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Ethereum 2.0 Phase 0 -- The Beacon Chain

NOTICE: This document is a work-in-progress for researchers and implementers. It reflects recent spec changes and takes precedence over the Python proof-of-concept implementation [python-poc].

Table of contents

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. In the initial deployment phases of Ethereum 2.0 the only mechanism to become a validator is to make a one-way ETH transaction to a deposit contract on Ethereum 1.0. Activation as a validator happens when Ethereum 1.0 deposit receipts are processed by the beacon chain, the activation balance is reached, and after a queuing process. 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 availability votes for a shard block, and simultaneously 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 Ethereum 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. 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 that the protocol recognizes it as representing the committee".
  • Proposer - the validator that creates a beacon chain block
  • Attester - a 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, which 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 get exactly one chance to make an attestation
  • Finalized, justified - see Casper FFG finalization [casper-ffg]
  • Withdrawal period - the number of slots between a validator exit and the validator balance being withdrawable
  • Genesis time - the Unix time of the genesis beacon chain block at slot 0

Constants

Misc

Name Value
SHARD_COUNT 2**10 (= 1,024)
TARGET_COMMITTEE_SIZE 2**7 (= 128)
MAX_BALANCE_CHURN_QUOTIENT 2**5 (= 32)
MAX_INDICES_PER_SLASHABLE_VOTE 2**12 (= 4,096)
MAX_EXIT_DEQUEUES_PER_EPOCH 2**2 (= 4)
SHUFFLE_ROUND_COUNT 90
  • For the safety of crosslinks TARGET_COMMITTEE_SIZE exceeds the recommended minimum committee size of 111; with sufficient active validators (at least SLOTS_PER_EPOCH * TARGET_COMMITTEE_SIZE), the shuffling algorithm ensures committee sizes 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_ADDRESS TBD
DEPOSIT_CONTRACT_TREE_DEPTH 2**5 (= 32)

Gwei values

Name Value Unit
MIN_DEPOSIT_AMOUNT 2**0 * 1e9 (= 1,000,000,000) Gwei
MAX_DEPOSIT_AMOUNT 2**5 * 1e9 (= 32,000,000,000) Gwei
FORK_CHOICE_BALANCE_INCREMENT 2**0 * 1e9 (= 1,000,000,000) Gwei
EJECTION_BALANCE 2**4 * 1e9 (= 16,000,000,000) Gwei

Initial values

Name Value
GENESIS_FORK_VERSION 0
GENESIS_SLOT 2**32
GENESIS_EPOCH slot_to_epoch(GENESIS_SLOT)
GENESIS_START_SHARD 0
FAR_FUTURE_EPOCH 2**64 - 1
ZERO_HASH int_to_bytes32(0)
EMPTY_SIGNATURE int_to_bytes96(0)
BLS_WITHDRAWAL_PREFIX_BYTE int_to_bytes1(0)
  • GENESIS_SLOT should be at least as large in terms of time as the largest of the time parameters or state list lengths below (ie. it should be at least as large as any value measured in slots, and at least SLOTS_PER_EPOCH times as large as any value measured in epochs).

Time parameters

Name Value Unit Duration
SECONDS_PER_SLOT 6 seconds 6 seconds
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
EPOCHS_PER_ETH1_VOTING_PERIOD 2**4 (= 16) epochs ~1.7 hours
SLOTS_PER_HISTORICAL_ROOT 2**13 (= 8,192) slots ~13 hours
MIN_VALIDATOR_WITHDRAWABILITY_DELAY 2**8 (= 256) epochs ~27 hours

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)
WHISTLEBLOWER_REWARD_QUOTIENT 2**9 (= 512)
ATTESTATION_INCLUSION_REWARD_QUOTIENT 2**3 (= 8)
INACTIVITY_PENALTY_QUOTIENT 2**24 (= 16,777,216)
MIN_PENALTY_QUOTIENT 2**5 (= 32)
  • The BASE_REWARD_QUOTIENT parameter dictates the per-epoch reward. It corresponds to ~2.54% annual interest assuming 10 million participating ETH in every epoch.
  • 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 to about 1/sqrt(e) ~= 60.6%. Indeed, the balance retained by offline validators 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 transactions 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 2**4 (= 16)

Signature domains

Name Value
DOMAIN_BEACON_BLOCK 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) objects.

Beacon chain transactions

Proposer slashings

ProposerSlashing
{
    # Proposer index
    'proposer_index': 'uint64',
    # First block header
    'header_1': BeaconBlockHeader,
    # Second block header
    'header_2': BeaconBlockHeader,
}

Attester slashings

AttesterSlashing
{
    # First slashable attestation
    'slashable_attestation_1': SlashableAttestation,
    # Second slashable attestation
    'slashable_attestation_2': SlashableAttestation,
}
SlashableAttestation
{
    # Validator indices
    'validator_indices': ['uint64'],
    # Attestation data
    'data': AttestationData,
    # Custody bitfield
    'custody_bitfield': 'bytes',
    # Aggregate signature
    'aggregate_signature': 'bytes96',
}

Attestations

Attestation
{
    # Attester aggregation bitfield
    'aggregation_bitfield': 'bytes',
    # Attestation data
    'data': AttestationData,
    # Custody bitfield
    'custody_bitfield': 'bytes',
    # BLS aggregate signature
    'aggregate_signature': 'bytes96',
}
AttestationData
{
    # Slot number
    'slot': 'uint64',
    # Shard number
    'shard': 'uint64',
    # Root of the signed beacon block
    'beacon_block_root': 'bytes32',
    # Root of the ancestor at the epoch boundary
    'epoch_boundary_root': 'bytes32',
    # Data from the shard since the last attestation
    'crosslink_data_root': 'bytes32',
    # Last crosslink
    'latest_crosslink': Crosslink,
    # Last justified epoch in the beacon state
    'justified_epoch': 'uint64',
    # Hash of the last justified beacon block
    'justified_block_root': 'bytes32',
}
AttestationDataAndCustodyBit
{
    # Attestation data
    'data': AttestationData,
    # Custody bit
    'custody_bit': 'bool',
}

Deposits

Deposit
{
    # Branch in the deposit tree
    'proof': ['bytes32'],
    # Index in the deposit tree
    'index': 'uint64',
    # Data
    'deposit_data': DepositData,
}
DepositData
{
    # Amount in Gwei
    'amount': 'uint64',
    # Timestamp from deposit contract
    'timestamp': 'uint64',
    # Deposit input
    'deposit_input': DepositInput,
}
DepositInput
{
    # BLS pubkey
    'pubkey': 'bytes48',
    # Withdrawal credentials
    'withdrawal_credentials': 'bytes32',
    # A BLS signature of this `DepositInput`
    'proof_of_possession': 'bytes96',
}

Voluntary exits

VoluntaryExit
{
    # Minimum epoch for processing exit
    'epoch': 'uint64',
    # Index of the exiting validator
    'validator_index': 'uint64',
    # Validator signature
    'signature': 'bytes96',
}

Transfers

Transfer
{
    # Sender index
    'sender': 'uint64',
    # Recipient index
    'to': '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 chain blocks

BeaconBlock

{
    # Header
    'slot': 'uint64',
    'previous_block_root': 'bytes32',
    'state_root': 'bytes32',
    'body': BeaconBlockBody,
    'signature': 'bytes96',
}

BeaconBlockHeader

{
    'slot': 'uint64',
    'previous_block_root': 'bytes32',
    'state_root': 'bytes32',
    'block_body_root': 'bytes32',
    'signature': 'bytes96',
}

BeaconBlockBody

{
    'randao_reveal': 'bytes96',
    'eth1_data': Eth1Data,
    'proposer_slashings': [ProposerSlashing],
    'attester_slashings': [AttesterSlashing],
    'attestations': [Attestation],
    'deposits': [Deposit],
    'voluntary_exits': [VoluntaryExit],
    'transfers': [Transfer],
}

Beacon chain state

BeaconState

{
    # Misc
    'slot': 'uint64',
    'genesis_time': 'uint64',
    'fork': Fork,  # For versioning hard forks

    # Validator registry
    'validator_registry': [Validator],
    'validator_balances': ['uint64'],
    'validator_registry_update_epoch': 'uint64',

    # Randomness and committees
    'latest_randao_mixes': ['bytes32'],
    'previous_shuffling_start_shard': 'uint64',
    'current_shuffling_start_shard': 'uint64',
    'previous_shuffling_epoch': 'uint64',
    'current_shuffling_epoch': 'uint64',
    'previous_shuffling_seed': 'bytes32',
    'current_shuffling_seed': 'bytes32',

    # Finality
    'previous_epoch_attestations': [PendingAttestation],
    'current_epoch_attestations': [PendingAttestation],
    'previous_justified_epoch': 'uint64',
    'justified_epoch': 'uint64',
    'justification_bitfield': 'uint64',
    'finalized_epoch': 'uint64',

    # Recent state
    'latest_crosslinks': [Crosslink],
    'latest_block_roots': ['bytes32'],
    'latest_state_roots': ['bytes32'],
    'latest_active_index_roots': ['bytes32'],
    'latest_slashed_balances': ['uint64'],  # 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': [Eth1DataVote],
    'deposit_index': 'uint64'
}

Validator

{
    # BLS public key
    'pubkey': 'bytes48',
    # Withdrawal credentials
    'withdrawal_credentials': 'bytes32',
    # Epoch when validator activated
    'activation_epoch': 'uint64',
    # Epoch when validator exited
    'exit_epoch': 'uint64',
    # Epoch when validator is eligible to withdraw
    'withdrawable_epoch': 'uint64',
    # Did the validator initiate an exit
    'initiated_exit': 'bool',
    # Was the validator slashed
    'slashed': 'bool',
}

Crosslink

{
    # Epoch number
    'epoch': 'uint64',
    # Shard data since the previous crosslink
    'crosslink_data_root': 'bytes32',
}

PendingAttestation

{
    # Attester aggregation bitfield
    'aggregation_bitfield': 'bytes',
    # Attestation data
    'data': AttestationData,
    # Custody bitfield
    'custody_bitfield': 'bytes',
    # Inclusion slot
    'inclusion_slot': 'uint64',
}

Fork

{
    # Previous fork version
    'previous_version': 'uint64',
    # Current fork version
    'current_version': 'uint64',
    # Fork epoch number
    'epoch': 'uint64',
}

Eth1Data

{
    # Root of the deposit tree
    'deposit_root': 'bytes32',
    # Block hash
    'block_hash': 'bytes32',
}

Eth1DataVote

{
    # Data being voted for
    'eth1_data': Eth1Data,
    # Vote count
    'vote_count': '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

def xor(bytes1: Bytes32, bytes2: Bytes32) -> Bytes32:
    return bytes(a ^ b for a, b in zip(bytes1, bytes2))

hash

The hash function is denoted by hash. In Phase 0 the beacon chain is deployed with the same hash function as Ethereum 1.0, i.e. Keccak-256 (also incorrectly known as SHA3).

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.

signed_root

def signed_root(object: SSZContainer) -> Bytes32 is a function defined in the SimpleSerialize spec to compute signed messages.

get_temporary_block_header

def get_temporary_block_header(block: BeaconBlock) -> BeaconBlockHeader:
    """
    Return the block header corresponding to a block with ``state_root`` set to ``ZERO_HASH``. 
    """
    return BeaconBlockHeader(
        slot=block.slot,
        previous_block_root=block.previous_block_root,
        state_root=ZERO_HASH,
        block_body_root=hash_tree_root(block.body),
        signature=block.signature,
    )

slot_to_epoch

def slot_to_epoch(slot: Slot) -> Epoch:
    """
    Return the epoch number of the given ``slot``.
    """
    return slot // SLOTS_PER_EPOCH

get_previous_epoch

def get_previous_epoch(state: BeaconState) -> Epoch:
    """`
    Return the previous epoch of the given ``state``.
    """
    return get_current_epoch(state) - 1

get_current_epoch

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

def get_epoch_start_slot(epoch: Epoch) -> Slot:
    """
    Return the starting slot of the given ``epoch``.
    """
    return epoch * SLOTS_PER_EPOCH

is_active_validator

def is_active_validator(validator: Validator, epoch: Epoch) -> bool:
    """
    Check if ``validator`` is active.
    """
    return validator.activation_epoch <= epoch < validator.exit_epoch

get_active_validator_indices

def get_active_validator_indices(validators: List[Validator], epoch: Epoch) -> List[ValidatorIndex]:
    """
    Get indices of active validators from ``validators``.
    """
    return [i for i, v in enumerate(validators) if is_active_validator(v, epoch)]

get_permuted_index

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

split

def split(values: List[Any], split_count: int) -> List[List[Any]]:
    """
    Splits ``values`` into ``split_count`` pieces.
    """
    list_length = len(values)
    return [
        values[(list_length * i // split_count): (list_length * (i + 1) // split_count)]
        for i in range(split_count)
    ]

get_epoch_committee_count

def get_epoch_committee_count(active_validator_count: int) -> int:
    """
    Return the number of committees in one epoch.
    """
    return max(
        1,
        min(
            SHARD_COUNT // SLOTS_PER_EPOCH,
            active_validator_count // SLOTS_PER_EPOCH // TARGET_COMMITTEE_SIZE,
        )
    ) * SLOTS_PER_EPOCH

get_shuffling

def get_shuffling(seed: Bytes32,
                  validators: List[Validator],
                  epoch: Epoch) -> List[List[ValidatorIndex]]:
    """
    Shuffle active validators and split into crosslink committees.
    Return a list of committees (each a list of validator indices).
    """
    # Shuffle active validator indices
    active_validator_indices = get_active_validator_indices(validators, epoch)
    length = len(active_validator_indices)
    shuffled_indices = [active_validator_indices[get_permuted_index(i, length, seed)] for i in range(length)]

    # Split the shuffled active validator indices
    return split(shuffled_indices, get_epoch_committee_count(length))

Invariant: if get_shuffling(seed, validators, epoch) returns some value x for some epoch <= get_current_epoch(state) + ACTIVATION_EXIT_DELAY, it should return the same value x for the same seed and epoch and possible future modifications of validators forever in phase 0, and until the ~1 year deletion delay in phase 2 and in the future.

Note: this definition and the next few definitions make heavy use of repetitive computing. Production implementations are expected to appropriately use caching/memoization to avoid redoing work.

get_previous_epoch_committee_count

def get_previous_epoch_committee_count(state: BeaconState) -> int:
    """
    Return the number of committees in the previous epoch of the given ``state``.
    """
    previous_active_validators = get_active_validator_indices(
        state.validator_registry,
        state.previous_shuffling_epoch,
    )
    return get_epoch_committee_count(len(previous_active_validators))

get_current_epoch_committee_count

def get_current_epoch_committee_count(state: BeaconState) -> int:
    """
    Return the number of committees in the current epoch of the given ``state``.
    """
    current_active_validators = get_active_validator_indices(
        state.validator_registry,
        state.current_shuffling_epoch,
    )
    return get_epoch_committee_count(len(current_active_validators))

get_next_epoch_committee_count

def get_next_epoch_committee_count(state: BeaconState) -> int:
    """
    Return the number of committees in the next epoch of the given ``state``.
    """
    next_active_validators = get_active_validator_indices(
        state.validator_registry,
        get_current_epoch(state) + 1,
    )
    return get_epoch_committee_count(len(next_active_validators))

get_crosslink_committees_at_slot

def get_crosslink_committees_at_slot(state: BeaconState,
                                     slot: Slot,
                                     registry_change: bool=False) -> List[Tuple[List[ValidatorIndex], Shard]]:
    """
    Return the list of ``(committee, shard)`` tuples for the ``slot``.

    Note: There are two possible shufflings for crosslink committees for a
    ``slot`` in the next epoch -- with and without a `registry_change`
    """
    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

    if epoch == current_epoch:
        committees_per_epoch = get_current_epoch_committee_count(state)
        seed = state.current_shuffling_seed
        shuffling_epoch = state.current_shuffling_epoch
        shuffling_start_shard = state.current_shuffling_start_shard
    elif epoch == previous_epoch:
        committees_per_epoch = get_previous_epoch_committee_count(state)
        seed = state.previous_shuffling_seed
        shuffling_epoch = state.previous_shuffling_epoch
        shuffling_start_shard = state.previous_shuffling_start_shard
    elif epoch == next_epoch:
        epochs_since_last_registry_update = current_epoch - state.validator_registry_update_epoch
        if registry_change:
            committees_per_epoch = get_next_epoch_committee_count(state)
            seed = generate_seed(state, next_epoch)
            shuffling_epoch = next_epoch
            current_committees_per_epoch = get_current_epoch_committee_count(state)
            shuffling_start_shard = (state.current_shuffling_start_shard + current_committees_per_epoch) % SHARD_COUNT
        elif epochs_since_last_registry_update > 1 and is_power_of_two(epochs_since_last_registry_update):
            committees_per_epoch = get_next_epoch_committee_count(state)
            seed = generate_seed(state, next_epoch)
            shuffling_epoch = next_epoch
            shuffling_start_shard = state.current_shuffling_start_shard
        else:
            committees_per_epoch = get_current_epoch_committee_count(state)
            seed = state.current_shuffling_seed
            shuffling_epoch = state.current_shuffling_epoch
            shuffling_start_shard = state.current_shuffling_start_shard

    shuffling = get_shuffling(
        seed,
        state.validator_registry,
        shuffling_epoch,
    )
    offset = slot % SLOTS_PER_EPOCH
    committees_per_slot = committees_per_epoch // SLOTS_PER_EPOCH
    slot_start_shard = (shuffling_start_shard + committees_per_slot * offset) % SHARD_COUNT

    return [
        (
            shuffling[committees_per_slot * offset + i],
            (slot_start_shard + i) % SHARD_COUNT,
        )
        for i in range(committees_per_slot)
    ]

get_block_root

def get_block_root(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(_, s) should always return hash_tree_root of the block in the beacon chain at slot s, and get_crosslink_committees_at_slot(_, s) should not change unless the validator registry changes.

get_state_root

def get_state_root(state: BeaconState,
                   slot: Slot) -> Bytes32:
    """
    Return the state root at a recent ``slot``.
    """
    assert slot < state.slot <= slot + SLOTS_PER_HISTORICAL_ROOT
    return state.latest_state_roots[slot % SLOTS_PER_HISTORICAL_ROOT]

get_randao_mix

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

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

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

def get_beacon_proposer_index(state: BeaconState,
                              slot: Slot,
                              registry_change: bool=False) -> ValidatorIndex:
    """
    Return the beacon proposer index 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

    first_committee, _ = get_crosslink_committees_at_slot(state, slot, registry_change)[0]
    return first_committee[slot % len(first_committee)]

merkle_root

def merkle_root(values: List[Bytes32]) -> Bytes32:
    """
    Merkleize ``values`` (where ``len(values)`` is a power of two) and return the Merkle root.
    Note that the leaves are not hashed.
    """
    o = [0] * len(values) + values
    for i in range(len(values) - 1, 0, -1):
        o[i] = hash(o[i * 2] + o[i * 2 + 1])
    return o[1]

verify_merkle_branch

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_attestation_participants

def get_attestation_participants(state: BeaconState,
                                 attestation_data: AttestationData,
                                 bitfield: bytes) -> List[ValidatorIndex]:
    """
    Return the participant indices at for the ``attestation_data`` and ``bitfield``.
    """
    # Find the committee in the list with the desired shard
    crosslink_committees = get_crosslink_committees_at_slot(state, attestation_data.slot)

    assert attestation_data.shard in [shard for _, shard in crosslink_committees]
    crosslink_committee = [committee for committee, shard in crosslink_committees if shard == attestation_data.shard][0]

    assert verify_bitfield(bitfield, len(crosslink_committee))

    # Find the participating attesters in the committee
    participants = []
    for i, validator_index in enumerate(crosslink_committee):
        aggregation_bit = get_bitfield_bit(bitfield, i)
        if aggregation_bit == 0b1:
            participants.append(validator_index)
    return participants

is_power_of_two

def is_power_of_two(value: int) -> bool:
    """
    Check if ``value`` is a power of two integer.
    """
    return (value > 0) and (value & (value - 1) == 0)

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

def bytes_to_int(data: bytes) -> int:
    return int.from_bytes(data, 'little')

get_effective_balance

def get_effective_balance(state: BeaconState, index: ValidatorIndex) -> Gwei:
    """
    Return the effective balance (also known as "balance at stake") for a validator with the given ``index``.
    """
    return min(state.validator_balances[index], MAX_DEPOSIT_AMOUNT)

get_total_balance

def get_total_balance(state: BeaconState, validators: List[ValidatorIndex]) -> Gwei:
    """
    Return the combined effective balance of an array of ``validators``.
    """
    return sum([get_effective_balance(state, i) for i in validators])

get_fork_version

def get_fork_version(fork: Fork,
                     epoch: Epoch) -> int:
    """
    Return the fork version of the given ``epoch``.
    """
    if epoch < fork.epoch:
        return fork.previous_version
    else:
        return fork.current_version

get_domain

def get_domain(fork: Fork,
               epoch: Epoch,
               domain_type: int) -> int:
    """
    Get the domain number that represents the fork meta and signature domain.
    """
    fork_version = get_fork_version(fork, epoch)
    return fork_version * 2**32 + domain_type

get_bitfield_bit

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

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

verify_slashable_attestation

def verify_slashable_attestation(state: BeaconState, slashable_attestation: SlashableAttestation) -> bool:
    """
    Verify validity of ``slashable_attestation`` fields.
    """
    if slashable_attestation.custody_bitfield != b'\x00' * len(slashable_attestation.custody_bitfield):  # [TO BE REMOVED IN PHASE 1]
        return False

    if len(slashable_attestation.validator_indices) == 0:
        return False

    for i in range(len(slashable_attestation.validator_indices) - 1):
        if slashable_attestation.validator_indices[i] >= slashable_attestation.validator_indices[i + 1]:
            return False

    if not verify_bitfield(slashable_attestation.custody_bitfield, len(slashable_attestation.validator_indices)):
        return False

    if len(slashable_attestation.validator_indices) > MAX_INDICES_PER_SLASHABLE_VOTE:
        return False

    custody_bit_0_indices = []
    custody_bit_1_indices = []
    for i, validator_index in enumerate(slashable_attestation.validator_indices):
        if get_bitfield_bit(slashable_attestation.custody_bitfield, i) == 0b0:
            custody_bit_0_indices.append(validator_index)
        else:
            custody_bit_1_indices.append(validator_index)

    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=slashable_attestation.data, custody_bit=0b0)),
            hash_tree_root(AttestationDataAndCustodyBit(data=slashable_attestation.data, custody_bit=0b1)),
        ],
        signature=slashable_attestation.aggregate_signature,
        domain=get_domain(state.fork, slot_to_epoch(slashable_attestation.data.slot), DOMAIN_ATTESTATION),
    )

is_double_vote

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

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.justified_epoch
    source_epoch_2 = attestation_data_2.justified_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

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

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

bls_verify

bls_verify is a function for verifying a BLS signature, defined in the BLS Signature spec.

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_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.

process_deposit

Used to add a validator or top up an existing validator's balance by some deposit amount:

def process_deposit(state: BeaconState, deposit: Deposit) -> None:
    """
    Process a deposit from Ethereum 1.0.
    Note that this function mutates ``state``.
    """
    deposit_input = deposit.deposit_data.deposit_input

    # Should equal 8 bytes for deposit_data.amount +
    #              8 bytes for deposit_data.timestamp +
    #              176 bytes for deposit_data.deposit_input
    # It should match the deposit_data in the eth1.0 deposit contract
    serialized_deposit_data = serialize(deposit.deposit_data)
    # Deposits must be processed in order
    assert deposit.index == state.deposit_index
    
    # Verify the Merkle branch
    merkle_branch_is_valid = verify_merkle_branch(
        leaf=hash(serialized_deposit_data),
        proof=deposit.proof,
        depth=DEPOSIT_CONTRACT_TREE_DEPTH,
        index=deposit.index,
        root=state.latest_eth1_data.deposit_root,
    )
    assert merkle_branch_is_valid
        
    # Increment the next deposit index we are expecting. Note that this
    # needs to be done here because while the deposit contract will never
    # create an invalid Merkle branch, it may admit an invalid deposit
    # object, and we need to be able to skip over it
    state.deposit_index += 1
    
    # Verify the proof of possession
    proof_is_valid = bls_verify(
        pubkey=deposit_input.pubkey,
        message_hash=signed_root(deposit_input, "proof_of_possession"),
        signature=deposit_input.proof_of_possession,
        domain=get_domain(
            state.fork,
            get_current_epoch(state),
            DOMAIN_DEPOSIT,
        )
    )
    
    if not proof_is_valid:
        return

    validator_pubkeys = [v.pubkey for v in state.validator_registry]
    pubkey = deposit_input.pubkey
    amount = deposit.deposit_data.amount
    withdrawal_credentials = deposit_input.withdrawal_credentials

    if pubkey not in validator_pubkeys:
        # Add new validator
        validator = Validator(
            pubkey=pubkey,
            withdrawal_credentials=withdrawal_credentials,
            activation_epoch=FAR_FUTURE_EPOCH,
            exit_epoch=FAR_FUTURE_EPOCH,
            withdrawable_epoch=FAR_FUTURE_EPOCH,
            initiated_exit=False,
            slashed=False,
        )

        # Note: In phase 2 registry indices that have been withdrawn for a long time will be recycled.
        state.validator_registry.append(validator)
        state.validator_balances.append(amount)
    else:
        # Increase balance by deposit amount
        index = validator_pubkeys.index(pubkey)
        assert state.validator_registry[index].withdrawal_credentials == withdrawal_credentials

        state.validator_balances[index] += amount

Routines for updating validator status

Note: All functions in this section mutate state.

activate_validator

def activate_validator(state: BeaconState, index: ValidatorIndex, is_genesis: bool) -> None:
    """
    Activate the validator of the given ``index``.
    Note that this function mutates ``state``.
    """
    validator = state.validator_registry[index]

    validator.activation_epoch = GENESIS_EPOCH if is_genesis else get_delayed_activation_exit_epoch(get_current_epoch(state))

initiate_validator_exit

def initiate_validator_exit(state: BeaconState, index: ValidatorIndex) -> None:
    """
    Initiate the validator of the given ``index``.
    Note that this function mutates ``state``.
    """
    validator = state.validator_registry[index]
    validator.initiated_exit = True

exit_validator

def exit_validator(state: BeaconState, index: ValidatorIndex) -> None:
    """
    Exit the validator of the given ``index``.
    Note that this function mutates ``state``.
    """
    validator = state.validator_registry[index]
    delayed_activation_exit_epoch = get_delayed_activation_exit_epoch(get_current_epoch(state))

    # The following updates only occur if not previous exited
    if validator.exit_epoch <= delayed_activation_exit_epoch:
        return
    else:
        validator.exit_epoch = delayed_activation_exit_epoch

slash_validator

def slash_validator(state: BeaconState, index: ValidatorIndex) -> None:
    """
    Slash the validator with index ``index``.
    Note that this function mutates ``state``.
    """
    validator = state.validator_registry[index]
    assert state.slot < get_epoch_start_slot(validator.withdrawable_epoch)  # [TO BE REMOVED IN PHASE 2]
    exit_validator(state, index)
    state.latest_slashed_balances[get_current_epoch(state) % LATEST_SLASHED_EXIT_LENGTH] += get_effective_balance(state, index)

    whistleblower_index = get_beacon_proposer_index(state, state.slot)
    whistleblower_reward = get_effective_balance(state, index) // WHISTLEBLOWER_REWARD_QUOTIENT
    state.validator_balances[whistleblower_index] += whistleblower_reward
    state.validator_balances[index] -= whistleblower_reward
    validator.slashed = True
    validator.withdrawable_epoch = get_current_epoch(state) + LATEST_SLASHED_EXIT_LENGTH

prepare_validator_for_withdrawal

def prepare_validator_for_withdrawal(state: BeaconState, index: ValidatorIndex) -> None:
    """
    Set the validator with the given ``index`` as withdrawable
    ``MIN_VALIDATOR_WITHDRAWABILITY_DELAY`` after the current epoch.
    Note that this function mutates ``state``.
    """
    validator = state.validator_registry[index]
    validator.withdrawable_epoch = get_current_epoch(state) + MIN_VALIDATOR_WITHDRAWABILITY_DELAY

Ethereum 1.0 deposit contract

The initial deployment phases of Ethereum 2.0 are implemented without consensus changes to Ethereum 1.0. A deposit contract at address DEPOSIT_CONTRACT_ADDRESS is added to Ethereum 1.0 for deposits of ETH to the beacon chain. Validator balances will be withdrawable to the shards in phase 2, i.e. when the EVM2.0 is deployed and the shards have state.

Deposit arguments

The deposit contract has a single deposit function which takes as argument a SimpleSerialize'd DepositInput.

Withdrawal credentials

One of the DepositInput fields is withdrawal_credentials. It is a commitment to credentials for withdrawals to shards. The first byte of withdrawal_credentials is a version number. As of now the only expected format is as follows:

  • withdrawal_credentials[:1] == BLS_WITHDRAWAL_PREFIX_BYTE
  • withdrawal_credentials[1:] == hash(withdrawal_pubkey)[1:] where withdrawal_pubkey is a BLS pubkey

The private key corresponding to withdrawal_pubkey will be required to initiate a withdrawal. It can be stored separately until a withdrawal is required, e.g. in cold storage.

Deposit logs

Every Ethereum 1.0 deposit, of size between MIN_DEPOSIT_AMOUNT and MAX_DEPOSIT_AMOUNT, emits a Deposit log for consumption by the beacon chain. The deposit contract does little validation, pushing most of the validator onboarding logic to the beacon chain. In particular, the proof of possession (a BLS12 signature) is not verified by the deposit contract.

Eth2Genesis log

When sufficiently many full deposits have been made the deposit contract emits the Eth2Genesis log. The beacon chain state may then be initialized by calling the get_genesis_beacon_state function (defined below) where:

  • genesis_time equals time in the Eth2Genesis log
  • latest_eth1_data.deposit_root equals deposit_root in the Eth2Genesis log
  • latest_eth1_data.block_hash equals the hash of the block that included the log
  • genesis_validator_deposits is a list of Deposit objects built according to the Deposit logs up to the deposit that triggered the Eth2Genesis log, processed in the order in which they were emitted (oldest to newest)

Vyper code

The source for the Vyper contract lives in a separate repository at https://github.com/ethereum/deposit_contract/blob/master/deposit_contract/contracts/validator_registration.v.py.

Note: to save ~10x on gas this contract uses a somewhat unintuitive progressive Merkle root calculation algo that requires only O(log(n)) storage. See https://github.com/ethereum/research/blob/master/beacon_chain_impl/progressive_merkle_tree.py for an implementation of the same algo in python tested for correctness.

For convenience, we provide the interface to the contract here:

  • __init__(): initializes the contract
  • get_deposit_root() -> bytes32: returns the current root of the deposit tree
  • deposit(bytes[512]): adds a deposit instance to the deposit tree, incorporating the input argument and the value transferred in the given call. Note: the amount of value transferred must be within MIN_DEPOSIT_AMOUNT and MAX_DEPOSIT_AMOUNT, inclusive. Each of these constants are specified in units of Gwei.

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.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 = get_empty_block().
  • Set genesis_block.state_root = hash_tree_root(genesis_state).
def get_empty_block() -> BeaconBlock:
    """
    Get an empty ``BeaconBlock``.
    """
    return BeaconBlock(
        slot=GENESIS_SLOT,
        previous_block_root=ZERO_HASH,
        state_root=ZERO_HASH,
        body=BeaconBlockBody(
            randao_reveal=EMPTY_SIGNATURE,
            eth1_data=Eth1Data(
                deposit_root=ZERO_HASH,
                block_hash=ZERO_HASH,
            ),
            proposer_slashings=[],
            attester_slashings=[],
            attestations=[],
            deposits=[],
            voluntary_exits=[],
            transfers=[],
        ),
        signature=EMPTY_SIGNATURE,
    )

def get_genesis_beacon_state(genesis_validator_deposits: List[Deposit],
                             genesis_time: int,
                             genesis_eth1_data: Eth1Data) -> BeaconState:
    """
    Get the genesis ``BeaconState``.
    """
    state = BeaconState(
        # Misc
        slot=GENESIS_SLOT,
        genesis_time=genesis_time,
        fork=Fork(
            previous_version=GENESIS_FORK_VERSION,
            current_version=GENESIS_FORK_VERSION,
            epoch=GENESIS_EPOCH,
        ),

        # Validator registry
        validator_registry=[],
        validator_balances=[],
        validator_registry_update_epoch=GENESIS_EPOCH,

        # Randomness and committees
        latest_randao_mixes=[ZERO_HASH for _ in range(LATEST_RANDAO_MIXES_LENGTH)],
        previous_shuffling_start_shard=GENESIS_START_SHARD,
        current_shuffling_start_shard=GENESIS_START_SHARD,
        previous_shuffling_epoch=GENESIS_EPOCH,
        current_shuffling_epoch=GENESIS_EPOCH,
        previous_shuffling_seed=ZERO_HASH,
        current_shuffling_seed=ZERO_HASH,

        # Finality
        previous_epoch_attestations=[],
        current_epoch_attestations=[],
        previous_justified_epoch=GENESIS_EPOCH,
        justified_epoch=GENESIS_EPOCH,
        justification_bitfield=0,
        finalized_epoch=GENESIS_EPOCH,

        # Recent state
        latest_crosslinks=[Crosslink(epoch=GENESIS_EPOCH, crosslink_data_root=ZERO_HASH) for _ in range(SHARD_COUNT)],
        latest_block_roots=[ZERO_HASH for _ in range(SLOTS_PER_HISTORICAL_ROOT)],
        latest_state_roots=[ZERO_HASH for _ in range(SLOTS_PER_HISTORICAL_ROOT)],
        latest_active_index_roots=[ZERO_HASH for _ in range(LATEST_ACTIVE_INDEX_ROOTS_LENGTH)],
        latest_slashed_balances=[0 for _ in range(LATEST_SLASHED_EXIT_LENGTH)],
        latest_block_header=get_temporary_block_header(get_empty_block()),
        historical_roots=[],

        # Ethereum 1.0 chain data
        latest_eth1_data=genesis_eth1_data,
        eth1_data_votes=[],
        deposit_index=0,
    )

    # Process genesis deposits
    for deposit in genesis_validator_deposits:
        process_deposit(state, deposit)

    # Process genesis activations
    for validator_index, _ in enumerate(state.validator_registry):
        if get_effective_balance(state, validator_index) >= MAX_DEPOSIT_AMOUNT:
            activate_validator(state, validator_index, is_genesis=True)

    genesis_active_index_root = hash_tree_root(get_active_validator_indices(state.validator_registry, GENESIS_EPOCH))
    for index in range(LATEST_ACTIVE_INDEX_ROOTS_LENGTH):
        state.latest_active_index_roots[index] = genesis_active_index_root
    state.current_shuffling_seed = generate_seed(state, GENESIS_EPOCH)

    return state

Beacon chain processing

The beacon chain is the system chain for Ethereum 2.0. The main responsibilities of the beacon chain are:

  • Store and maintain the registry of validators
  • Process crosslinks (see above)
  • Process its per-block consensus, as well as the finality gadget

Processing the beacon chain is similar to processing the Ethereum 1.0 chain. Clients download and process blocks, and maintain a view of what is the current "canonical chain", terminating at the current "head". However, because of the beacon chain's relationship with Ethereum 1.0, and because it is a proof-of-stake chain, there are differences.

For a beacon chain block, block, to be processed by a node, the following conditions must be met:

  • The parent block with root block.previous_block_root has been processed and accepted.
  • An Ethereum 1.0 block pointed to by the state.latest_eth1_data.block_hash has been processed and accepted.
  • The node's Unix time is greater than or equal to state.genesis_time + (block.slot - GENESIS_SLOT) * SECONDS_PER_SLOT. (Note that leap seconds mean that slots will occasionally last SECONDS_PER_SLOT + 1 or SECONDS_PER_SLOT - 1 seconds, possibly several times a year.)

If these conditions are not met, the client should delay processing the beacon block until the conditions are all satisfied.

Beacon block production is significantly different because of the proof of stake mechanism. A client simply checks what it thinks is the canonical chain when it should create a block, and looks up what its slot number is; when the slot arrives, it either proposes or attests to a block as required. Note that this requires each node to have a clock that is roughly (i.e. within SECONDS_PER_SLOT seconds) synchronized with the other nodes.

Beacon chain fork choice rule

The beacon chain fork choice rule is a hybrid that combines justification and finality with Latest Message Driven (LMD) Greediest Heaviest Observed SubTree (GHOST). At any point in time a validator v subjectively calculates the beacon chain head as follows.

  • Abstractly define Store as the type of storage object for the chain data and store be the set of attestations and blocks that the validator v has observed and verified (in particular, block ancestors must be recursively verified). Attestations not yet included in any chain are still included in store.
  • Let finalized_head be the finalized block with the highest epoch. (A block B is finalized if there is a descendant of B in store the processing of which sets B as finalized.)
  • Let justified_head be the descendant of finalized_head with the highest epoch that has been justified for at least 1 epoch. (A block B is justified if there is a descendant of B in store the processing of which sets B as justified.) If no such descendant exists set justified_head to finalized_head.
  • Let get_ancestor(store: Store, block: BeaconBlock, slot: Slot) -> BeaconBlock be the ancestor of block with slot number slot. The get_ancestor function can be defined recursively as:
def get_ancestor(store: Store, block: BeaconBlock, slot: Slot) -> BeaconBlock:
    """
    Get the ancestor of ``block`` with slot number ``slot``; return ``None`` if not found.
    """
    if block.slot == slot:
        return block
    elif block.slot < slot:
        return None
    else:
        return get_ancestor(store, store.get_parent(block), slot)
  • Let get_latest_attestation(store: Store, validator_index: ValidatorIndex) -> Attestation be the attestation with the highest slot number in store from the validator with the given validator_index. If several such attestations exist, use the one the validator v observed first.
  • Let get_latest_attestation_target(store: Store, validator_index: ValidatorIndex) -> BeaconBlock be the target block in the attestation get_latest_attestation(store, validator_index).
  • Let get_children(store: Store, block: BeaconBlock) -> List[BeaconBlock] returns the child blocks of the given block.
  • Let justified_head_state be the resulting BeaconState object from processing the chain up to the justified_head.
  • The head is lmd_ghost(store, justified_head_state, justified_head) where the function lmd_ghost is defined below. Note that the implementation below is suboptimal; there are implementations that compute the head in time logarithmic in slot count.
def lmd_ghost(store: Store, start_state: BeaconState, start_block: BeaconBlock) -> BeaconBlock:
    """
    Execute the LMD-GHOST algorithm to find the head ``BeaconBlock``.
    """
    validators = start_state.validator_registry
    active_validator_indices = get_active_validator_indices(validators, slot_to_epoch(start_state.slot))
    attestation_targets = [
        (validator_index, get_latest_attestation_target(store, validator_index))
        for validator_index in active_validator_indices
    ]

    def get_vote_count(block: BeaconBlock) -> int:
        return sum(
            get_effective_balance(start_state.validator_balances[validator_index]) // FORK_CHOICE_BALANCE_INCREMENT
            for validator_index, target in attestation_targets
            if get_ancestor(store, target, block.slot) == block
        )

    head = start_block
    while 1:
        children = get_children(store, head)
        if len(children) == 0:
            return head
        head = max(children, key=get_vote_count)

Beacon chain state transition function

We now define the state transition function. At a high level the state transition is made up of three parts:

  1. The per-slot transitions, which happens at the start of every slot.
  2. The per-block transitions, which happens at every block.
  3. The per-epoch transitions, which happens at the end of the last slot of every epoch (i.e. (state.slot + 1) % SLOTS_PER_EPOCH == 0).

The per-slot transitions focus on the slot counter and block roots records updates; the per-block transitions generally focus on verifying aggregate signatures and saving temporary records relating to the per-block activity in the BeaconState; the per-epoch transitions focus on the validator registry, including adjusting balances and activating and exiting validators, as well as processing crosslinks and managing block justification/finalization.

Beacon blocks that trigger unhandled Python exceptions (e.g. out-of-range list accesses) and failed asserts 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 per-slot and per-epoch sections once for each skipped slot and then once for the slot containing the new block.

Per-slot processing

At every slot > GENESIS_SLOT run the following function:

def advance_slot(state: BeaconState) -> None:
    state.latest_state_roots[state.slot % SLOTS_PER_HISTORICAL_ROOT] = hash_tree_root(state)
    state.latest_block_roots[state.slot % SLOTS_PER_HISTORICAL_ROOT] = get_block_root(state, state.slot - 1)
    state.slot += 1
    if state.latest_block_header.state_root == ZERO_HASH:
        state.latest_block_header.state_root = get_state_root(state, 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

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 == hash_tree_root(state.latest_block_header)
    # Save previous block root
    state.latest_block_roots[(state.slot - 1) % SLOTS_PER_HISTORICAL_ROOT] = block.previous_block_root
    # Save current block as the new latest block
    state.latest_block_header = get_temporary_block_header(block)
    # Verify proposer signature
    proposer = state.validator_registry[get_beacon_proposer_index(state, state.slot)]
    assert bls_verify(
        pubkey=proposer.pubkey,
        message_hash=signed_root(block, "signature"),
        signature=block.signature,
        domain=get_domain(state.fork, get_current_epoch(state), DOMAIN_BEACON_BLOCK)
    )

RANDAO

def process_randao(state: BeaconState, block: BeaconBlock) -> None:
    proposer = state.validator_registry[get_beacon_proposer_index(state, state.slot)]
    # Verify that the provided randao value is valid
    assert bls_verify(
        pubkey=proposer.pubkey,
        message_hash=hash_tree_root(get_current_epoch(state)),
        signature=block.body.randao_reveal,
        domain=get_domain(state.fork, get_current_epoch(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

def process_eth1_data(state: BeaconState, block: BeaconBlock) -> None:
    for eth1_data_vote in state.eth1_data_votes:
        # If someone else has already voted for the same hash, add to its counter
        if eth1_data_vote.eth1_data == block.body.eth1_data:
            eth1_data_vote.vote_count += 1
            return
    # If we're seeing this hash for the first time, make a new counter
    state.eth1_data_votes.append(Eth1DataVote(eth1_data=block.body.eth1_data, vote_count=1))

Transactions

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:

def process_proposer_slashing(state: BeaconState,
                              proposer_slashing: ProposerSlashing) -> None:
    """
    Process ``ProposerSlashing`` transaction.
    Note that this function mutates ``state``.
    """
    proposer = state.validator_registry[proposer_slashing.proposer_index]
    # Verify that the slot is the same
    assert proposer_slashing.header_1.slot == proposer_slashing.header_2.slot
    # Verify that the shard is the same (or that both proposals are beacon chain proposals)
    assert proposer_slashing.header_1.shard == proposer_slashing.header_2.shard
    # But the roots are different!
    assert proposer_slashing.header_1.block_root != proposer_slashing.header_2.block_root
    # Proposer is not yet slashed
    assert proposer.slashed is False
    # Signatures are valid
    for header in (proposer_slashing.header_1, proposer_slashing.header_2):
        assert bls_verify(
            pubkey=proposer.pubkey,
            message_hash=signed_root(header, "signature"),           
            signature=header.signature,
            domain=get_domain(state.fork, slot_to_epoch(header.slot), DOMAIN_PROPOSAL)
        )
    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:

def process_attester_slashing(state: BeaconState,
                              attester_slashing: AttesterSlashing) -> None:
    """
    Process ``AttesterSlashing`` transaction.
    Note that this function mutates ``state``.
    """
    attestation1 = attester_slashing.slashable_attestation_1  
    attestation2 = attester_slashing.slashable_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_slashable_attestation(state, attestation1)
    assert verify_slashable_attestation(state, attestation2)
    slashable_indices = [
        index for index in attestation1.validator_indices
        if (
            index in attestation2.validator_indices and
            state.validator_registry[index].slashed is False
        )
    ]
    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:

def process_attestation(state: BeaconState, attestation: Attestation) -> None:
    """
    Process ``Attestation`` transaction.
    Note that this function mutates ``state``.
    """
    # Can't submit attestations that are too far in history (or in prehistory) 
    assert attestation.data.slot >= GENESIS_SLOT
    assert state.slot < attestation.data.slot + SLOTS_PER_EPOCH
    # Can't submit attestations too quickly
    assert attestation.data.slot + MIN_ATTESTATION_INCLUSION_DELAY <= state.slot
    # Verify that the justified epoch is correct, case 1: current epoch attestations
    if slot_to_epoch(attestation.data.slot + 1) >= get_current_epoch(state):
        assert attestation.data.justified_epoch == state.justified_epoch
    # Case 2: previous epoch attestations
    else:
        assert attestation.data.justified_epoch == state.previous_justified_epoch
    # Check that the justified block root is correct
    assert attestation.data.justified_block_root == get_block_root(
        state, get_epoch_start_slot(attestation.data.justified_epoch)
    )
    # Check that the crosslink data is valid
    acceptable_crosslink_data = {
        # Case 1: Latest crosslink matches the one in the state
        attestation.data.latest_crosslink,
        # Case 2: State has already been updated, state's latest crosslink matches the crosslink
        # the attestation is trying to create
        Crosslink(
            crosslink_data_root=attestation.data.crosslink_data_root,
            epoch=slot_to_epoch(attestation.data.slot)
        )
    }
    assert state.latest_crosslinks[attestation.data.shard] in acceptable_crosslink_data
    # Attestation must be nonempty!
    assert attestation.aggregation_bitfield != b'\x00' * len(attestation.aggregation_bitfield)
    # Custody must be empty (to be removed in phase 1)
    assert attestation.custody_bitfield == b'\x00' * len(attestation.custody_bitfield)
    # Get the committee for the specific shard that this attestation is for
    crosslink_committee = [
        committee for committee, shard in get_crosslink_committees_at_slot(state, attestation.data.slot)
        if shard == attestation.data.shard
    ][0]
    # Custody bitfield must be a subset of the attestation bitfield
    for i in range(len(crosslink_committee)):
        if get_bitfield_bit(attestation.aggregation_bitfield, i) == 0b0:
            assert get_bitfield_bit(attestation.custody_bitfield, i) == 0b0
    # Verify aggregate signature
    participants = get_attestation_participants(state, attestation.data, attestation.aggregation_bitfield)
    custody_bit_1_participants = get_attestation_participants(state, attestation.data, attestation.custody_bitfield)
    custody_bit_0_participants = [i for i in participants if i not in custody_bit_1_participants]

    assert bls_verify_multiple(
        pubkeys=[
            bls_aggregate_pubkeys([state.validator_registry[i].pubkey for i in custody_bit_0_participants]),
            bls_aggregate_pubkeys([state.validator_registry[i].pubkey for i in custody_bit_1_participants]),
        ],
        message_hashes=[
            hash_tree_root(AttestationDataAndCustodyBit(data=attestation.data, custody_bit=0b0)),
            hash_tree_root(AttestationDataAndCustodyBit(data=attestation.data, custody_bit=0b1)),
        ],
        signature=attestation.aggregate_signature,
        domain=get_domain(state.fork, slot_to_epoch(attestation.data.slot), DOMAIN_ATTESTATION),
    )
    # Crosslink data root is zero (to be removed in phase 1)
    assert attestation.data.crosslink_data_root == ZERO_HASH
    # Apply the attestation
    pending_attestation = PendingAttestation(
        data=attestation.data,
        aggregation_bitfield=attestation.aggregation_bitfield,
        custody_bitfield=attestation.custody_bitfield,
        inclusion_slot=state.slot
    )
    if slot_to_epoch(attestation.data.slot) == get_current_epoch(state):
        state.current_epoch_attestations.append(pending_attestation)
    elif slot_to_epoch(attestation.data.slot) == get_previous_epoch(state):
        state.previous_epoch_attestations.append(pending_attestation)
Deposits

Verify that len(block.body.deposits) <= MAX_DEPOSITS.

For each deposit in block.body.deposits, run process_deposit(state, deposit).

Voluntary exits

Verify that len(block.body.voluntary_exits) <= MAX_VOLUNTARY_EXITS.

For each exit in block.body.voluntary_exits, run the following function:

def process_exit(state: BeaconState, exit: VoluntaryExit) -> None:
    """
    Process ``VoluntaryExit`` transaction.
    Note that this function mutates ``state``.
    """
    validator = state.validator_registry[exit.validator_index]
    # Verify the validator has not yet exited
    assert validator.exit_epoch > get_delayed_activation_exit_epoch(get_current_epoch(state))
    # Exits must specify an epoch when they become valid; they are not valid before then
    assert get_current_epoch(state) >= exit.epoch
    # Verify signature
    assert bls_verify(
        pubkey=validator.pubkey,
        message_hash=signed_root(exit, "signature"),
        signature=exit.signature,
        domain=get_domain(state.fork, exit.epoch, DOMAIN_VOLUNTARY_EXIT)
    )
    # Run the exit
    initiate_validator_exit(state, exit.validator_index)
Transfers

Note: Transfers are a temporary functionality for phases 0 and 1, to be removed in phase 2.

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:

def process_transfer(state: BeaconState, transfer: Transfer) -> None:
    """
    Process ``Transfer`` transaction.
    Note that this function mutates ``state``.
    """
    # Verify the amount and fee aren't individually too big (for anti-overflow purposes)
    assert state.validator_balances[transfer.sender] >= max(transfer.amount, transfer.fee)
    # Verify that we have enough ETH to send, and that after the transfer the balance will be either
    # exactly zero or at least MIN_DEPOSIT_AMOUNT
    assert (
        state.validator_balances[transfer.sender] == transfer.amount + transfer.fee or
        state.validator_balances[transfer.sender] >= transfer.amount + transfer.fee + MIN_DEPOSIT_AMOUNT
    )
    # A transfer is valid in only one slot
    assert state.slot == transfer.slot
    # Only withdrawn or not-yet-deposited accounts can transfer
    assert (
        get_current_epoch(state) >= state.validator_registry[transfer.sender].withdrawable_epoch or
        state.validator_registry[transfer.sender].activation_epoch == FAR_FUTURE_EPOCH
    )
    # 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(
        pubkey=transfer.pubkey,
        message_hash=signed_root(transfer, "signature"),
        signature=transfer.signature,
        domain=get_domain(state.fork, slot_to_epoch(transfer.slot), DOMAIN_TRANSFER)
    )
    # Process the transfer
    state.validator_balances[transfer.sender] -= transfer.amount + transfer.fee
    state.validator_balances[transfer.to] += transfer.amount
    state.validator_balances[get_beacon_proposer_index(state, state.slot)] += transfer.fee

Per-epoch processing

The steps below happen when (state.slot + 1) % SLOTS_PER_EPOCH == 0.

Helper functions

We define some helper functions:

def get_current_total_balance(state: BeaconState) -> Gwei:
    return get_total_balance(state, get_active_validator_indices(state.validator_registry, get_current_epoch(state)))

def get_previous_total_balance(state: BeaconState) -> Gwei:
    return get_total_balance(state, get_active_validator_indices(state.validator_registry, get_previous_epoch(state)))

def get_attesting_indices(state: BeaconState, attestations: List[PendingAttestation]) -> List[ValidatorIndex]:
    output = set()
    for a in attestations:
        output = output.union([get_attestation_participants(state, a.data, a.aggregation_bitfield)])
    return sorted(list(output))

def get_attesting_balance(state: BeaconState, attestations: List[PendingAttestation]) -> List[ValidatorIndex]:
    return get_total_balance(state, get_attesting_indices(state, attestations))

def get_current_epoch_boundary_attestations(state: BeaconState) -> List[PendingAttestation]:
    return [
        a for a in state.current_epoch_attestations if 
        a.data.epoch_boundary_root == get_block_root(state, get_epoch_start_slot(get_current_epoch(state)))
    ]

def get_previous_epoch_boundary_attestations(state: BeaconState) -> List[PendingAttestation]:
    return [
        a for a in state.previous_epoch_attestations if 
        a.data.epoch_boundary_root == get_block_root(state, get_epoch_start_slot(get_previous_epoch(state)))
    ]

def get_previous_epoch_matching_head_attestations(state: BeaconState) -> List[PendingAttestation]:
    return [
        a for a in state.previous_epoch_attestations if
        a.data.beacon_block_root == get_block_root(state, a.data.slot)
    ]

Note: Total balances computed for the previous epoch might be marginally different than the actual total balances during the previous epoch transition. Due to the tight bound on validator churn each epoch and small per-epoch rewards/penalties, the potential balance difference is very low and only marginally affects consensus safety.

def get_winning_root_and_participants(state: BeaconState, shard: Shard) -> Tuple[Bytes32, List[ValidatorIndex]]:
    all_attestations = state.current_epoch_attestations + state.previous_epoch_attestations
    valid_attestations = [
        a for a in all_attestations if a.data.latest_crosslink == state.latest_crosslinks[shard]
    ]
    all_roots = [a.data.crosslink_data_root for a in valid_attestations]

    # handle when no attestations for shard available
    if len(all_roots) == 0:
        return ZERO_HASH, []

    def get_attestations_for(root) -> List[PendingAttestation]:
        return [a for a in valid_attestations if a.data.crosslink_data_root == root]

    winning_root = max(all_roots, key=lambda r: get_attesting_balance(state, get_attestations_for(r)))

    return winning_root, get_attesting_indices(state, get_attestations_for(winning_root))

def earliest_attestation(state: BeaconState, validator_index: ValidatorIndex) -> PendingAttestation:
    return min([
        a for a in state.previous_epoch_attestations if
        validator_index in get_attestation_participants(state, a.data, a.aggregation_bitfield)
    ], key=lambda a: a.inclusion_slot)

def inclusion_slot(state: BeaconState, validator_index: ValidatorIndex) -> Slot:
    return earliest_attestation(state, validator_index).inclusion_slot

def inclusion_distance(state: BeaconState, validator_index: ValidatorIndex) -> int:
    attestation = earliest_attestation(state, validator_index)
    return attestation.inclusion_slot - attestation.data.slot

Justification

Run the following function:

def update_justification_and_finalization(state: BeaconState) -> None:
    new_justified_epoch = state.justified_epoch
    # Rotate the justification bitfield up one epoch to make room for the current epoch
    state.justification_bitfield <<= 1
    # If the previous epoch gets justified, fill the second last bit
    previous_boundary_attesting_balance = get_attesting_balance(state, get_previous_epoch_boundary_attestations(state))
    if previous_boundary_attesting_balance * 3 >= get_previous_total_balance(state) * 2:
        new_justified_epoch = get_current_epoch(state) - 1
        state.justification_bitfield |= 2
    # If the current epoch gets justified, fill the last bit
    current_boundary_attesting_balance = get_attesting_balance(state, get_current_epoch_boundary_attestations(state))
    if current_boundary_attesting_balance * 3 >= get_current_total_balance(state) * 2:
        new_justified_epoch = get_current_epoch(state)
        state.justification_bitfield |= 1
        
    # Process finalizations
    bitfield = state.justification_bitfield
    current_epoch = get_current_epoch(state)
    # The 2nd/3rd/4th most recent epochs are all justified, the 2nd using the 4th as source
    if (bitfield >> 1) % 8 == 0b111 and state.previous_justified_epoch == current_epoch - 3:
        state.finalized_epoch = state.previous_justified_epoch
    # The 2nd/3rd most recent epochs are both justified, the 2nd using the 3rd as source
    if (bitfield >> 1) % 4 == 0b11 and state.previous_justified_epoch == current_epoch - 2:
        state.finalized_epoch = state.previous_justified_epoch
    # The 1st/2nd/3rd most recent epochs are all justified, the 1st using the 3rd as source
    if (bitfield >> 0) % 8 == 0b111 and state.justified_epoch == current_epoch - 2:
        state.finalized_epoch = state.justified_epoch
    # The 1st/2nd most recent epochs are both justified, the 1st using the 2nd as source        
    if (bitfield >> 0) % 4 == 0b11 and state.justified_epoch == current_epoch - 1:
        state.finalized_epoch = state.justified_epoch
        
    # Rotate justified epochs
    state.previous_justified_epoch = state.justified_epoch
    state.justified_epoch = new_justified_epoch

Crosslinks

Run the following function:

def process_crosslinks(state: BeaconState) -> None:
    current_epoch = get_current_epoch(state)
    previous_epoch = current_epoch - 1
    next_epoch = current_epoch + 1
    for slot in range(get_epoch_start_slot(previous_epoch), get_epoch_start_slot(next_epoch)):
        for crosslink_committee, shard in get_crosslink_committees_at_slot(state, slot):
            winning_root, participants = get_winning_root_and_participants(state, shard)
            participating_balance = get_total_balance(state, participants)
            total_balance = get_total_balance(state, crosslink_committee)
            if 3 * participating_balance >= 2 * total_balance:
                state.latest_crosslinks[shard] = Crosslink(
                    epoch=slot_to_epoch(slot),
                    crosslink_data_root=winning_root
                )

Eth1 data

Run the following function:

def maybe_reset_eth1_period(state: BeaconState) -> None:
    if (get_current_epoch(state) + 1) % EPOCHS_PER_ETH1_VOTING_PERIOD == 0:
        for eth1_data_vote in state.eth1_data_votes:
            # If a majority of all votes were for a particular eth1_data value,
            # then set that as the new canonical value
            if eth1_data_vote.vote_count * 2 > EPOCHS_PER_ETH1_VOTING_PERIOD * SLOTS_PER_EPOCH:
                state.latest_eth1_data = eth1_data_vote.eth1_data
        state.eth1_data_votes = []

Rewards and penalties

First, we define some additional helpers:

def get_base_reward(state: BeaconState, index: ValidatorIndex) -> Gwei:
    adjusted_quotient = integer_squareroot(get_previous_total_balance(state)) // BASE_REWARD_QUOTIENT
    return get_effective_balance(state, index) // adjusted_quotient // 5

def get_inactivity_penalty(state: BeaconState, index: ValidatorIndex) -> Gwei:
    epochs_since_finality = get_current_epoch(state) + 1 - state.finalized_epoch
    return (
        get_base_reward(state, index) +
        get_effective_balance(state, index) * epochs_since_finality // INACTIVITY_PENALTY_QUOTIENT // 2
    )

Note: When applying penalties in the following balance recalculations implementers should make sure the uint64 does not underflow.

Justification and finalization
def get_justification_and_finalization_deltas(state: BeaconState) -> Tuple[List[Gwei], List[Gwei]]:
    epochs_since_finality = get_current_epoch(state) + 1 - state.finalized_epoch
    if epochs_since_finality <= 4:
        return compute_normal_justification_and_finalization_deltas(state)
    else:
        return compute_inactivity_leak_deltas(state)

When blocks are finalizing normally...

def compute_normal_justification_and_finalization_deltas(state: BeaconState) -> Tuple[List[Gwei], List[Gwei]]:
    # deltas[0] for rewards
    # deltas[1] for penalties
    deltas = [
        [0 for index in range(len(state.validator_registry))],
        [0 for index in range(len(state.validator_registry))]
    ]
    # Some helper variables
    boundary_attestations = get_previous_epoch_boundary_attestations(state)
    boundary_attesting_balance = get_attesting_balance(state, boundary_attestations)
    total_balance = get_previous_total_balance(state)
    total_attesting_balance = get_attesting_balance(state, state.previous_epoch_attestations)
    matching_head_attestations = get_previous_epoch_matching_head_attestations(state)
    matching_head_balance = get_attesting_balance(state, matching_head_attestations)
    # Process rewards or penalties for all validators
    for index in get_active_validator_indices(state.validator_registry, get_previous_epoch(state)):
        # Expected FFG source
        if index in get_attesting_indices(state, state.previous_epoch_attestations):
            deltas[0][index] += get_base_reward(state, index) * total_attesting_balance // total_balance
            # Inclusion speed bonus
            deltas[0][index] += (
                get_base_reward(state, index) * MIN_ATTESTATION_INCLUSION_DELAY //
                inclusion_distance(state, index)
            )
        else:
            deltas[1][index] += get_base_reward(state, index)
        # Expected FFG target
        if index in get_attesting_indices(state, boundary_attestations):
            deltas[0][index] += get_base_reward(state, index) * boundary_attesting_balance // total_balance
        else:
            deltas[1][index] += get_base_reward(state, index)
        # Expected head
        if index in get_attesting_indices(state, matching_head_attestations):
            deltas[0][index] += get_base_reward(state, index) * matching_head_balance // total_balance
        else:
            deltas[1][index] += get_base_reward(state, index)
        # Proposer bonus
        proposer_index = get_beacon_proposer_index(state, inclusion_slot(state, index))
        deltas[0][proposer_index] += base_reward(state, index) // ATTESTATION_INCLUSION_REWARD_QUOTIENT
    return deltas

When blocks are not finalizing normally...

def compute_inactivity_leak_deltas(state: BeaconState) -> Tuple[List[Gwei], List[Gwei]]:
    # deltas[0] for rewards
    # deltas[1] for penalties
    deltas = [
        [0 for index in range(len(state.validator_registry))],
        [0 for index in range(len(state.validator_registry))]
    ]
    boundary_attestations = get_previous_epoch_boundary_attestations(state)
    matching_head_attestations = get_previous_epoch_matching_head_attestations(state)
    active_validator_indices = get_active_validator_indices(state.validator_registry, get_previous_epoch(state))
    epochs_since_finality = get_current_epoch(state) + 1 - state.finalized_epoch
    for index in active_validator_indices:
        if index not in get_attesting_indices(state, state.previous_epoch_attestations):
            deltas[1][index] += get_inactivity_penalty(state, index, epochs_since_finality)
        else:
            # If a validator did attest, apply a small penalty for getting attestations included late
            deltas[0][index] += (
                base_reward(state, index) * MIN_ATTESTATION_INCLUSION_DELAY //
                inclusion_distance(state, index)
            )
            deltas[1][index] += base_reward(state, index)
        if index not in get_attesting_indices(state, boundary_attestations):
            deltas[1][index] += get_inactivity_penalty(state, index, epochs_since_finality)
        if index not in get_attesting_indices(state, matching_head_attestations):
            deltas[1][index] += get_base_reward(state, index)
    # Penalize slashed-but-inactive validators as though they were active but offline
    for index in range(len(state.validator_registry)):
        if index not in active_validator_indices and state.validator_registry[index].slashed:
            deltas[1][index] += (
                2 * get_inactivity_penalty(state, index, epochs_since_finality) +
                get_base_reward(state, index)
            )
    return deltas
Crosslinks
def get_crosslink_deltas(state: BeaconState) -> Tuple[List[Gwei], List[Gwei]]:
    # deltas[0] for rewards
    # deltas[1] for penalties
    deltas = [
        [0 for index in range(len(state.validator_registry))],
        [0 for index in range(len(state.validator_registry))]
    ]
    previous_epoch_start_slot = get_epoch_start_slot(get_previous_epoch(state))
    current_epoch_start_slot = get_epoch_start_slot(get_current_epoch(state))
    for slot in range(previous_epoch_start_slot, current_epoch_start_slot):
        for crosslink_committee, shard in get_crosslink_committees_at_slot(state, slot):
            winning_root, participants = get_winning_root_and_participants(state, shard)
            participating_balance = get_total_balance(state, participants)
            total_balance = get_total_balance(state, crossling_committee)
            for index in crosslink_committee:
                if index in participants:
                    deltas[0][index] += get_base_reward(state, index) * participating_balance // total_balance
                else:
                    deltas[1][index] += get_base_reward(state, index)
    return deltas

Apply rewards

Run the following:

def apply_rewards(state: BeaconState) -> None:
    deltas1 = get_justification_and_finalization_deltas(state)
    deltas2 = get_crosslink_deltas(state)
    for i in range(len(state.validator_registry)):
        state.validator_balances[i] = max(
            0,
            state.validator_balances[i] + deltas1[0][i] + deltas2[0][i] - deltas1[1][i] - deltas2[1][i]
        )

Ejections

  • Run process_ejections(state).
def process_ejections(state: BeaconState) -> None:
    """
    Iterate through the validator registry
    and eject active validators with balance below ``EJECTION_BALANCE``.
    """
    for index in get_active_validator_indices(state.validator_registry, get_current_epoch(state)):
        if state.validator_balances[index] < EJECTION_BALANCE:
            exit_validator(state, index)

Validator registry and shuffling seed data

def should_update_validator_registry(state: BeaconState) -> bool:
    # Must have finalized a new block
    if state.finalized_epoch <= state.validator_registry_update_epoch:
        return False
    # Must have processed new crosslinks on all shards of the current epoch
    shards_to_check = [
        (state.current_shuffling_start_shard + i) % SHARD_COUNT
        for i in range(get_current_epoch_committee_count(state))
    ]
    for shard in shards_to_check:
        if state.latest_crosslinks[shard].epoch <= state.validator_registry_update_epoch:
            return False
    return True

def update_validator_registry(state: BeaconState) -> None:
    """
    Update validator registry.
    Note that this function mutates ``state``.
    """
    current_epoch = get_current_epoch(state)
    # The active validators
    active_validator_indices = get_active_validator_indices(state.validator_registry, current_epoch)
    # The total effective balance of active validators
    total_balance = get_total_balance(state, active_validator_indices)

    # The maximum balance churn in Gwei (for deposits and exits separately)
    max_balance_churn = max(
        MAX_DEPOSIT_AMOUNT,
        total_balance // (2 * MAX_BALANCE_CHURN_QUOTIENT)
    )

    # Activate validators within the allowable balance churn
    balance_churn = 0
    for index, validator in enumerate(state.validator_registry):
        if validator.activation_epoch == FAR_FUTURE_EPOCH and state.validator_balances[index] >= MAX_DEPOSIT_AMOUNT:
            # Check the balance churn would be within the allowance
            balance_churn += get_effective_balance(state, index)
            if balance_churn > max_balance_churn:
                break

            # Activate validator
            activate_validator(state, index, is_genesis=False)

    # Exit validators within the allowable balance churn
    balance_churn = 0
    for index, validator in enumerate(state.validator_registry):
        if validator.exit_epoch == FAR_FUTURE_EPOCH and validator.initiated_exit:
            # Check the balance churn would be within the allowance
            balance_churn += get_effective_balance(state, index)
            if balance_churn > max_balance_churn:
                break

            # Exit validator
            exit_validator(state, index)

    state.validator_registry_update_epoch = current_epoch

Run the following function:

def update_registry_and_shuffling_data(state: BeaconState) -> None:
    # First set previous shuffling data to current shuffling data
    state.previous_shuffling_epoch = state.current_shuffling_epoch
    state.previous_shuffling_start_shard = state.current_shuffling_start_shard
    state.previous_shuffling_seed = state.current_shuffling_seed
    current_epoch = get_current_epoch(state)
    next_epoch = current_epoch + 1
    # Check if we should update, and if so, update
    if should_update_validator_registry(state):
        update_validator_registry(state)
        # If we update the registry, update the shuffling data and shards as well
        state.current_shuffling_epoch = next_epoch
        state.current_shuffling_start_shard = (
            state.current_shuffling_start_shard +
            get_current_epoch_committee_count(state) % SHARD_COUNT
        )
        state.current_shuffling_seed = generate_seed(state, state.current_shuffling_epoch)
    else:
        # If processing at least one crosslink keeps failing, then reshuffle every power of two,
        # but don't update the current_shuffling_start_shard
        epochs_since_last_registry_update = current_epoch - state.validator_registry_update_epoch
        if epochs_since_last_registry_update > 1 and is_power_of_two(epochs_since_last_registry_update):
            state.current_shuffling_epoch = next_epoch
            state.current_shuffling_seed = generate_seed(state, state.current_shuffling_epoch)

Invariant: the active index root that is hashed into the shuffling seed actually is the hash_tree_root of the validator set that is used for that epoch.

Slashings and exit queue

Run process_slashings(state) and process_exit_queue(state):

def process_slashings(state: BeaconState) -> None:
    """
    Process the slashings.
    Note that this function mutates ``state``.
    """
    current_epoch = get_current_epoch(state)
    active_validator_indices = get_active_validator_indices(state.validator_registry, current_epoch)
    total_balance = get_total_balance(state, active_validator_indices)

    # Compute `total_penalties`
    epoch_index = current_epoch % LATEST_SLASHED_EXIT_LENGTH
    total_at_start = state.latest_slashed_balances[(epoch_index + 1) % LATEST_SLASHED_EXIT_LENGTH]
    total_at_end = state.latest_slashed_balances[epoch_index]
    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(
                get_effective_balance(state, index) * min(total_penalties * 3, total_balance) // total_balance,
                get_effective_balance(state, index) // MIN_PENALTY_QUOTIENT
            )
            state.validator_balances[index] -= penalty

def process_exit_queue(state: BeaconState) -> None:
    """
    Process the exit queue.
    Note that this function mutates ``state``.
    """
    def eligible(index):
        validator = state.validator_registry[index]
        # Filter out dequeued validators
        if validator.withdrawable_epoch != FAR_FUTURE_EPOCH:
            return False
        # Dequeue if the minimum amount of time has passed
        else:
            return get_current_epoch(state) >= validator.exit_epoch + MIN_VALIDATOR_WITHDRAWABILITY_DELAY

    eligible_indices = filter(eligible, list(range(len(state.validator_registry))))
    # Sort in order of exit epoch, and validators that exit within the same epoch exit in order of validator index
    sorted_indices = sorted(eligible_indices, key=lambda index: state.validator_registry[index].exit_epoch)
    for dequeues, index in enumerate(sorted_indices):
        if dequeues >= MAX_EXIT_DEQUEUES_PER_EPOCH:
            break
        prepare_validator_for_withdrawal(state, index)

Final updates

Run the following function:

def finish_epoch_update(state: BeaconState) -> None:
    current_epoch = get_current_epoch(state)
    next_epoch = current_epoch + 1
    # 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.validator_registry, 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:
        state.historical_roots.append(merkle_root(state.latest_block_roots + state.latest_state_roots))
    # Rotate current/previous epoch attestations
    state.previous_epoch_attestations = state.current_epoch_attestations
    state.current_epoch_attestations = []

State root verification

Verify block.state_root == hash_tree_root(state) if there exists a block for the slot being processed.

References

This section is divided into Normative and Informative references. Normative references are those that must be read in order to implement this specification, while Informative references are merely that, information. An example of the former might be the details of a required consensus algorithm, and an example of the latter might be a pointer to research that demonstrates why a particular consensus algorithm might be better suited for inclusion in the standard than another.

Normative

Informative

casper-ffg
  Casper the Friendly Finality Gadget. V. Buterin and V. Griffith. URL: https://arxiv.org/abs/1710.09437

python-poc
  Python proof-of-concept implementation. Ethereum Foundation. URL: https://github.com/ethereum/beacon_chain

Copyright

Copyright and related rights waived via CC0.