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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 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 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 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 get exactly one chance to make an attestation.
  • Finalized, justified - see the Casper FFG paper.
  • 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

Note: the default mainnet values for the constants are included here for spec-design purposes. The different configurations for mainnet, testnets, and yaml-based testing can be found in the configs/constant_presets/ directory. These configurations are updated for releases, but may be out of sync during dev changes.

Misc

Name Value
SHARD_COUNT 2**10 (= 1,024)
TARGET_COMMITTEE_SIZE 2**7 (= 128)
MAX_INDICES_PER_ATTESTATION 2**12 (= 4,096)
MIN_PER_EPOCH_CHURN_LIMIT 2**2 (= 4)
CHURN_LIMIT_QUOTIENT 2**16 (= 65,536)
BASE_REWARDS_PER_EPOCH 5
SHUFFLE_ROUND_COUNT 90
  • For the safety of crosslinks TARGET_COMMITTEE_SIZE exceeds the recommended minimum committee size of 111; with sufficient active validators (at least SLOTS_PER_EPOCH * TARGET_COMMITTEE_SIZE), the shuffling algorithm ensures committee sizes of at least TARGET_COMMITTEE_SIZE. (Unbiasable randomness with a Verifiable Delay Function (VDF) will improve committee robustness and lower the safe minimum committee size.)

Deposit contract

Name Value
DEPOSIT_CONTRACT_ADDRESS TBD
DEPOSIT_CONTRACT_TREE_DEPTH 2**5 (= 32)

Gwei values

Name Value Unit
MIN_DEPOSIT_AMOUNT 2**0 * 10**9 (= 1,000,000,000) Gwei
MAX_EFFECTIVE_BALANCE 2**5 * 10**9 (= 32,000,000,000) Gwei
EJECTION_BALANCE 2**4 * 10**9 (= 16,000,000,000) Gwei
EFFECTIVE_BALANCE_INCREMENT 2**0 * 10**9 (= 1,000,000,000) Gwei

Initial values

Name Value
GENESIS_SLOT 0
GENESIS_EPOCH 0
FAR_FUTURE_EPOCH 2**64 - 1
ZERO_HASH int_to_bytes32(0)
BLS_WITHDRAWAL_PREFIX_BYTE int_to_bytes1(0)

Time parameters

Name Value Unit Duration
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
SLOTS_PER_ETH1_VOTING_PERIOD 2**10 (= 1,024) slots ~1.7 hours
SLOTS_PER_HISTORICAL_ROOT 2**13 (= 8,192) slots ~13 hours
MIN_VALIDATOR_WITHDRAWABILITY_DELAY 2**8 (= 256) epochs ~27 hours
PERSISTENT_COMMITTEE_PERIOD 2**11 (= 2,048) epochs 9 days
MAX_CROSSLINK_EPOCHS 2**6 (= 64) epochs ~7 hours
MIN_EPOCHS_TO_INACTIVITY_PENALTY 2**2 (= 4) epochs 25.6 minutes
  • MAX_CROSSLINK_EPOCHS should be a small constant times SHARD_COUNT // SLOTS_PER_EPOCH

State list lengths

Name Value Unit Duration
LATEST_RANDAO_MIXES_LENGTH 2**13 (= 8,192) epochs ~36 days
LATEST_ACTIVE_INDEX_ROOTS_LENGTH 2**13 (= 8,192) epochs ~36 days
LATEST_SLASHED_EXIT_LENGTH 2**13 (= 8,192) epochs ~36 days

Reward and penalty quotients

Name Value
BASE_REWARD_QUOTIENT 2**5 (= 32)
WHISTLEBLOWING_REWARD_QUOTIENT 2**9 (= 512)
PROPOSER_REWARD_QUOTIENT 2**3 (= 8)
INACTIVITY_PENALTY_QUOTIENT 2**25 (= 33,554,432)
MIN_SLASHING_PENALTY_QUOTIENT 2**5 (= 32)
  • The BASE_REWARD_QUOTIENT is NOT final. Once all other protocol details are finalized it will be adjusted, to target a theoretical maximum total issuance of 2**21 ETH per year if 2**27 ETH is validating (and therefore 2**20 per year if 2**25 ETH is validating, etc etc)
  • The INACTIVITY_PENALTY_QUOTIENT equals INVERSE_SQRT_E_DROP_TIME**2 where INVERSE_SQRT_E_DROP_TIME := 2**12 epochs (~18 days) is the time it takes the inactivity penalty to reduce the balance of non-participating validators 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 operations per block

Name Value
MAX_PROPOSER_SLASHINGS 2**4 (= 16)
MAX_ATTESTER_SLASHINGS 2**0 (= 1)
MAX_ATTESTATIONS 2**7 (= 128)
MAX_DEPOSITS 2**4 (= 16)
MAX_VOLUNTARY_EXITS 2**4 (= 16)
MAX_TRANSFERS 0

Signature domains

Name Value
DOMAIN_BEACON_PROPOSER 0
DOMAIN_RANDAO 1
DOMAIN_ATTESTATION 2
DOMAIN_DEPOSIT 3
DOMAIN_VOLUNTARY_EXIT 4
DOMAIN_TRANSFER 5

Data structures

The following data structures are defined as SimpleSerialize (SSZ) objects.

The types are defined topologically to aid in facilitating an executable version of the spec.

Misc dependencies

Fork

{
    # Previous fork version
    'previous_version': 'bytes4',
    # Current fork version
    'current_version': 'bytes4',
    # Fork epoch number
    'epoch': 'uint64',
}

Crosslink

{
    # Epoch number
    'epoch': 'uint64',
    # Root of the previous crosslink
    'previous_crosslink_root': 'bytes32',
    # Root of the crosslinked shard data since the previous crosslink
    'crosslink_data_root': 'bytes32',
}

Eth1Data

{
    # Root of the deposit tree
    'deposit_root': 'bytes32',
    # Total number of deposits
    'deposit_count': 'uint64',
    # Block hash
    'block_hash': 'bytes32',
}

AttestationData

{
    # LMD GHOST vote
    'slot': 'uint64',
    'beacon_block_root': 'bytes32',

    # FFG vote
    'source_epoch': 'uint64',
    'source_root': 'bytes32',
    'target_root': 'bytes32',

    # Crosslink vote
    'shard': 'uint64',
    'previous_crosslink_root': 'bytes32',
    'crosslink_data_root': 'bytes32',
}

AttestationDataAndCustodyBit

{
    # Attestation data
    'data': AttestationData,
    # Custody bit
    'custody_bit': 'bool',
}

IndexedAttestation

{
    # Validator indices
    'custody_bit_0_indices': ['uint64'],
    'custody_bit_1_indices': ['uint64'],
    # Attestation data
    'data': AttestationData,
    # Aggregate signature
    'signature': 'bytes96',
}

DepositData

{
    # BLS pubkey
    'pubkey': 'bytes48',
    # Withdrawal credentials
    'withdrawal_credentials': 'bytes32',
    # Amount in Gwei
    'amount': 'uint64',
    # Container self-signature
    'signature': 'bytes96',
}

BeaconBlockHeader

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

Validator

{
    # BLS public key
    'pubkey': 'bytes48',
    # Withdrawal credentials
    'withdrawal_credentials': 'bytes32',
    # Epoch when became eligible for activation
    'activation_eligibility_epoch': 'uint64',
    # Epoch when validator activated
    'activation_epoch': 'uint64',
    # Epoch when validator exited
    'exit_epoch': 'uint64',
    # Epoch when validator is eligible to withdraw
    'withdrawable_epoch': 'uint64',
    # Was the validator slashed
    'slashed': 'bool',
    # Effective balance
    'effective_balance': 'uint64',
}

PendingAttestation

{
    # Attester aggregation bitfield
    'aggregation_bitfield': 'bytes',
    # Attestation data
    'data': AttestationData,
    # Inclusion slot
    'inclusion_slot': 'uint64',
    # Proposer index
    'proposer_index': 'uint64',
}

HistoricalBatch

{
    # Block roots
    'block_roots': ['bytes32', SLOTS_PER_HISTORICAL_ROOT],
    # State roots
    'state_roots': ['bytes32', SLOTS_PER_HISTORICAL_ROOT],
}

Beacon operations

ProposerSlashing

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

AttesterSlashing

{
    # First attestation
    'attestation_1': IndexedAttestation,
    # Second attestation
    'attestation_2': IndexedAttestation,
}

Attestation

{
    # Attester aggregation bitfield
    'aggregation_bitfield': 'bytes',
    # Attestation data
    'data': AttestationData,
    # Custody bitfield
    'custody_bitfield': 'bytes',
    # BLS aggregate signature
    'signature': 'bytes96',
}

Deposit

{
    # Branch in the deposit tree
    'proof': ['bytes32', DEPOSIT_CONTRACT_TREE_DEPTH],
    # Index in the deposit tree
    'index': 'uint64',
    # Data
    'data': DepositData,
}

VoluntaryExit

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

Transfer

{
    # Sender index
    'sender': 'uint64',
    # Recipient index
    'recipient': 'uint64',
    # Amount in Gwei
    'amount': 'uint64',
    # Fee in Gwei for block proposer
    'fee': 'uint64',
    # Inclusion slot
    'slot': 'uint64',
    # Sender withdrawal pubkey
    'pubkey': 'bytes48',
    # Sender signature
    'signature': 'bytes96',
}

Beacon blocks

BeaconBlockBody

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

BeaconBlock

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

Beacon state

BeaconState

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

    # Validator registry
    'validator_registry': [Validator],
    'balances': ['uint64'],

    # Randomness and committees
    'latest_randao_mixes': ['bytes32', LATEST_RANDAO_MIXES_LENGTH],
    'latest_start_shard': 'uint64',

    # Finality
    'previous_epoch_attestations': [PendingAttestation],
    'current_epoch_attestations': [PendingAttestation],
    'previous_justified_epoch': 'uint64',
    'current_justified_epoch': 'uint64',
    'previous_justified_root': 'bytes32',
    'current_justified_root': 'bytes32',
    'justification_bitfield': 'uint64',
    'finalized_epoch': 'uint64',
    'finalized_root': 'bytes32',

    # Recent state
    'current_crosslinks': [Crosslink, SHARD_COUNT],
    'previous_crosslinks': [Crosslink, SHARD_COUNT],
    'latest_block_roots': ['bytes32', SLOTS_PER_HISTORICAL_ROOT],
    'latest_state_roots': ['bytes32', SLOTS_PER_HISTORICAL_ROOT],
    'latest_active_index_roots': ['bytes32', LATEST_ACTIVE_INDEX_ROOTS_LENGTH],
    'latest_slashed_balances': ['uint64', LATEST_SLASHED_EXIT_LENGTH],  # Balances slashed at every withdrawal period
    'latest_block_header': BeaconBlockHeader,  # `latest_block_header.state_root == ZERO_HASH` temporarily
    'historical_roots': ['bytes32'],

    # Ethereum 1.0 chain data
    'latest_eth1_data': Eth1Data,
    'eth1_data_votes': [Eth1Data],
    'deposit_index': 'uint64',
}

Custom Types

We define the following Python custom types for type hinting and readability:

Name SSZ equivalent Description
Slot uint64 a slot number
Epoch uint64 an epoch number
Shard uint64 a shard number
ValidatorIndex uint64 a validator registry index
Gwei uint64 an amount in Gwei
Bytes32 bytes32 32 bytes of binary data
BLSPubkey bytes48 a BLS12-381 public key
BLSSignature bytes96 a BLS12-381 signature

Helper functions

Note: The definitions below are for specification purposes and are not necessarily optimal implementations.

xor

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

hash

The hash function is SHA256.

Note: We aim to migrate to a S[T/N]ARK-friendly hash function in a future Ethereum 2.0 deployment phase.

hash_tree_root

def hash_tree_root(object: SSZSerializable) -> Bytes32 is a function for hashing objects into a single root utilizing a hash tree structure. hash_tree_root is defined in the SimpleSerialize spec.

signing_root

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

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 the current epoch if it's genesis epoch.
    """
    current_epoch = get_current_epoch(state)
    return (current_epoch - 1) if current_epoch > GENESIS_EPOCH else current_epoch

get_current_epoch

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

is_slashable_validator

def is_slashable_validator(validator: Validator, epoch: Epoch) -> bool:
    """
    Check if ``validator`` is slashable.
    """
    return validator.slashed is False and (validator.activation_epoch <= epoch < validator.withdrawable_epoch)

get_active_validator_indices

def get_active_validator_indices(state: BeaconState, epoch: Epoch) -> List[ValidatorIndex]:
    """
    Get active validator indices at ``epoch``.
    """
    return [i for i, v in enumerate(state.validator_registry) if is_active_validator(v, epoch)]

increase_balance

def increase_balance(state: BeaconState, index: ValidatorIndex, delta: Gwei) -> None:
    """
    Increase validator balance by ``delta``.
    """
    state.balances[index] += delta

decrease_balance

def decrease_balance(state: BeaconState, index: ValidatorIndex, delta: Gwei) -> None:
    """
    Decrease validator balance by ``delta`` with underflow protection.
    """
    state.balances[index] = 0 if delta > state.balances[index] else state.balances[index] - delta

get_permuted_index

def get_permuted_index(index: int, list_size: int, seed: Bytes32) -> int:
    """
    Return `p(index)` in a pseudorandom permutation `p` of `0...list_size - 1` with ``seed`` as entropy.

    Utilizes 'swap or not' shuffling found in
    https://link.springer.com/content/pdf/10.1007%2F978-3-642-32009-5_1.pdf
    See the 'generalized domain' algorithm on page 3.
    """
    assert index < list_size
    assert list_size <= 2**40

    for round in range(SHUFFLE_ROUND_COUNT):
        pivot = bytes_to_int(hash(seed + int_to_bytes1(round))[0:8]) % list_size
        flip = (pivot - index) % list_size
        position = max(index, flip)
        source = hash(seed + int_to_bytes1(round) + int_to_bytes4(position // 256))
        byte = source[(position % 256) // 8]
        bit = (byte >> (position % 8)) % 2
        index = flip if bit else index

    return index

get_split_offset

def get_split_offset(list_size: int, chunks: int, index: int) -> int:
    """
    Returns a value such that for a list L, chunk count k and index i,
    split(L, k)[i] == L[get_split_offset(len(L), k, i): get_split_offset(len(L), k, i+1)]
    """
    return (list_size * index) // chunks

get_epoch_committee_count

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

def get_shard_delta(state: BeaconState, epoch: Epoch) -> int:
    """
    Return the number of shards to increment ``state.latest_start_shard`` during ``epoch``.
    """
    return min(get_epoch_committee_count(state, epoch), SHARD_COUNT - SHARD_COUNT // SLOTS_PER_EPOCH)

compute_committee

def compute_committee(validator_indices: List[ValidatorIndex],
                      seed: Bytes32,
                      index: int,
                      total_committees: int) -> List[ValidatorIndex]:
    """
    Return the ``index``'th shuffled committee out of a total ``total_committees``
    using ``validator_indices`` and ``seed``.
    """
    start_offset = get_split_offset(len(validator_indices), total_committees, index)
    end_offset = get_split_offset(len(validator_indices), total_committees, index + 1)
    return [
        validator_indices[get_permuted_index(i, len(validator_indices), seed)]
        for i in range(start_offset, end_offset)
    ]

Note: this definition and the next few definitions are highly inefficient as algorithms, as they re-calculate many sub-expressions. Production implementations are expected to appropriately use caching/memoization to avoid redoing work.

get_crosslink_committees_at_slot

def get_crosslink_committees_at_slot(state: BeaconState,
                                     slot: Slot) -> List[Tuple[List[ValidatorIndex], Shard]]:
    """
    Return the list of ``(committee, shard)`` tuples for the ``slot``.
    """
    epoch = slot_to_epoch(slot)
    current_epoch = get_current_epoch(state)
    previous_epoch = get_previous_epoch(state)
    next_epoch = current_epoch + 1

    assert previous_epoch <= epoch <= next_epoch
    indices = get_active_validator_indices(state, epoch)

    if epoch == current_epoch:
        start_shard = state.latest_start_shard
    elif epoch == previous_epoch:
        previous_shard_delta = get_shard_delta(state, previous_epoch)
        start_shard = (state.latest_start_shard - previous_shard_delta) % SHARD_COUNT
    elif epoch == next_epoch:
        current_shard_delta = get_shard_delta(state, current_epoch)
        start_shard = (state.latest_start_shard + current_shard_delta) % SHARD_COUNT

    committees_per_epoch = get_epoch_committee_count(state, epoch)
    committees_per_slot = committees_per_epoch // SLOTS_PER_EPOCH
    offset = slot % SLOTS_PER_EPOCH
    slot_start_shard = (start_shard + committees_per_slot * offset) % SHARD_COUNT
    seed = generate_seed(state, epoch)

    return [
        (
            compute_committee(indices, seed, committees_per_slot * offset + i, committees_per_epoch),
            (slot_start_shard + i) % SHARD_COUNT,
        )
        for i in range(committees_per_slot)
    ]

get_block_root_at_slot

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

get_block_root

def get_block_root(state: BeaconState,
                   epoch: Epoch) -> Bytes32:
    """
    Return the block root at a recent ``epoch``.
    """
    return get_block_root_at_slot(state, get_epoch_start_slot(epoch))

get_randao_mix

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) -> ValidatorIndex:
    """
    Return the beacon proposer index at ``state.slot``.
    """
    current_epoch = get_current_epoch(state)
    first_committee, _ = get_crosslink_committees_at_slot(state, state.slot)[0]
    MAX_RANDOM_BYTE = 2**8 - 1
    i = 0
    while True:
        candidate_index = first_committee[(current_epoch + i) % len(first_committee)]
        random_byte = hash(generate_seed(state, current_epoch) + int_to_bytes8(i // 32))[i % 32]
        effective_balance = state.validator_registry[candidate_index].effective_balance
        if effective_balance * MAX_RANDOM_BYTE >= MAX_EFFECTIVE_BALANCE * random_byte:
            return candidate_index
        i += 1

verify_merkle_branch

def verify_merkle_branch(leaf: Bytes32, proof: List[Bytes32], depth: int, index: int, root: Bytes32) -> bool:
    """
    Verify that the given ``leaf`` is on the merkle branch ``proof``
    starting with the given ``root``.
    """
    value = leaf
    for i in range(depth):
        if index // (2**i) % 2:
            value = hash(proof[i] + value)
        else:
            value = hash(value + proof[i])
    return value == root

get_attesting_indices

def get_attesting_indices(state: BeaconState,
                          attestation_data: AttestationData,
                          bitfield: bytes) -> List[ValidatorIndex]:
    """
    Return the sorted attesting indices corresponding to ``attestation_data`` and ``bitfield``.
    """
    crosslink_committees = get_crosslink_committees_at_slot(state, attestation_data.slot)
    crosslink_committee = [committee for committee, shard in crosslink_committees if shard == attestation_data.shard][0]
    assert verify_bitfield(bitfield, len(crosslink_committee))
    return sorted([index for i, index in enumerate(crosslink_committee) if get_bitfield_bit(bitfield, i) == 0b1])

int_to_bytes1, int_to_bytes2, ...

int_to_bytes1(x): return x.to_bytes(1, 'little'), int_to_bytes2(x): return x.to_bytes(2, 'little'), and so on for all integers, particularly 1, 2, 3, 4, 8, 32, 48, 96.

bytes_to_int

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

get_total_balance

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

get_domain

def get_domain(state: BeaconState,
               domain_type: int,
               message_epoch: int=None) -> int:
    """
    Return the signature domain (fork version concatenated with domain type) of a message.
    """
    epoch = get_current_epoch(state) if message_epoch is None else message_epoch
    fork_version = state.fork.previous_version if epoch < state.fork.epoch else state.fork.current_version
    return bytes_to_int(fork_version + int_to_bytes4(domain_type))

get_bitfield_bit

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

convert_to_indexed

def convert_to_indexed(state: BeaconState, attestation: Attestation) -> IndexedAttestation:
    """
    Convert ``attestation`` to (almost) indexed-verifiable form.
    """
    attesting_indices = get_attesting_indices(state, attestation.data, attestation.aggregation_bitfield)
    custody_bit_1_indices = get_attesting_indices(state, attestation.data, attestation.custody_bitfield)
    custody_bit_0_indices = [index for index in attesting_indices if index not in custody_bit_1_indices]

    return IndexedAttestation(
        custody_bit_0_indices=custody_bit_0_indices,
        custody_bit_1_indices=custody_bit_1_indices,
        data=attestation.data,
        signature=attestation.signature,
    )

verify_indexed_attestation

def verify_indexed_attestation(state: BeaconState, indexed_attestation: IndexedAttestation) -> bool:
    """
    Verify validity of ``indexed_attestation`` fields.
    """
    custody_bit_0_indices = indexed_attestation.custody_bit_0_indices
    custody_bit_1_indices = indexed_attestation.custody_bit_1_indices

    # Ensure no duplicate indices across custody bits
    assert len(set(custody_bit_0_indices).intersection(set(custody_bit_1_indices))) == 0

    if len(custody_bit_1_indices) > 0:  # [TO BE REMOVED IN PHASE 1]
        return False

    if not (1 <= len(custody_bit_0_indices) + len(custody_bit_1_indices) <= MAX_INDICES_PER_ATTESTATION):
        return False

    if custody_bit_0_indices != sorted(custody_bit_0_indices):
        return False

    if custody_bit_1_indices != sorted(custody_bit_1_indices):
        return False

    return bls_verify_multiple(
        pubkeys=[
            bls_aggregate_pubkeys([state.validator_registry[i].pubkey for i in custody_bit_0_indices]),
            bls_aggregate_pubkeys([state.validator_registry[i].pubkey for i in custody_bit_1_indices]),
        ],
        message_hashes=[
            hash_tree_root(AttestationDataAndCustodyBit(data=indexed_attestation.data, custody_bit=0b0)),
            hash_tree_root(AttestationDataAndCustodyBit(data=indexed_attestation.data, custody_bit=0b1)),
        ],
        signature=indexed_attestation.signature,
        domain=get_domain(state, DOMAIN_ATTESTATION, slot_to_epoch(indexed_attestation.data.slot)),
    )

is_double_vote

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.source_epoch
    source_epoch_2 = attestation_data_2.source_epoch
    target_epoch_1 = slot_to_epoch(attestation_data_1.slot)
    target_epoch_2 = slot_to_epoch(attestation_data_2.slot)

    return source_epoch_1 < source_epoch_2 and target_epoch_2 < target_epoch_1

integer_squareroot

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

get_churn_limit

def get_churn_limit(state: BeaconState) -> int:
    return max(
        MIN_PER_EPOCH_CHURN_LIMIT,
        len(get_active_validator_indices(state, get_current_epoch(state))) // CHURN_LIMIT_QUOTIENT
    )

bls_verify

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

bls_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.

Routines for updating validator status

Note: All functions in this section mutate 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``.
    """
    # Return if validator already initiated exit
    validator = state.validator_registry[index]
    if validator.exit_epoch != FAR_FUTURE_EPOCH:
        return

    # Compute exit queue epoch
    exit_epochs = [v.exit_epoch for v in state.validator_registry if v.exit_epoch != FAR_FUTURE_EPOCH]
    exit_queue_epoch = max(exit_epochs + [get_delayed_activation_exit_epoch(get_current_epoch(state))])
    exit_queue_churn = len([v for v in state.validator_registry if v.exit_epoch == exit_queue_epoch])
    if exit_queue_churn >= get_churn_limit(state):
        exit_queue_epoch += 1

    # Set validator exit epoch and withdrawable epoch
    validator.exit_epoch = exit_queue_epoch
    validator.withdrawable_epoch = validator.exit_epoch + MIN_VALIDATOR_WITHDRAWABILITY_DELAY

slash_validator

def slash_validator(state: BeaconState, slashed_index: ValidatorIndex, whistleblower_index: ValidatorIndex=None) -> None:
    """
    Slash the validator with index ``slashed_index``.
    Note that this function mutates ``state``.
    """
    current_epoch = get_current_epoch(state)
    initiate_validator_exit(state, slashed_index)
    state.validator_registry[slashed_index].slashed = True
    state.validator_registry[slashed_index].withdrawable_epoch = current_epoch + LATEST_SLASHED_EXIT_LENGTH
    slashed_balance = state.validator_registry[slashed_index].effective_balance
    state.latest_slashed_balances[current_epoch % LATEST_SLASHED_EXIT_LENGTH] += slashed_balance

    proposer_index = get_beacon_proposer_index(state)
    if whistleblower_index is None:
        whistleblower_index = proposer_index
    whistleblowing_reward = slashed_balance // WHISTLEBLOWING_REWARD_QUOTIENT
    proposer_reward = whistleblowing_reward // PROPOSER_REWARD_QUOTIENT
    increase_balance(state, proposer_index, proposer_reward)
    increase_balance(state, whistleblower_index, whistleblowing_reward - proposer_reward)
    decrease_balance(state, slashed_index, whistleblowing_reward)

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 the DepositData elements.

Withdrawal credentials

One of the DepositData 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 at least MIN_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 BLS12-381 signature) is not verified by the deposit contract.

Eth2Genesis log

When a sufficient amount of 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.deposit_count equals deposit_count 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(pubkey: bytes[48], withdrawal_credentials: bytes[32], signature: bytes[96]): adds a deposit instance to the deposit tree, incorporating the input arguments and the value transferred in the given call. Note: the amount of value transferred must be at least MIN_DEPOSIT_AMOUNT. 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.deposit_count is the deposit_count contained in the Eth2Genesis log.
    • genesis_eth1_data.block_hash is the hash of the Ethereum 1.0 block that emitted the Eth2Genesis log.
  • Let genesis_state = get_genesis_beacon_state(genesis_validator_deposits, genesis_time, genesis_eth1_data).
  • Let genesis_block = BeaconBlock(state_root=hash_tree_root(genesis_state)).
def get_genesis_beacon_state(genesis_validator_deposits: List[Deposit],
                             genesis_time: int,
                             genesis_eth1_data: Eth1Data) -> BeaconState:
    """
    Get the genesis ``BeaconState``.
    """
    state = BeaconState(genesis_time=genesis_time, latest_eth1_data=genesis_eth1_data)

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

    # Process genesis activations
    for index, validator in enumerate(state.validator_registry):
        if validator.effective_balance >= MAX_EFFECTIVE_BALANCE:
            validator.activation_eligibility_epoch = GENESIS_EPOCH
            validator.activation_epoch = GENESIS_EPOCH

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

    return state

Beacon chain processing

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

  • 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 * 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, index: ValidatorIndex) -> Attestation be the attestation with the highest slot number in store from the validator with the given index. If several such attestations exist, use the one the validator v observed first.
  • Let get_latest_attestation_target(store: Store, index: ValidatorIndex) -> BeaconBlock be the target block in the attestation get_latest_attestation(store, 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 = [(i, get_latest_attestation_target(store, i)) for i in active_validator_indices]

    # Use the effective balance for fork choice voting to reduce recomputations and save bandwidth
    def get_vote_count(block: BeaconBlock) -> int:
        return sum(
            start_state.validator_registry[validator_index].effective_balance
            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
        # Ties broken by favoring block with lexicographically higher root
        head = max(children, key=lambda x: (get_vote_count(x), hash_tree_root(x)))

Beacon chain state transition function

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

  1. State caching, which happens at the start of every slot.
  2. The per-epoch transitions, which happens at the start of the first slot of every epoch.
  3. The per-slot transitions, which happens at every slot.
  4. The per-block transitions, which happens at every block.

Transition section notes:

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

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

State caching

At every slot > GENESIS_SLOT run the following function:

def cache_state(state: BeaconState) -> None:
    # Cache latest known state root (for previous slot)
    latest_state_root = hash_tree_root(state)
    state.latest_state_roots[state.slot % SLOTS_PER_HISTORICAL_ROOT] = latest_state_root

    # Store latest known state root (for previous slot) in latest_block_header if it is empty
    if state.latest_block_header.state_root == ZERO_HASH:
        state.latest_block_header.state_root = latest_state_root

    # Cache latest known block root (for previous slot)
    latest_block_root = signing_root(state.latest_block_header)
    state.latest_block_roots[state.slot % SLOTS_PER_HISTORICAL_ROOT] = latest_block_root

Per-epoch processing

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

Helper functions

We define epoch transition helper functions:

def get_total_active_balance(state: BeaconState) -> Gwei:
    return get_total_balance(state, get_active_validator_indices(state, get_current_epoch(state)))

def get_matching_source_attestations(state: BeaconState, epoch: Epoch) -> List[PendingAttestation]:
    assert epoch in (get_current_epoch(state), get_previous_epoch(state))
    return state.current_epoch_attestations if epoch == get_current_epoch(state) else state.previous_epoch_attestations

def get_matching_target_attestations(state: BeaconState, epoch: Epoch) -> List[PendingAttestation]:
    return [
        a for a in get_matching_source_attestations(state, epoch)
        if a.data.target_root == get_block_root(state, epoch)
    ]

def get_matching_head_attestations(state: BeaconState, epoch: Epoch) -> List[PendingAttestation]:
    return [
        a for a in get_matching_source_attestations(state, epoch)
        if a.data.beacon_block_root == get_block_root_at_slot(state, a.data.slot)
    ]

def get_unslashed_attesting_indices(state: BeaconState, attestations: List[PendingAttestation]) -> List[ValidatorIndex]:
    output = set()
    for a in attestations:
        output = output.union(get_attesting_indices(state, a.data, a.aggregation_bitfield))
    return sorted(filter(lambda index: not state.validator_registry[index].slashed, list(output)))

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

def get_crosslink_from_attestation_data(state: BeaconState, data: AttestationData) -> Crosslink:
    return Crosslink(
        epoch=min(slot_to_epoch(data.slot), state.current_crosslinks[data.shard].epoch + MAX_CROSSLINK_EPOCHS),
        previous_crosslink_root=data.previous_crosslink_root,
        crosslink_data_root=data.crosslink_data_root,
    )

def get_winning_crosslink_and_attesting_indices(state: BeaconState, shard: Shard, epoch: Epoch) -> Tuple[Crosslink, List[ValidatorIndex]]:
    shard_attestations = [a for a in get_matching_source_attestations(state, epoch) if a.data.shard == shard]
    shard_crosslinks = [get_crosslink_from_attestation_data(state, a.data) for a in shard_attestations]
    candidate_crosslinks = [
        c for c in shard_crosslinks
        if hash_tree_root(state.current_crosslinks[shard]) in (c.previous_crosslink_root, hash_tree_root(c))
    ]
    if len(candidate_crosslinks) == 0:
        return Crosslink(epoch=GENESIS_EPOCH, previous_crosslink_root=ZERO_HASH, crosslink_data_root=ZERO_HASH), []

    def get_attestations_for(crosslink: Crosslink) -> List[PendingAttestation]:
        return [a for a in shard_attestations if get_crosslink_from_attestation_data(state, a.data) == crosslink]
    # Winning crosslink has the crosslink data root with the most balance voting for it (ties broken lexicographically)
    winning_crosslink = max(candidate_crosslinks, key=lambda crosslink: (
        get_attesting_balance(state, get_attestations_for(crosslink)), crosslink.crosslink_data_root
    ))

    return winning_crosslink, get_unslashed_attesting_indices(state, get_attestations_for(winning_crosslink))

def get_earliest_attestation(state: BeaconState, attestations: List[PendingAttestation], index: ValidatorIndex) -> PendingAttestation:
    return min([
        a for a in attestations if index in get_attesting_indices(state, a.data, a.aggregation_bitfield)
    ], key=lambda a: a.inclusion_slot)

Justification and finalization

Run the following function:

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 = get_previous_epoch(state)
        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 = get_current_epoch(state)
        state.current_justified_root = get_block_root(state, state.current_justified_epoch)
        state.justification_bitfield |= (1 << 0)

    # Process finalizations
    bitfield = state.justification_bitfield
    # The 2nd/3rd/4th most recent epochs are justified, the 2nd using the 4th as source
    if (bitfield >> 1) % 8 == 0b111 and old_previous_justified_epoch == current_epoch - 3:
        state.finalized_epoch = old_previous_justified_epoch
        state.finalized_root = get_block_root(state, state.finalized_epoch)
    # The 2nd/3rd most recent epochs are justified, the 2nd using the 3rd as source
    if (bitfield >> 1) % 4 == 0b11 and old_previous_justified_epoch == current_epoch - 2:
        state.finalized_epoch = old_previous_justified_epoch
        state.finalized_root = get_block_root(state, state.finalized_epoch)
    # The 1st/2nd/3rd most recent epochs are justified, the 1st using the 3rd as source
    if (bitfield >> 0) % 8 == 0b111 and old_current_justified_epoch == current_epoch - 2:
        state.finalized_epoch = old_current_justified_epoch
        state.finalized_root = get_block_root(state, state.finalized_epoch)
    # The 1st/2nd most recent epochs are justified, the 1st using the 2nd as source
    if (bitfield >> 0) % 4 == 0b11 and old_current_justified_epoch == current_epoch - 1:
        state.finalized_epoch = old_current_justified_epoch
        state.finalized_root = get_block_root(state, state.finalized_epoch)

Crosslinks

Run the following function:

def process_crosslinks(state: BeaconState) -> None:
    state.previous_crosslinks = [c for c in state.current_crosslinks]
    previous_epoch = get_previous_epoch(state)
    next_epoch = get_current_epoch(state) + 1
    for slot in range(get_epoch_start_slot(previous_epoch), get_epoch_start_slot(next_epoch)):
        epoch = slot_to_epoch(slot)
        for crosslink_committee, shard in get_crosslink_committees_at_slot(state, slot):
            winning_crosslink, attesting_indices = get_winning_crosslink_and_attesting_indices(state, shard, epoch)
            if 3 * get_total_balance(state, attesting_indices) >= 2 * get_total_balance(state, crosslink_committee):
                state.current_crosslinks[shard] = winning_crosslink

Rewards and penalties

First, we define additional helpers:

def get_base_reward(state: BeaconState, index: ValidatorIndex) -> Gwei:
    adjusted_quotient = integer_squareroot(get_total_active_balance(state)) // BASE_REWARD_QUOTIENT
    if adjusted_quotient == 0:
        return 0
    return state.validator_registry[index].effective_balance // adjusted_quotient // BASE_REWARDS_PER_EPOCH


def get_attestation_deltas(state: BeaconState) -> Tuple[List[Gwei], List[Gwei]]:
    previous_epoch = get_previous_epoch(state)
    total_balance = get_total_active_balance(state)
    rewards = [0 for index in range(len(state.validator_registry))]
    penalties = [0 for index in range(len(state.validator_registry))]
    eligible_validator_indices = [
        index for index, v in enumerate(state.validator_registry)
        if is_active_validator(v, previous_epoch) or (v.slashed and previous_epoch + 1 < v.withdrawable_epoch)
    ]

    # Micro-incentives for matching FFG source, FFG target, and head
    matching_source_attestations = get_matching_source_attestations(state, previous_epoch)
    matching_target_attestations = get_matching_target_attestations(state, previous_epoch)
    matching_head_attestations = get_matching_head_attestations(state, previous_epoch)
    for attestations in (matching_source_attestations, matching_target_attestations, matching_head_attestations):
        unslashed_attesting_indices = get_unslashed_attesting_indices(state, attestations)
        attesting_balance = get_attesting_balance(state, attestations)
        for index in eligible_validator_indices:
            if index in unslashed_attesting_indices:
                rewards[index] += get_base_reward(state, index) * attesting_balance // total_balance
            else:
                penalties[index] += get_base_reward(state, index)

    # Proposer and inclusion delay micro-rewards
    for index in get_unslashed_attesting_indices(state, matching_source_attestations):
        earliest_attestation = get_earliest_attestation(state, matching_source_attestations, index)
        rewards[earliest_attestation.proposer_index] += get_base_reward(state, index) // PROPOSER_REWARD_QUOTIENT
        inclusion_delay = earliest_attestation.inclusion_slot - earliest_attestation.data.slot
        rewards[index] += get_base_reward(state, index) * MIN_ATTESTATION_INCLUSION_DELAY // inclusion_delay

    # Inactivity penalty
    finality_delay = previous_epoch - state.finalized_epoch
    if finality_delay > MIN_EPOCHS_TO_INACTIVITY_PENALTY:
        matching_target_attesting_indices = get_unslashed_attesting_indices(state, matching_target_attestations)
        for index in eligible_validator_indices:
            penalties[index] += BASE_REWARDS_PER_EPOCH * get_base_reward(state, index)
            if index not in matching_target_attesting_indices:
                penalties[index] += state.validator_registry[index].effective_balance * finality_delay // INACTIVITY_PENALTY_QUOTIENT

    return [rewards, penalties]

def get_crosslink_deltas(state: BeaconState) -> Tuple[List[Gwei], List[Gwei]]:
    rewards = [0 for index in range(len(state.validator_registry))]
    penalties = [0 for index in range(len(state.validator_registry))]
    for slot in range(get_epoch_start_slot(get_previous_epoch(state)), get_epoch_start_slot(get_current_epoch(state))):
        epoch = slot_to_epoch(slot)
        for crosslink_committee, shard in get_crosslink_committees_at_slot(state, slot):
            winning_crosslink, attesting_indices = get_winning_crosslink_and_attesting_indices(state, shard, epoch)
            attesting_balance = get_total_balance(state, attesting_indices)
            committee_balance = get_total_balance(state, crosslink_committee)
            for index in crosslink_committee:
                base_reward = get_base_reward(state, index)
                if index in attesting_indices:
                    rewards[index] += base_reward * attesting_balance // committee_balance
                else:
                    penalties[index] += base_reward
    return [rewards, penalties]

Run the following function:

def process_rewards_and_penalties(state: BeaconState) -> None:
    if get_current_epoch(state) == GENESIS_EPOCH:
        return

    rewards1, penalties1 = get_attestation_deltas(state)
    rewards2, penalties2 = get_crosslink_deltas(state)
    for i in range(len(state.validator_registry)):
        increase_balance(state, i, rewards1[i] + rewards2[i])
        decrease_balance(state, i, penalties1[i] + penalties2[i])

Registry updates

Run the following function:

def process_registry_updates(state: BeaconState) -> None:
    # Process activation eligibility and ejections
    for index, validator in enumerate(state.validator_registry):
        if validator.activation_eligibility_epoch == FAR_FUTURE_EPOCH and validator.effective_balance >= MAX_EFFECTIVE_BALANCE:
            validator.activation_eligibility_epoch = get_current_epoch(state)

        if is_active_validator(validator, get_current_epoch(state)) and validator.effective_balance <= EJECTION_BALANCE:
            initiate_validator_exit(state, index)

    # Queue validators eligible for activation and not dequeued for activation prior to finalized epoch
    activation_queue = sorted([
        index for index, validator in enumerate(state.validator_registry) if
        validator.activation_eligibility_epoch != FAR_FUTURE_EPOCH and
        validator.activation_epoch >= get_delayed_activation_exit_epoch(state.finalized_epoch)
    ], key=lambda index: state.validator_registry[index].activation_eligibility_epoch)
    # Dequeued validators for activation up to churn limit (without resetting activation epoch)
    for index in activation_queue[:get_churn_limit(state)]:
        if validator.activation_epoch == FAR_FUTURE_EPOCH:
            validator.activation_epoch = get_delayed_activation_exit_epoch(get_current_epoch(state))

Slashings

Run the following function:

def process_slashings(state: BeaconState) -> None:
    current_epoch = get_current_epoch(state)
    active_validator_indices = get_active_validator_indices(state, current_epoch)
    total_balance = get_total_balance(state, active_validator_indices)

    # Compute `total_penalties`
    total_at_start = state.latest_slashed_balances[(current_epoch + 1) % LATEST_SLASHED_EXIT_LENGTH]
    total_at_end = state.latest_slashed_balances[current_epoch % LATEST_SLASHED_EXIT_LENGTH]
    total_penalties = total_at_end - total_at_start

    for index, validator in enumerate(state.validator_registry):
        if validator.slashed and current_epoch == validator.withdrawable_epoch - LATEST_SLASHED_EXIT_LENGTH // 2:
            penalty = max(
                validator.effective_balance * min(total_penalties * 3, total_balance) // total_balance,
                validator.effective_balance // MIN_SLASHING_PENALTY_QUOTIENT
            )
            decrease_balance(state, index, penalty)

Final updates

Run the following function:

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 % SLOTS_PER_ETH1_VOTING_PERIOD == 0:
        state.eth1_data_votes = []
    # Update effective balances with hysteresis
    for index, validator in enumerate(state.validator_registry):
        balance = min(state.balances[index], MAX_EFFECTIVE_BALANCE)
        HALF_INCREMENT = EFFECTIVE_BALANCE_INCREMENT // 2
        if balance < validator.effective_balance or validator.effective_balance + 3 * HALF_INCREMENT < balance:
            validator.effective_balance = balance - balance % EFFECTIVE_BALANCE_INCREMENT
    # Update start shard
    state.latest_start_shard = (state.latest_start_shard + get_shard_delta(state, current_epoch)) % SHARD_COUNT
    # Set active index root
    index_root_position = (next_epoch + ACTIVATION_EXIT_DELAY) % LATEST_ACTIVE_INDEX_ROOTS_LENGTH
    state.latest_active_index_roots[index_root_position] = hash_tree_root(
        get_active_validator_indices(state, next_epoch + ACTIVATION_EXIT_DELAY)
    )
    # Set total slashed balances
    state.latest_slashed_balances[next_epoch % LATEST_SLASHED_EXIT_LENGTH] = (
        state.latest_slashed_balances[current_epoch % LATEST_SLASHED_EXIT_LENGTH]
    )
    # Set randao mix
    state.latest_randao_mixes[next_epoch % LATEST_RANDAO_MIXES_LENGTH] = get_randao_mix(state, current_epoch)
    # Set historical root accumulator
    if next_epoch % (SLOTS_PER_HISTORICAL_ROOT // SLOTS_PER_EPOCH) == 0:
        historical_batch = HistoricalBatch(
            block_roots=state.latest_block_roots,
            state_roots=state.latest_state_roots,
        )
        state.historical_roots.append(hash_tree_root(historical_batch))
    # Rotate current/previous epoch attestations
    state.previous_epoch_attestations = state.current_epoch_attestations
    state.current_epoch_attestations = []

Per-slot processing

At every slot > GENESIS_SLOT run the following function:

def advance_slot(state: BeaconState) -> None:
    state.slot += 1

Per-block processing

For every block except the genesis block, run process_block_header(state, block), process_randao(state, block) and process_eth1_data(state, block).

Block header

def process_block_header(state: BeaconState, block: BeaconBlock) -> None:
    # Verify that the slots match
    assert block.slot == state.slot
    # Verify that the parent matches
    assert block.previous_block_root == signing_root(state.latest_block_header)
    # Save current block as the new latest block
    state.latest_block_header = BeaconBlockHeader(
        slot=block.slot,
        previous_block_root=block.previous_block_root,
        block_body_root=hash_tree_root(block.body),
    )
    # Verify proposer is not slashed
    proposer = state.validator_registry[get_beacon_proposer_index(state)]
    assert not proposer.slashed
    # Verify proposer signature
    assert bls_verify(proposer.pubkey, signing_root(block), block.signature, get_domain(state, DOMAIN_BEACON_PROPOSER))

RANDAO

def process_randao(state: BeaconState, block: BeaconBlock) -> None:
    proposer = state.validator_registry[get_beacon_proposer_index(state)]
    # Verify that the provided randao value is valid
    assert bls_verify(proposer.pubkey, hash_tree_root(get_current_epoch(state)), block.body.randao_reveal, get_domain(state, DOMAIN_RANDAO))
    # Mix it in
    state.latest_randao_mixes[get_current_epoch(state) % LATEST_RANDAO_MIXES_LENGTH] = (
        xor(get_randao_mix(state, get_current_epoch(state)),
            hash(block.body.randao_reveal))
    )

Eth1 data

def process_eth1_data(state: BeaconState, block: BeaconBlock) -> None:
    state.eth1_data_votes.append(block.body.eth1_data)
    if state.eth1_data_votes.count(block.body.eth1_data) * 2 > SLOTS_PER_ETH1_VOTING_PERIOD:
        state.latest_eth1_data = block.body.eth1_data

Operations

Proposer slashings

Verify that len(block.body.proposer_slashings) <= MAX_PROPOSER_SLASHINGS.

For each proposer_slashing in block.body.proposer_slashings, run the following function:

def process_proposer_slashing(state: BeaconState,
                              proposer_slashing: ProposerSlashing) -> None:
    """
    Process ``ProposerSlashing`` operation.
    Note that this function mutates ``state``.
    """
    proposer = state.validator_registry[proposer_slashing.proposer_index]
    # Verify that the epoch is the same
    assert slot_to_epoch(proposer_slashing.header_1.slot) == slot_to_epoch(proposer_slashing.header_2.slot)
    # But the headers are different
    assert proposer_slashing.header_1 != proposer_slashing.header_2
    # Check proposer is slashable
    assert is_slashable_validator(proposer, get_current_epoch(state))
    # Signatures are valid
    for header in (proposer_slashing.header_1, proposer_slashing.header_2):
        domain = get_domain(state, DOMAIN_BEACON_PROPOSER, slot_to_epoch(header.slot))
        assert bls_verify(proposer.pubkey, signing_root(header), header.signature, domain)

    slash_validator(state, proposer_slashing.proposer_index)
Attester slashings

Verify that len(block.body.attester_slashings) <= MAX_ATTESTER_SLASHINGS.

For each attester_slashing in block.body.attester_slashings, run the following function:

def process_attester_slashing(state: BeaconState,
                              attester_slashing: AttesterSlashing) -> None:
    """
    Process ``AttesterSlashing`` operation.
    Note that this function mutates ``state``.
    """
    attestation1 = attester_slashing.attestation_1
    attestation2 = attester_slashing.attestation_2
    # Check that the attestations are conflicting
    assert attestation1.data != attestation2.data
    assert (
        is_double_vote(attestation1.data, attestation2.data) or
        is_surround_vote(attestation1.data, attestation2.data)
    )

    assert verify_indexed_attestation(state, attestation1)
    assert verify_indexed_attestation(state, attestation2)
    attesting_indices_1 = attestation1.custody_bit_0_indices + attestation1.custody_bit_1_indices
    attesting_indices_2 = attestation2.custody_bit_0_indices + attestation2.custody_bit_1_indices
    slashable_indices = [
        index for index in attesting_indices_1
        if (
            index in attesting_indices_2 and
            is_slashable_validator(state.validator_registry[index], get_current_epoch(state))
        )
    ]
    assert len(slashable_indices) >= 1
    for index in slashable_indices:
        slash_validator(state, index)
Attestations

Verify that len(block.body.attestations) <= MAX_ATTESTATIONS.

For each attestation in block.body.attestations, run the following function:

def process_attestation(state: BeaconState, attestation: Attestation) -> None:
    """
    Process ``Attestation`` operation.
    Note that this function mutates ``state``.
    """
    data = attestation.data
    min_slot = state.slot - SLOTS_PER_EPOCH if get_current_epoch(state) > GENESIS_EPOCH else GENESIS_SLOT
    assert min_slot <= data.slot <= state.slot - MIN_ATTESTATION_INCLUSION_DELAY

    # Check target epoch, source epoch, source root, and source crosslink
    target_epoch = slot_to_epoch(data.slot)
    assert (target_epoch, data.source_epoch, data.source_root, data.previous_crosslink_root) in {
        (get_current_epoch(state), state.current_justified_epoch, state.current_justified_root, hash_tree_root(state.current_crosslinks[data.shard])),
        (get_previous_epoch(state), state.previous_justified_epoch, state.previous_justified_root, hash_tree_root(state.previous_crosslinks[data.shard])),
    }

    # Check crosslink data root
    assert data.crosslink_data_root == ZERO_HASH  # [to be removed in phase 1]

    # Check signature and bitfields
    assert verify_indexed_attestation(state, convert_to_indexed(state, attestation))

    # Cache pending attestation
    pending_attestation = PendingAttestation(
        data=data,
        aggregation_bitfield=attestation.aggregation_bitfield,
        inclusion_slot=state.slot,
        proposer_index=get_beacon_proposer_index(state),
    )
    if target_epoch == get_current_epoch(state):
        state.current_epoch_attestations.append(pending_attestation)
    else:
        state.previous_epoch_attestations.append(pending_attestation)
Deposits

Verify that len(block.body.deposits) == min(MAX_DEPOSITS, state.latest_eth1_data.deposit_count - state.deposit_index).

For each deposit in block.body.deposits, run the following function:

def process_deposit(state: BeaconState, deposit: Deposit) -> None:
    """
    Process an Eth1 deposit, registering a validator or increasing its balance.
    Note that this function mutates ``state``.
    """
    # Verify the Merkle branch
    assert verify_merkle_branch(
        leaf=hash_tree_root(deposit.data),
        proof=deposit.proof,
        depth=DEPOSIT_CONTRACT_TREE_DEPTH,
        index=deposit.index,
        root=state.latest_eth1_data.deposit_root,
    )

    # Deposits must be processed in order
    assert deposit.index == state.deposit_index
    state.deposit_index += 1

    pubkey = deposit.data.pubkey
    amount = deposit.data.amount
    validator_pubkeys = [v.pubkey for v in state.validator_registry]
    if pubkey not in validator_pubkeys:
        # Verify the deposit signature (proof of possession)
        if not bls_verify(pubkey, signing_root(deposit.data), deposit.data.signature, get_domain(state, DOMAIN_DEPOSIT)):
            return

        # Add validator and balance entries
        state.validator_registry.append(Validator(
            pubkey=pubkey,
            withdrawal_credentials=deposit.data.withdrawal_credentials,
            activation_eligibility_epoch=FAR_FUTURE_EPOCH,
            activation_epoch=FAR_FUTURE_EPOCH,
            exit_epoch=FAR_FUTURE_EPOCH,
            withdrawable_epoch=FAR_FUTURE_EPOCH,
            effective_balance=amount - amount % EFFECTIVE_BALANCE_INCREMENT
        ))
        state.balances.append(amount)
    else:
        # Increase balance by deposit amount
        index = validator_pubkeys.index(pubkey)
        increase_balance(state, index, amount)
Voluntary exits

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

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

def process_voluntary_exit(state: BeaconState, exit: VoluntaryExit) -> None:
    """
    Process ``VoluntaryExit`` operation.
    Note that this function mutates ``state``.
    """
    validator = state.validator_registry[exit.validator_index]
    # Verify the validator is active
    assert is_active_validator(validator, get_current_epoch(state))
    # Verify the validator has not yet exited
    assert validator.exit_epoch == FAR_FUTURE_EPOCH
    # Exits must specify an epoch when they become valid; they are not valid before then
    assert get_current_epoch(state) >= exit.epoch
    # Verify the validator has been active long enough
    assert get_current_epoch(state) - validator.activation_epoch >= PERSISTENT_COMMITTEE_PERIOD
    # Verify signature
    domain = get_domain(state, DOMAIN_VOLUNTARY_EXIT, exit.epoch)
    assert bls_verify(validator.pubkey, signing_root(exit), exit.signature, domain)
    # Initiate exit
    initiate_validator_exit(state, exit.validator_index)
Transfers

Verify that len(block.body.transfers) <= MAX_TRANSFERS and that all transfers are distinct.

For each transfer in block.body.transfers, run the following function:

def process_transfer(state: BeaconState, transfer: Transfer) -> None:
    """
    Process ``Transfer`` operation.
    Note that this function mutates ``state``.
    """
    # Verify the amount and fee are not individually too big (for anti-overflow purposes)
    assert state.balances[transfer.sender] >= max(transfer.amount, transfer.fee)
    # A transfer is valid in only one slot
    assert state.slot == transfer.slot
    # Sender must be not yet eligible for activation, withdrawn, or transfer balance over MAX_EFFECTIVE_BALANCE
    assert (
        state.validator_registry[transfer.sender].activation_eligibility_epoch == FAR_FUTURE_EPOCH or
        get_current_epoch(state) >= state.validator_registry[transfer.sender].withdrawable_epoch or
        transfer.amount + transfer.fee + MAX_EFFECTIVE_BALANCE <= state.balances[transfer.sender]
    )
    # Verify that the pubkey is valid
    assert (
        state.validator_registry[transfer.sender].withdrawal_credentials ==
        BLS_WITHDRAWAL_PREFIX_BYTE + hash(transfer.pubkey)[1:]
    )
    # Verify that the signature is valid
    assert bls_verify(transfer.pubkey, signing_root(transfer), transfer.signature, get_domain(state, DOMAIN_TRANSFER))
    # Process the transfer
    decrease_balance(state, transfer.sender, transfer.amount + transfer.fee)
    increase_balance(state, transfer.recipient, transfer.amount)
    increase_balance(state, get_beacon_proposer_index(state), transfer.fee)
    # Verify balances are not dust
    assert not (0 < state.balances[transfer.sender] < MIN_DEPOSIT_AMOUNT)
    assert not (0 < state.balances[transfer.recipient] < MIN_DEPOSIT_AMOUNT)

State root verification

Verify the block's state_root by running the following function:

def verify_block_state_root(state: BeaconState, block: BeaconBlock) -> None:
    assert block.state_root == hash_tree_root(state)