= 8,192) | epochs | ~36 days |
| `HISTORICAL_ROOTS_LIMIT` | `2**24` (= 16,777,216) | historical roots | ~26,131 years |
| `VALIDATOR_REGISTRY_LIMIT` | `2**40` (= 1,099,511,627,776) | validator spots | |
### Rewards and penalties
| Name | Value |
| - | - |
| `BASE_REWARD_FACTOR` | `2**6` (= 64) |
| `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 `INACTIVITY_PENALTY_QUOTIENT` equals `INVERSE_SQRT_E_DROP_TIME**2` where `INVERSE_SQRT_E_DROP_TIME := 2**12 epochs` (about 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` |
## Containers
The following types are [SimpleSerialize (SSZ)](../simple-serialize.md) containers.
*Note*: The definitions are ordered topologically to facilitate execution of the spec.
*Note*: Fields missing in container instantiations default to their zero value.
### Misc dependencies
#### `Fork`
```python
class Fork(Container):
previous_version: Version
current_version: Version
epoch: Epoch # Epoch of latest fork
```
#### `Validator`
```python
class Validator(Container):
pubkey: BLSPubkey
withdrawal_credentials: Hash # Commitment to pubkey for withdrawals and transfers
effective_balance: Gwei # Balance at stake
slashed: Bool
# Status epochs
activation_eligibility_epoch: Epoch # When criteria for activation were met
activation_epoch: Epoch
exit_epoch: Epoch
withdrawable_epoch: Epoch # When validator can withdraw or transfer funds
```
#### `Crosslink`
```python
class Crosslink(Container):
shard: Shard
parent_root: Hash
# Crosslinking data
start_epoch: Epoch
end_epoch: Epoch
data_root: Hash
```
#### `AttestationData`
```python
class AttestationData(Container):
# LMD GHOST vote
beacon_block_root: Hash
# FFG vote
source_epoch: Epoch
source_root: Hash
target_epoch: Epoch
target_root: Hash
# Crosslink vote
crosslink: Crosslink
```
#### `AttestationDataAndCustodyBit`
```python
class AttestationDataAndCustodyBit(Container):
data: AttestationData
custody_bit: Bit # Challengeable bit (SSZ-bool, 1 byte) for the custody of crosslink data
```
#### `IndexedAttestation`
```python
class IndexedAttestation(Container):
custody_bit_0_indices: List[ValidatorIndex, MAX_INDICES_PER_ATTESTATION] # Indices with custody bit equal to 0
custody_bit_1_indices: List[ValidatorIndex, MAX_INDICES_PER_ATTESTATION] # Indices with custody bit equal to 1
data: AttestationData
signature: BLSSignature
```
#### `PendingAttestation`
```python
class PendingAttestation(Container):
aggregation_bitfield: Bytes[MAX_INDICES_PER_ATTESTATION // 8]
data: AttestationData
inclusion_delay: Slot
proposer_index: ValidatorIndex
```
#### `Eth1Data`
```python
class Eth1Data(Container):
deposit_root: Hash
deposit_count: uint64
block_hash: Hash
```
#### `HistoricalBatch`
```python
class HistoricalBatch(Container):
block_roots: Vector[Hash, SLOTS_PER_HISTORICAL_ROOT]
state_roots: Vector[Hash, SLOTS_PER_HISTORICAL_ROOT]
```
#### `DepositData`
```python
class DepositData(Container):
pubkey: BLSPubkey
withdrawal_credentials: Hash
amount: Gwei
signature: BLSSignature
```
#### `BeaconBlockHeader`
```python
class BeaconBlockHeader(Container):
slot: Slot
parent_root: Hash
state_root: Hash
body_root: Hash
signature: BLSSignature
```
### Beacon operations
#### `ProposerSlashing`
```python
class ProposerSlashing(Container):
proposer_index: ValidatorIndex
header_1: BeaconBlockHeader
header_2: BeaconBlockHeader
```
#### `AttesterSlashing`
```python
class AttesterSlashing(Container):
attestation_1: IndexedAttestation
attestation_2: IndexedAttestation
```
#### `Attestation`
```python
class Attestation(Container):
aggregation_bitfield: Bytes[MAX_INDICES_PER_ATTESTATION // 8]
data: AttestationData
custody_bitfield: Bytes[MAX_INDICES_PER_ATTESTATION // 8]
signature: BLSSignature
```
#### `Deposit`
```python
class Deposit(Container):
proof: Vector[Hash, DEPOSIT_CONTRACT_TREE_DEPTH] # Merkle path to deposit root
data: DepositData
```
#### `VoluntaryExit`
```python
class VoluntaryExit(Container):
epoch: Epoch # Earliest epoch when voluntary exit can be processed
validator_index: ValidatorIndex
signature: BLSSignature
```
#### `Transfer`
```python
class Transfer(Container):
sender: ValidatorIndex
recipient: ValidatorIndex
amount: Gwei
fee: Gwei
slot: Slot # Slot at which transfer must be processed
pubkey: BLSPubkey # Withdrawal pubkey
signature: BLSSignature # Signature checked against withdrawal pubkey
```
### Beacon blocks
#### `BeaconBlockBody`
```python
class BeaconBlockBody(Container):
randao_reveal: BLSSignature
eth1_data: Eth1Data # Eth1 data vote
graffiti: Bytes32 # Arbitrary data
# Operations
proposer_slashings: List[ProposerSlashing, MAX_PROPOSER_SLASHINGS]
attester_slashings: List[AttesterSlashing, MAX_ATTESTER_SLASHINGS]
attestations: List[Attestation, MAX_ATTESTATIONS]
deposits: List[Deposit, MAX_DEPOSITS]
voluntary_exits: List[VoluntaryExit, MAX_VOLUNTARY_EXITS]
transfers: List[Transfer, MAX_TRANSFERS]
```
#### `BeaconBlock`
```python
class BeaconBlock(Container):
slot: Slot
parent_root: Hash
state_root: Hash
body: BeaconBlockBody
signature: BLSSignature
```
### Beacon state
#### `BeaconState`
```python
class BeaconState(Container):
# Versioning
genesis_time: uint64
slot: Slot
fork: Fork
# History
latest_block_header: BeaconBlockHeader
block_roots: Vector[Hash, SLOTS_PER_HISTORICAL_ROOT]
state_roots: Vector[Hash, SLOTS_PER_HISTORICAL_ROOT]
historical_roots: List[Hash, HISTORICAL_ROOTS_LIMIT]
# Eth1
eth1_data: Eth1Data
eth1_data_votes: List[Eth1Data, SLOTS_PER_ETH1_VOTING_PERIOD]
eth1_deposit_index: uint64
# Registry
validators: List[Validator, VALIDATOR_REGISTRY_LIMIT]
balances: List[Gwei, VALIDATOR_REGISTRY_LIMIT]
# Shuffling
start_shard: Shard
randao_mixes: Vector[Hash, EPOCHS_PER_HISTORICAL_VECTOR]
active_index_roots: Vector[Hash, EPOCHS_PER_HISTORICAL_VECTOR] # Active registry digests for light clients
# Slashings
slashed_balances: Vector[Gwei, EPOCHS_PER_SLASHED_BALANCES_VECTOR] # Sums of slashed effective balances
# Attestations
previous_epoch_attestations: List[PendingAttestation, MAX_ATTESTATIONS * SLOTS_PER_EPOCH]
current_epoch_attestations: List[PendingAttestation, MAX_ATTESTATIONS * SLOTS_PER_EPOCH]
# Crosslinks
previous_crosslinks: Vector[Crosslink, SHARD_COUNT] # Previous epoch snapshot
current_crosslinks: Vector[Crosslink, SHARD_COUNT]
# Justification
previous_justified_epoch: Epoch # Previous epoch snapshot
previous_justified_root: Hash # Previous epoch snapshot
current_justified_epoch: Epoch
current_justified_root: Hash
justification_bitfield: uint64 # Bit set for every recent justified epoch
# Finality
finalized_epoch: Epoch
finalized_root: Hash
```
## 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 Bytes32(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) -> Hash` is a function for hashing objects into a single root by utilizing a hash tree structure, as defined in the [SimpleSerialize spec](../simple-serialize.md#merkleization).
### `signing_root`
`def signing_root(object: Container) -> Hash` is a function for computing signing messages, as defined in the [SimpleSerialize spec](../simple-serialize.md#self-signed-containers).
### `calc_merkle_tree_from_leaves`
```python
zerohashes = [ZERO_HASH]
for layer in range(1, 100):
zerohashes.append(hash(zerohashes[layer - 1] + zerohashes[layer - 1]))
def calc_merkle_tree_from_leaves(values: Sequence[Hash], layer_count: int=32) -> Sequence[Sequence[Hash]]:
values = list(values)
tree = [values[::]]
for h in range(layer_count):
if len(values) % 2 == 1:
values.append(zerohashes[h])
values = [hash(values[i] + values[i + 1]) for i in range(0, len(values), 2)]
tree.append(values[::])
return tree
```
### `get_merkle_root`
```python
def get_merkle_root(values: Sequence[Hash], pad_to: int=1) -> Hash:
layer_count = int(log2(pad_to))
if len(values) == 0:
return zerohashes[layer_count]
return calc_merkle_tree_from_leaves(values, layer_count)[-1][0]
```
### `bls_domain`
```python
def bls_domain(domain_type: int, fork_version: bytes=b'\x00\x00\x00\x00') -> int:
"""
Return the bls domain given by the ``domain_type`` and optional 4 byte ``fork_version`` (defaults to zero).
"""
return bytes_to_int(int_to_bytes(domain_type, length=4) + fork_version)
```
### `slot_to_epoch`
```python
def slot_to_epoch(slot: Slot) -> Epoch:
"""
Return the epoch number of the given ``slot``.
"""
return Epoch(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 GENESIS_EPOCH if current_epoch == GENESIS_EPOCH else Epoch(current_epoch - 1)
```
### `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 Slot(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 (not validator.slashed) and (validator.activation_epoch <= epoch < validator.withdrawable_epoch)
```
### `get_active_validator_indices`
```python
def get_active_validator_indices(state: BeaconState, epoch: Epoch) -> Sequence[ValidatorIndex]:
"""
Get active validator indices at ``epoch``.
"""
return [ValidatorIndex(i) for i, v in enumerate(state.validators) 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_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.start_shard`` during ``epoch``.
"""
return min(get_epoch_committee_count(state, epoch), SHARD_COUNT - SHARD_COUNT // SLOTS_PER_EPOCH)
```
### `get_epoch_start_shard`
```python
def get_epoch_start_shard(state: BeaconState, epoch: Epoch) -> Shard:
assert epoch <= get_current_epoch(state) + 1
check_epoch = Epoch(get_current_epoch(state) + 1)
shard = Shard((state.start_shard + get_shard_delta(state, get_current_epoch(state))) % SHARD_COUNT)
while check_epoch > epoch:
check_epoch -= Epoch(1)
shard = Shard((shard + SHARD_COUNT - get_shard_delta(state, check_epoch)) % SHARD_COUNT)
return shard
```
### `get_attestation_data_slot`
```python
def get_attestation_data_slot(state: BeaconState, data: AttestationData) -> Slot:
committee_count = get_epoch_committee_count(state, data.target_epoch)
offset = (data.crosslink.shard + SHARD_COUNT - get_epoch_start_shard(state, data.target_epoch)) % SHARD_COUNT
return Slot(get_epoch_start_slot(data.target_epoch) + offset // (committee_count // SLOTS_PER_EPOCH))
```
### `get_block_root_at_slot`
```python
def get_block_root_at_slot(state: BeaconState,
slot: Slot) -> Hash:
"""
Return the block root at a recent ``slot``.
"""
assert slot < state.slot <= slot + SLOTS_PER_HISTORICAL_ROOT
return state.block_roots[slot % SLOTS_PER_HISTORICAL_ROOT]
```
### `get_block_root`
```python
def get_block_root(state: BeaconState,
epoch: Epoch) -> Hash:
"""
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) -> Hash:
"""
Return the randao mix at a recent ``epoch``.
``epoch`` expected to be between (current_epoch - EPOCHS_PER_HISTORICAL_VECTOR, current_epoch].
"""
return state.randao_mixes[epoch % EPOCHS_PER_HISTORICAL_VECTOR]
```
### `get_active_index_root`
```python
def get_active_index_root(state: BeaconState,
epoch: Epoch) -> Hash:
"""
Return the index root at a recent ``epoch``.
``epoch`` expected to be between
(current_epoch - EPOCHS_PER_HISTORICAL_VECTOR + ACTIVATION_EXIT_DELAY, current_epoch + ACTIVATION_EXIT_DELAY].
"""
return state.active_index_roots[epoch % EPOCHS_PER_HISTORICAL_VECTOR]
```
### `generate_seed`
```python
def generate_seed(state: BeaconState,
epoch: Epoch) -> Hash:
"""
Generate a seed for the given ``epoch``.
"""
return hash(
get_randao_mix(state, Epoch(epoch + EPOCHS_PER_HISTORICAL_VECTOR - MIN_SEED_LOOKAHEAD)) +
get_active_index_root(state, epoch) +
int_to_bytes(epoch, length=32)
)
```
### `get_beacon_proposer_index`
```python
def get_beacon_proposer_index(state: BeaconState) -> ValidatorIndex:
"""
Return the current beacon proposer index.
"""
epoch = get_current_epoch(state)
committees_per_slot = get_epoch_committee_count(state, epoch) // SLOTS_PER_EPOCH
offset = committees_per_slot * (state.slot % SLOTS_PER_EPOCH)
shard = Shard((get_epoch_start_shard(state, epoch) + offset) % SHARD_COUNT)
first_committee = get_crosslink_committee(state, epoch, shard)
MAX_RANDOM_BYTE = 2**8 - 1
seed = generate_seed(state, epoch)
i = 0
while True:
candidate_index = first_committee[(epoch + i) % len(first_committee)]
random_byte = hash(seed + int_to_bytes(i // 32, length=8))[i % 32]
effective_balance = state.validators[candidate_index].effective_balance
if effective_balance * MAX_RANDOM_BYTE >= MAX_EFFECTIVE_BALANCE * random_byte:
return ValidatorIndex(candidate_index)
i += 1
```
### `verify_merkle_branch`
```python
def verify_merkle_branch(leaf: Hash, proof: Sequence[Hash], depth: int, index: int, root: Hash) -> 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_shuffled_index`
```python
def get_shuffled_index(index: ValidatorIndex, index_count: int, seed: Hash) -> ValidatorIndex:
"""
Return the shuffled validator index corresponding to ``seed`` (and ``index_count``).
"""
assert index < index_count
assert index_count <= 2**40
# Swap or not (https://link.springer.com/content/pdf/10.1007%2F978-3-642-32009-5_1.pdf)
# See the 'generalized domain' algorithm on page 3
for current_round in range(SHUFFLE_ROUND_COUNT):
pivot = bytes_to_int(hash(seed + int_to_bytes(current_round, length=1))[0:8]) % index_count
flip = ValidatorIndex((pivot + index_count - index) % index_count)
position = max(index, flip)
source = hash(
seed + int_to_bytes(current_round, length=1) +
int_to_bytes(position // 256, length=4)
)
byte = source[(position % 256) // 8]
bit = (byte >> (position % 8)) % 2
index = flip if bit else index
return ValidatorIndex(index)
```
### `compute_committee`
```python
def compute_committee(indices: Sequence[ValidatorIndex],
seed: Hash, index: int, count: int) -> Sequence[ValidatorIndex]:
start = (len(indices) * index) // count
end = (len(indices) * (index + 1)) // count
return [indices[get_shuffled_index(ValidatorIndex(i), len(indices), seed)] for i in range(start, end)]
```
### `get_crosslink_committee`
```python
def get_crosslink_committee(state: BeaconState, epoch: Epoch, shard: Shard) -> Sequence[ValidatorIndex]:
return compute_committee(
indices=get_active_validator_indices(state, epoch),
seed=generate_seed(state, epoch),
index=(shard + SHARD_COUNT - get_epoch_start_shard(state, epoch)) % SHARD_COUNT,
count=get_epoch_committee_count(state, epoch),
)
```
### `get_attesting_indices`
```python
def get_attesting_indices(state: BeaconState,
attestation_data: AttestationData,
bitfield: bytes) -> Sequence[ValidatorIndex]:
"""
Return the sorted attesting indices corresponding to ``attestation_data`` and ``bitfield``.
"""
committee = get_crosslink_committee(state, attestation_data.target_epoch, attestation_data.crosslink.shard)
assert verify_bitfield(bitfield, len(committee))
return sorted([index for i, index in enumerate(committee) if get_bitfield_bit(bitfield, i) == 0b1])
```
### `int_to_bytes`
```python
def int_to_bytes(integer: int, length: int) -> bytes:
return integer.to_bytes(length, 'little')
```
### `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: Set[ValidatorIndex]) -> Gwei:
"""
Return the combined effective balance of the ``indices``. (1 Gwei minimum to avoid divisions by zero.)
"""
return Gwei(max(sum([state.validators[index].effective_balance for index in indices]), 1))
```
### `get_domain`
```python
def get_domain(state: BeaconState,
domain_type: int,
message_epoch: Epoch=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 bls_domain(domain_type, fork_version)
```
### `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)
assert set(custody_bit_1_indices).issubset(attesting_indices)
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,
)
```
### `validate_indexed_attestation`
```python
def validate_indexed_attestation(state: BeaconState, indexed_attestation: IndexedAttestation) -> None:
"""
Verify validity of ``indexed_attestation``.
"""
bit_0_indices = indexed_attestation.custody_bit_0_indices
bit_1_indices = indexed_attestation.custody_bit_1_indices
# Verify no index has custody bit equal to 1 [to be removed in phase 1]
assert len(bit_1_indices) == 0
# Verify max number of indices
assert len(bit_0_indices) + len(bit_1_indices) <= MAX_INDICES_PER_ATTESTATION
# Verify index sets are disjoint
assert len(set(bit_0_indices).intersection(bit_1_indices)) == 0
# Verify indices are sorted
assert bit_0_indices == sorted(bit_0_indices) and bit_1_indices == sorted(bit_1_indices)
# Verify aggregate signature
assert bls_verify_multiple(
pubkeys=[
bls_aggregate_pubkeys([state.validators[i].pubkey for i in bit_0_indices]),
bls_aggregate_pubkeys([state.validators[i].pubkey for i in 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, indexed_attestation.data.target_epoch),
)
```
### `is_slashable_attestation_data`
```python
def is_slashable_attestation_data(data_1: AttestationData, data_2: AttestationData) -> bool:
"""
Check if ``data_1`` and ``data_2`` are slashable according to Casper FFG rules.
"""
return (
# Double vote
(data_1 != data_2 and data_1.target_epoch == data_2.target_epoch) or
# Surround vote
(data_1.source_epoch < data_2.source_epoch and data_2.target_epoch < data_1.target_epoch)
)
```
### `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(epoch + 1 + ACTIVATION_EXIT_DELAY)
```
### `get_churn_limit`
```python
def get_churn_limit(state: BeaconState) -> int:
"""
Return the churn limit based on the active validator count.
"""
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, as 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, as 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, as 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 exit of the validator of the given ``index``.
"""
# Return if validator already initiated exit
validator = state.validators[index]
if validator.exit_epoch != FAR_FUTURE_EPOCH:
return
# Compute exit queue epoch
exit_epochs = [v.exit_epoch for v in state.validators 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.validators if v.exit_epoch == exit_queue_epoch])
if exit_queue_churn >= get_churn_limit(state):
exit_queue_epoch += Epoch(1)
# Set validator exit epoch and withdrawable epoch
validator.exit_epoch = exit_queue_epoch
validator.withdrawable_epoch = 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``.
"""
current_epoch = get_current_epoch(state)
initiate_validator_exit(state, slashed_index)
state.validators[slashed_index].slashed = True
state.validators[slashed_index].withdrawable_epoch = Epoch(current_epoch + EPOCHS_PER_SLASHED_BALANCES_VECTOR)
slashed_balance = state.validators[slashed_index].effective_balance
state.slashed_balances[current_epoch % EPOCHS_PER_SLASHED_BALANCES_VECTOR] += slashed_balance
proposer_index = get_beacon_proposer_index(state)
if whistleblower_index is None:
whistleblower_index = proposer_index
whistleblowing_reward = Gwei(slashed_balance // WHISTLEBLOWING_REWARD_QUOTIENT)
proposer_reward = Gwei(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)
```
## Genesis
### Genesis trigger
Before genesis has been triggered and for every Eth 1.0 block call `is_genesis_trigger(deposits, time)` where:
* `deposits` is the list of all deposits up to the Eth 1.0 block, ordered chronologically
* `time` is the Unix time of the Eth 1.0 block
When `is_genesis_trigger(deposits, time) is True` for the first time let:
* `genesis_deposits = deposits`
* `genesis_time = time - time % SECONDS_PER_DAY + 2 * SECONDS_PER_DAY` where `SECONDS_PER_DAY = 86400`
* `genesis_eth1_block_hash` is the Eth 1.0 block hash that emitted the log for the last deposit in `deposits`
*Note*: The function `is_genesis_trigger` has yet to be agreed upon by the community, and can be updated as necessary. We define the following testing placeholder:
```python
def is_genesis_trigger(deposits: List[Deposit, 2**DEPOSIT_CONTRACT_TREE_DEPTH], time: uint64) -> bool:
SECONDS_PER_DAY = 86400
# Do not deploy too early
if time - time % SECONDS_PER_DAY + 2 * SECONDS_PER_DAY < MIN_GENESIS_TIME:
return False
# Process deposits
state = BeaconState()
leaves = list(map(lambda deposit: hash_tree_root(deposit.data), deposits))
for deposit_index, deposit in enumerate(deposits):
state.eth1_data.deposit_root = get_merkle_root(leaves[:deposit_index + 1], 2**DEPOSIT_CONTRACT_TREE_DEPTH)
state.eth1_data.deposit_count = deposit_index + 1
state.eth1_deposit_index = deposit_index
process_deposit(state, deposit)
# Count active validators at genesis
active_validator_count = 0
for validator in state.validators:
if validator.effective_balance == MAX_EFFECTIVE_BALANCE:
active_validator_count += 1
# Check effective balance to trigger genesis
return active_validator_count >= GENESIS_ACTIVE_VALIDATOR_COUNT
```
### Genesis state
Let `genesis_state = get_genesis_beacon_state(genesis_deposits, genesis_time, genesis_eth1_block_hash)`.
```python
def get_genesis_beacon_state(deposits: Sequence[Deposit], genesis_time: int, genesis_eth1_block_hash: Hash) -> BeaconState:
state = BeaconState(
genesis_time=genesis_time,
eth1_data=Eth1Data(block_hash=genesis_eth1_block_hash),
latest_block_header=BeaconBlockHeader(body_root=hash_tree_root(BeaconBlockBody())),
)
# Process genesis deposits
leaves = list(map(lambda deposit: hash_tree_root(deposit.data), deposits))
for deposit_index, deposit in enumerate(deposits):
state.eth1_data.deposit_root = get_merkle_root(leaves[:deposit_index + 1], 2**DEPOSIT_CONTRACT_TREE_DEPTH)
state.eth1_data.deposit_count = deposit_index + 1
state.eth1_deposit_index = deposit_index
process_deposit(state, deposit)
# Process genesis activations
for validator in state.validators:
if validator.effective_balance == MAX_EFFECTIVE_BALANCE:
validator.activation_eligibility_epoch = GENESIS_EPOCH
validator.activation_epoch = GENESIS_EPOCH
# Populate active_index_roots
genesis_active_index_root = hash_tree_root(
List[ValidatorIndex, VALIDATOR_REGISTRY_LIMIT](get_active_validator_indices(state, GENESIS_EPOCH))
)
for index in range(EPOCHS_PER_HISTORICAL_VECTOR):
state.active_index_roots[index] = genesis_active_index_root
return state
```
### Genesis block
Let `genesis_block = BeaconBlock(state_root=hash_tree_root(genesis_state))`.
## Beacon chain state transition function
The post-state corresponding to a pre-state `state` and a block `block` is defined as `state_transition(state, block)`. State transitions that trigger an unhandled excpetion (e.g. a failed `assert` or an out-of-range list access) are considered invalid.
```python
def state_transition(state: BeaconState, block: BeaconBlock, validate_state_root: bool=False) -> BeaconState:
# Process slots (including those with no blocks) since block
process_slots(state, block.slot)
# Process block
process_block(state, block)
# Validate state root (`validate_state_root == True` in production)
if validate_state_root:
assert block.state_root == hash_tree_root(state)
# Return post-state
return state
```
```python
def process_slots(state: BeaconState, slot: Slot) -> None:
assert state.slot <= slot
while state.slot < slot:
process_slot(state)
# Process epoch on the first slot of the next epoch
if (state.slot + 1) % SLOTS_PER_EPOCH == 0:
process_epoch(state)
state.slot += Slot(1)
```
```python
def process_slot(state: BeaconState) -> None:
# Cache state root
previous_state_root = hash_tree_root(state)
state.state_roots[state.slot % SLOTS_PER_HISTORICAL_ROOT] = previous_state_root
# Cache latest block header state root
if state.latest_block_header.state_root == ZERO_HASH:
state.latest_block_header.state_root = previous_state_root
# Cache block root
previous_block_root = signing_root(state.latest_block_header)
state.block_roots[state.slot % SLOTS_PER_HISTORICAL_ROOT] = previous_block_root
```
### Epoch processing
*Note*: The `# @LabelHere` lines below are placeholders to show that code will be inserted here in a future phase.
```python
def process_epoch(state: BeaconState) -> None:
process_justification_and_finalization(state)
process_crosslinks(state)
process_rewards_and_penalties(state)
process_registry_updates(state)
# @process_reveal_deadlines
# @process_challenge_deadlines
process_slashings(state)
process_final_updates(state)
# @after_process_final_updates
```
#### Helper functions
```python
def get_total_active_balance(state: BeaconState) -> Gwei:
return get_total_balance(state, set(get_active_validator_indices(state, get_current_epoch(state))))
```
```python
def get_matching_source_attestations(state: BeaconState, epoch: Epoch) -> Sequence[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) -> Sequence[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) -> Sequence[PendingAttestation]:
return [
a for a in get_matching_source_attestations(state, epoch)
if a.data.beacon_block_root == get_block_root_at_slot(state, get_attestation_data_slot(state, a.data))
]
```
```python
def get_unslashed_attesting_indices(state: BeaconState,
attestations: Sequence[PendingAttestation]) -> Set[ValidatorIndex]:
output = set() # type: Set[ValidatorIndex]
for a in attestations:
output = output.union(get_attesting_indices(state, a.data, a.aggregation_bitfield))
return set(filter(lambda index: not state.validators[index].slashed, list(output)))
```
```python
def get_attesting_balance(state: BeaconState, attestations: Sequence[PendingAttestation]) -> Gwei:
return get_total_balance(state, get_unslashed_attesting_indices(state, attestations))
```
```python
def get_winning_crosslink_and_attesting_indices(state: BeaconState,
epoch: Epoch,
shard: Shard) -> Tuple[Crosslink, Set[ValidatorIndex]]:
attestations = [a for a in get_matching_source_attestations(state, epoch) if a.data.crosslink.shard == shard]
crosslinks = list(filter(
lambda c: hash_tree_root(state.current_crosslinks[shard]) in (c.parent_root, hash_tree_root(c)),
[a.data.crosslink for a in attestations]
))
# Winning crosslink has the crosslink data root with the most balance voting for it (ties broken lexicographically)
winning_crosslink = max(crosslinks, key=lambda c: (
get_attesting_balance(state, [a for a in attestations if a.data.crosslink == c]), c.data_root
), default=Crosslink())
winning_attestations = [a for a in attestations if a.data.crosslink == winning_crosslink]
return winning_crosslink, get_unslashed_attesting_indices(state, winning_attestations)
```
#### Justification and finalization
```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 + 3 == current_epoch:
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 + 2 == current_epoch:
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 + 2 == current_epoch:
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 + 1 == current_epoch:
state.finalized_epoch = old_current_justified_epoch
state.finalized_root = get_block_root(state, state.finalized_epoch)
```
#### Crosslinks
```python
def process_crosslinks(state: BeaconState) -> None:
state.previous_crosslinks = [c for c in state.current_crosslinks]
for epoch in (get_previous_epoch(state), get_current_epoch(state)):
for offset in range(get_epoch_committee_count(state, epoch)):
shard = Shard((get_epoch_start_shard(state, epoch) + offset) % SHARD_COUNT)
crosslink_committee = set(get_crosslink_committee(state, epoch, shard))
winning_crosslink, attesting_indices = get_winning_crosslink_and_attesting_indices(state, epoch, shard)
if 3 * get_total_balance(state, attesting_indices) >= 2 * get_total_balance(state, crosslink_committee):
state.current_crosslinks[shard] = winning_crosslink
```
#### Rewards and penalties
```python
def get_base_reward(state: BeaconState, index: ValidatorIndex) -> Gwei:
total_balance = get_total_active_balance(state)
effective_balance = state.validators[index].effective_balance
return Gwei(effective_balance * BASE_REWARD_FACTOR // integer_squareroot(total_balance) // BASE_REWARDS_PER_EPOCH)
```
```python
def get_attestation_deltas(state: BeaconState) -> Tuple[Sequence[Gwei], Sequence[Gwei]]:
previous_epoch = get_previous_epoch(state)
total_balance = get_total_active_balance(state)
rewards = [Gwei(0) for _ in range(len(state.validators))]
penalties = [Gwei(0) for _ in range(len(state.validators))]
eligible_validator_indices = [
ValidatorIndex(index) for index, v in enumerate(state.validators)
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_total_balance(state, unslashed_attesting_indices)
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):
index = ValidatorIndex(index)
attestation = min([
a for a in matching_source_attestations
if index in get_attesting_indices(state, a.data, a.aggregation_bitfield)
], key=lambda a: a.inclusion_delay)
proposer_reward = Gwei(get_base_reward(state, index) // PROPOSER_REWARD_QUOTIENT)
rewards[attestation.proposer_index] += proposer_reward
max_attester_reward = get_base_reward(state, index) - proposer_reward
rewards[index] += Gwei(max_attester_reward * MIN_ATTESTATION_INCLUSION_DELAY // attestation.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:
index = ValidatorIndex(index)
penalties[index] += Gwei(BASE_REWARDS_PER_EPOCH * get_base_reward(state, index))
if index not in matching_target_attesting_indices:
penalties[index] += Gwei(
state.validators[index].effective_balance * finality_delay // INACTIVITY_PENALTY_QUOTIENT
)
return rewards, penalties
```
```python
def get_crosslink_deltas(state: BeaconState) -> Tuple[Sequence[Gwei], Sequence[Gwei]]:
rewards = [Gwei(0) for _ in range(len(state.validators))]
penalties = [Gwei(0) for _ in range(len(state.validators))]
epoch = get_previous_epoch(state)
for offset in range(get_epoch_committee_count(state, epoch)):
shard = Shard((get_epoch_start_shard(state, epoch) + offset) % SHARD_COUNT)
crosslink_committee = set(get_crosslink_committee(state, epoch, shard))
winning_crosslink, attesting_indices = get_winning_crosslink_and_attesting_indices(state, epoch, shard)
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
```
```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 index in range(len(state.validators)):
increase_balance(state, ValidatorIndex(index), rewards1[index] + rewards2[index])
decrease_balance(state, ValidatorIndex(index), penalties1[index] + penalties2[index])
```
#### Registry updates
```python
def process_registry_updates(state: BeaconState) -> None:
# Process activation eligibility and ejections
for index, validator in enumerate(state.validators):
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, ValidatorIndex(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.validators) if
validator.activation_eligibility_epoch != FAR_FUTURE_EPOCH and
validator.activation_epoch >= get_delayed_activation_exit_epoch(state.finalized_epoch)
], key=lambda index: state.validators[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)]:
validator = state.validators[index]
if validator.activation_epoch == FAR_FUTURE_EPOCH:
validator.activation_epoch = get_delayed_activation_exit_epoch(get_current_epoch(state))
```
#### Slashings
```python
def process_slashings(state: BeaconState) -> None:
epoch = get_current_epoch(state)
total_balance = get_total_active_balance(state)
# Compute slashed balances in the current epoch
total_at_start = state.slashed_balances[(epoch + 1) % EPOCHS_PER_SLASHED_BALANCES_VECTOR]
total_at_end = state.slashed_balances[epoch % EPOCHS_PER_SLASHED_BALANCES_VECTOR]
total_penalties = total_at_end - total_at_start
for index, validator in enumerate(state.validators):
if validator.slashed and epoch + EPOCHS_PER_SLASHED_BALANCES_VECTOR // 2 == validator.withdrawable_epoch:
penalty = max(
validator.effective_balance * min(total_penalties * 3, total_balance) // total_balance,
validator.effective_balance // MIN_SLASHING_PENALTY_QUOTIENT
)
decrease_balance(state, ValidatorIndex(index), penalty)
```
#### Final updates
```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.validators):
balance = state.balances[index]
HALF_INCREMENT = EFFECTIVE_BALANCE_INCREMENT // 2
if balance < validator.effective_balance or validator.effective_balance + 3 * HALF_INCREMENT < balance:
validator.effective_balance = min(balance - balance % EFFECTIVE_BALANCE_INCREMENT, MAX_EFFECTIVE_BALANCE)
# Update start shard
state.start_shard = Shard((state.start_shard + get_shard_delta(state, current_epoch)) % SHARD_COUNT)
# Set active index root
index_root_position = (next_epoch + ACTIVATION_EXIT_DELAY) % EPOCHS_PER_HISTORICAL_VECTOR
state.active_index_roots[index_root_position] = hash_tree_root(
List[ValidatorIndex, VALIDATOR_REGISTRY_LIMIT](
get_active_validator_indices(state, Epoch(next_epoch + ACTIVATION_EXIT_DELAY))
)
)
# Set total slashed balances
state.slashed_balances[next_epoch % EPOCHS_PER_SLASHED_BALANCES_VECTOR] = (
state.slashed_balances[current_epoch % EPOCHS_PER_SLASHED_BALANCES_VECTOR]
)
# Set randao mix
state.randao_mixes[next_epoch % EPOCHS_PER_HISTORICAL_VECTOR] = 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.block_roots,
state_roots=state.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 = []
```
### Block processing
```python
def process_block(state: BeaconState, block: BeaconBlock) -> None:
process_block_header(state, block)
process_randao(state, block.body)
process_eth1_data(state, block.body)
process_operations(state, block.body)
```
#### 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.parent_root == signing_root(state.latest_block_header)
# Save current block as the new latest block
state.latest_block_header = BeaconBlockHeader(
slot=block.slot,
parent_root=block.parent_root,
state_root=ZERO_HASH, # Overwritten in next `process_slot` call
body_root=hash_tree_root(block.body),
)
# Verify proposer is not slashed
proposer = state.validators[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, body: BeaconBlockBody) -> None:
epoch = get_current_epoch(state)
# Verify RANDAO reveal
proposer = state.validators[get_beacon_proposer_index(state)]
assert bls_verify(proposer.pubkey, hash_tree_root(epoch), body.randao_reveal, get_domain(state, DOMAIN_RANDAO))
# Mix in RANDAO reveal
mix = xor(get_randao_mix(state, epoch), hash(body.randao_reveal))
state.randao_mixes[epoch % EPOCHS_PER_HISTORICAL_VECTOR] = mix
```
#### Eth1 data
```python
def process_eth1_data(state: BeaconState, body: BeaconBlockBody) -> None:
state.eth1_data_votes.append(body.eth1_data)
if state.eth1_data_votes.count(body.eth1_data) * 2 > SLOTS_PER_ETH1_VOTING_PERIOD:
state.eth1_data = body.eth1_data
```
#### Operations
```python
def process_operations(state: BeaconState, body: BeaconBlockBody) -> None:
# Verify that outstanding deposits are processed up to the maximum number of deposits
assert len(body.deposits) == min(MAX_DEPOSITS, state.eth1_data.deposit_count - state.eth1_deposit_index)
# Verify that there are no duplicate transfers
assert len(body.transfers) == len(set(body.transfers))
all_operations = (
(body.proposer_slashings, process_proposer_slashing),
(body.attester_slashings, process_attester_slashing),
(body.attestations, process_attestation),
(body.deposits, process_deposit),
(body.voluntary_exits, process_voluntary_exit),
(body.transfers, process_transfer),
) # type: Sequence[Tuple[List, Callable]]
for operations, function in all_operations:
for operation in operations:
function(state, operation)
```
##### Proposer slashings
```python
def process_proposer_slashing(state: BeaconState, proposer_slashing: ProposerSlashing) -> None:
"""
Process ``ProposerSlashing`` operation.
"""
proposer = state.validators[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
```python
def process_attester_slashing(state: BeaconState, attester_slashing: AttesterSlashing) -> None:
"""
Process ``AttesterSlashing`` operation.
"""
attestation_1 = attester_slashing.attestation_1
attestation_2 = attester_slashing.attestation_2
assert is_slashable_attestation_data(attestation_1.data, attestation_2.data)
validate_indexed_attestation(state, attestation_1)
validate_indexed_attestation(state, attestation_2)
slashed_any = False
attesting_indices_1 = attestation_1.custody_bit_0_indices + attestation_1.custody_bit_1_indices
attesting_indices_2 = attestation_2.custody_bit_0_indices + attestation_2.custody_bit_1_indices
for index in sorted(set(attesting_indices_1).intersection(attesting_indices_2)):
if is_slashable_validator(state.validators[index], get_current_epoch(state)):
slash_validator(state, index)
slashed_any = True
assert slashed_any
```
##### Attestations
```python
def process_attestation(state: BeaconState, attestation: Attestation) -> None:
"""
Process ``Attestation`` operation.
"""
data = attestation.data
assert data.crosslink.shard < SHARD_COUNT
assert data.target_epoch in (get_previous_epoch(state), get_current_epoch(state))
attestation_slot = get_attestation_data_slot(state, data)
assert attestation_slot + MIN_ATTESTATION_INCLUSION_DELAY <= state.slot <= attestation_slot + SLOTS_PER_EPOCH
pending_attestation = PendingAttestation(
data=data,
aggregation_bitfield=attestation.aggregation_bitfield,
inclusion_delay=state.slot - attestation_slot,
proposer_index=get_beacon_proposer_index(state),
)
if data.target_epoch == get_current_epoch(state):
ffg_data = (state.current_justified_epoch, state.current_justified_root, get_current_epoch(state))
parent_crosslink = state.current_crosslinks[data.crosslink.shard]
state.current_epoch_attestations.append(pending_attestation)
else:
ffg_data = (state.previous_justified_epoch, state.previous_justified_root, get_previous_epoch(state))
parent_crosslink = state.previous_crosslinks[data.crosslink.shard]
state.previous_epoch_attestations.append(pending_attestation)
# Check FFG data, crosslink data, and signature
assert ffg_data == (data.source_epoch, data.source_root, data.target_epoch)
assert data.crosslink.start_epoch == parent_crosslink.end_epoch
assert data.crosslink.end_epoch == min(data.target_epoch, parent_crosslink.end_epoch + MAX_EPOCHS_PER_CROSSLINK)
assert data.crosslink.parent_root == hash_tree_root(parent_crosslink)
assert data.crosslink.data_root == ZERO_HASH # [to be removed in phase 1]
validate_indexed_attestation(state, convert_to_indexed(state, attestation))
```
##### Deposits
```python
def process_deposit(state: BeaconState, deposit: Deposit) -> None:
"""
Process an Eth1 deposit, registering a validator or increasing its balance.
"""
# Verify the Merkle branch
assert verify_merkle_branch(
leaf=hash_tree_root(deposit.data),
proof=deposit.proof,
depth=DEPOSIT_CONTRACT_TREE_DEPTH,
index=state.eth1_deposit_index,
root=state.eth1_data.deposit_root,
)
# Deposits must be processed in order
state.eth1_deposit_index += 1
pubkey = deposit.data.pubkey
amount = deposit.data.amount
validator_pubkeys = [v.pubkey for v in state.validators]
if pubkey not in validator_pubkeys:
# Verify the deposit signature (proof of possession).
# Invalid signatures are allowed by the deposit contract,
# and hence included on-chain, but must not be processed.
# Note: Deposits are valid across forks, hence the deposit domain is retrieved directly from `bls_domain`
if not bls_verify(
pubkey, signing_root(deposit.data), deposit.data.signature, bls_domain(DOMAIN_DEPOSIT)
):
return
# Add validator and balance entries
state.validators.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=min(amount - amount % EFFECTIVE_BALANCE_INCREMENT, MAX_EFFECTIVE_BALANCE)
))
state.balances.append(amount)
else:
# Increase balance by deposit amount
index = ValidatorIndex(validator_pubkeys.index(pubkey))
increase_balance(state, index, amount)
```
##### Voluntary exits
```python
def process_voluntary_exit(state: BeaconState, exit: VoluntaryExit) -> None:
"""
Process ``VoluntaryExit`` operation.
"""
validator = state.validators[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
```python
def process_transfer(state: BeaconState, transfer: Transfer) -> None:
"""
Process ``Transfer`` operation.
"""
# Verify the balance the covers amount and fee (with overflow protection)
assert state.balances[transfer.sender] >= max(transfer.amount + transfer.fee, transfer.amount, transfer.fee)
# A transfer is valid in only one slot
assert state.slot == transfer.slot
# Sender must satisfy at least one of the following conditions in the parenthesis:
assert (
# * Has not been activated
state.validators[transfer.sender].activation_eligibility_epoch == FAR_FUTURE_EPOCH or
# * Is withdrawable
get_current_epoch(state) >= state.validators[transfer.sender].withdrawable_epoch or
# * Balance after transfer is more than the effective balance threshold
transfer.amount + transfer.fee + MAX_EFFECTIVE_BALANCE <= state.balances[transfer.sender]
)
# Verify that the pubkey is valid
assert (
state.validators[transfer.sender].withdrawal_credentials ==
int_to_bytes(BLS_WITHDRAWAL_PREFIX, length=1) + 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)
```