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Ethereum 2.0 Phase 0 -- The Beacon Chain

NOTICE: This document is a work in progress for researchers and implementers.

Table of contents

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_TREE_DEPTH 2**5 (= 32)

Gwei values

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

Initial values

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

Time parameters

Name Value Unit Duration
MIN_ATTESTATION_INCLUSION_DELAY 2**2 (= 4) slots 24 seconds
SLOTS_PER_EPOCH 2**6 (= 64) slots 6.4 minutes
MIN_SEED_LOOKAHEAD 2**0 (= 1) epochs 6.4 minutes
ACTIVATION_EXIT_DELAY 2**2 (= 4) epochs 25.6 minutes
SLOTS_PER_ETH1_VOTING_PERIOD 2**10 (= 1,024) slots ~1.7 hours
SLOTS_PER_HISTORICAL_ROOT 2**13 (= 8,192) slots ~13 hours
MIN_VALIDATOR_WITHDRAWABILITY_DELAY 2**8 (= 256) epochs ~27 hours
PERSISTENT_COMMITTEE_PERIOD 2**11 (= 2,048) epochs 9 days
MAX_CROSSLINK_EPOCHS 2**6 (= 64) epochs ~7 hours
MIN_EPOCHS_TO_INACTIVITY_PENALTY 2**2 (= 4) epochs 25.6 minutes
  • MAX_CROSSLINK_EPOCHS should be a small constant times SHARD_COUNT // SLOTS_PER_EPOCH

State list lengths

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

Reward and penalty quotients

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

{
    # Shard number
    'shard': 'uint64',
    # 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
    'beacon_block_root': 'bytes32',

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

    # Crosslink vote
    'crosslink': Crosslink,
}

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 delay
    'inclusion_delay': '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,
    'graffiti': 'bytes32',
    '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_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)

get_epoch_start_shard

def get_epoch_start_shard(state: BeaconState, epoch: Epoch) -> Shard:
    assert epoch <= get_current_epoch(state) + 1
    check_epoch = get_current_epoch(state) + 1
    shard = (state.latest_start_shard + get_shard_delta(state, get_current_epoch(state))) % SHARD_COUNT
    while check_epoch > epoch:
        check_epoch -= 1
        shard = (shard + SHARD_COUNT - get_shard_delta(state, check_epoch)) % SHARD_COUNT
    return shard

get_attestation_slot

def get_attestation_slot(state: BeaconState, attestation: Attestation) -> Slot:
    epoch = attestation.data.target_epoch
    committee_count = get_epoch_committee_count(state, epoch)
    offset = (attestation.data.crosslink.shard + SHARD_COUNT - get_epoch_start_shard(state, epoch)) % SHARD_COUNT
    return get_epoch_start_slot(epoch) + offset // (committee_count // SLOTS_PER_EPOCH)

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``.
    ``epoch`` expected to be between (current_epoch - LATEST_RANDAO_MIXES_LENGTH, current_epoch].
    """
    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``.
    ``epoch`` expected to be between
    (current_epoch - LATEST_ACTIVE_INDEX_ROOTS_LENGTH + ACTIVATION_EXIT_DELAY, current_epoch + 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 + LATEST_RANDAO_MIXES_LENGTH - 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 current beacon proposer index.
    """
    epoch = get_current_epoch(state)
    committees_per_slot = get_epoch_committee_count(state, epoch) // SLOTS_PER_EPOCH
    offset = committees_per_slot * (state.slot % SLOTS_PER_EPOCH)
    shard = (get_epoch_start_shard(state, epoch) + offset) % SHARD_COUNT
    first_committee = get_crosslink_committee(state, epoch, shard)
    MAX_RANDOM_BYTE = 2**8 - 1
    seed = generate_seed(state, epoch)
    i = 0
    while True:
        candidate_index = first_committee[(epoch + i) % len(first_committee)]
        random_byte = hash(seed + int_to_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_shuffled_index

def get_shuffled_index(index: ValidatorIndex, index_count: int, seed: Bytes32) -> ValidatorIndex:
    """
    Return the shuffled validator index corresponding to ``seed`` (and ``index_count``).
    """
    assert index < index_count
    assert index_count <= 2**40

    # Swap or not (https://link.springer.com/content/pdf/10.1007%2F978-3-642-32009-5_1.pdf)
    # See the 'generalized domain' algorithm on page 3
    for round in range(SHUFFLE_ROUND_COUNT):
        pivot = bytes_to_int(hash(seed + int_to_bytes1(round))[0:8]) % index_count
        flip = (pivot - index) % index_count
        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

compute_committee

def compute_committee(indices: List[ValidatorIndex], seed: Bytes32, index: int, count: int) -> List[ValidatorIndex]:
    start = (len(indices) * index) // count
    end = (len(indices) * (index + 1)) // count
    return [indices[get_shuffled_index(i, len(indices), seed)] for i in range(start, end)]

get_crosslink_committee

def get_crosslink_committee(state: BeaconState, epoch: Epoch, shard: Shard) -> List[ValidatorIndex]:
    return compute_committee(
        indices=get_active_validator_indices(state, epoch),
        seed=generate_seed(state, epoch),
        index=(shard + SHARD_COUNT - get_epoch_start_shard(state, epoch)) % SHARD_COUNT,
        count=get_epoch_committee_count(state, epoch),
    )

get_attesting_indices

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

int_to_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, indexed_attestation.data.target_epoch),
    )

is_slashable_attestation_data

def is_slashable_attestation_data(data_1: AttestationData, data_2: AttestationData) -> bool:
    """
    Check if ``data_1`` and ``data_2`` are slashable according to Casper FFG rules.
    """
    return (
        # Double vote
        (data_1 != data_2 and data_1.target_epoch == data_2.target_epoch) or
        # Surround vote
        (data_1.source_epoch < data_2.source_epoch and data_2.target_epoch < data_1.target_epoch)
    )

integer_squareroot

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 the churn limit based on the active validator count.
    """
    return max(
        MIN_PER_EPOCH_CHURN_LIMIT,
        len(get_active_validator_indices(state, get_current_epoch(state))) // CHURN_LIMIT_QUOTIENT
    )

bls_verify

bls_verify is a function for verifying a BLS signature, 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``.
    """
    # 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``.
    """
    current_epoch = get_current_epoch(state)
    initiate_validator_exit(state, slashed_index)
    state.validator_registry[slashed_index].slashed = True
    state.validator_registry[slashed_index].withdrawable_epoch = current_epoch + LATEST_SLASHED_EXIT_LENGTH
    slashed_balance = state.validator_registry[slashed_index].effective_balance
    state.latest_slashed_balances[current_epoch % LATEST_SLASHED_EXIT_LENGTH] += slashed_balance

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

On genesis

When enough full deposits have been made to the deposit contract, an Eth2Genesis log is emitted. Construct a corresponding genesis_state and genesis_block as follows:

  • Let genesis_validator_deposits be the list of deposits, ordered chronologically, up to and including the deposit that triggered the Eth2Genesis log.
  • Let genesis_time be the timestamp specified in the Eth2Genesis log.
  • Let genesis_eth1_data be the Eth1Data object where:
    • genesis_eth1_data.deposit_root is the deposit_root contained in the Eth2Genesis log.
    • genesis_eth1_data.deposit_count is the deposit_count contained in the Eth2Genesis log.
    • genesis_eth1_data.block_hash is the hash of the Ethereum 1.0 block that emitted the Eth2Genesis log.
  • Let genesis_state = get_genesis_beacon_state(genesis_validator_deposits, genesis_time, genesis_eth1_data).
  • Let genesis_block = BeaconBlock(state_root=hash_tree_root(genesis_state)).
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 validator in state.validator_registry:
        if validator.effective_balance >= MAX_EFFECTIVE_BALANCE:
            validator.activation_eligibility_epoch = GENESIS_EPOCH
            validator.activation_epoch = GENESIS_EPOCH

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

    return state

Beacon chain state transition function

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

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

Transition section notes:

  • The state caching caches the state root of the previous slot and updates block and state roots records.
  • The per-epoch transitions focus on the validator 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, get_attestation_slot(state, a))
    ]

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_winning_crosslink_and_attesting_indices(state: BeaconState, epoch: Epoch, shard: Shard) -> Tuple[Crosslink, List[ValidatorIndex]]:
    shard_attestations = [a for a in get_matching_source_attestations(state, epoch) if a.data.crosslink.shard == shard]
    candidate_crosslinks = [
        c for c in [a.data.crosslink for a in shard_attestations]
        if hash_tree_root(state.current_crosslinks[shard]) in (c.previous_crosslink_root, hash_tree_root(c))
    ]
    if len(candidate_crosslinks) == 0:
        return Crosslink(), []

    def get_attestations_for(crosslink: Crosslink) -> List[PendingAttestation]:
        return [a for a in shard_attestations if a.data.crosslink == 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))

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 = previous_epoch
        state.current_justified_root = get_block_root(state, state.current_justified_epoch)
        state.justification_bitfield |= (1 << 1)
    current_epoch_matching_target_balance = get_attesting_balance(state, get_matching_target_attestations(state, current_epoch))
    if current_epoch_matching_target_balance * 3 >= get_total_active_balance(state) * 2:
        state.current_justified_epoch = current_epoch
        state.current_justified_root = get_block_root(state, state.current_justified_epoch)
        state.justification_bitfield |= (1 << 0)

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

Crosslinks

Run the following function:

def process_crosslinks(state: BeaconState) -> None:
    state.previous_crosslinks = [c for c in state.current_crosslinks]
    for epoch in (get_previous_epoch(state), get_current_epoch(state)):
        for offset in range(get_epoch_committee_count(state, epoch)):
            shard = (get_epoch_start_shard(state, epoch) + offset) % SHARD_COUNT
            crosslink_committee = get_crosslink_committee(state, epoch, shard)
            winning_crosslink, attesting_indices = get_winning_crosslink_and_attesting_indices(state, epoch, shard)
            if 3 * get_total_balance(state, attesting_indices) >= 2 * get_total_balance(state, crosslink_committee):
                state.current_crosslinks[shard] = winning_crosslink

Rewards and penalties

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 _ in range(len(state.validator_registry))]
    penalties = [0 for _ 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):
        attestation = min([
            a for a in matching_source_attestations
            if index in get_attesting_indices(state, a.data, a.aggregation_bitfield)
        ], key=lambda a: a.inclusion_delay)
        rewards[attestation.proposer_index] += get_base_reward(state, index) // PROPOSER_REWARD_QUOTIENT
        rewards[index] += get_base_reward(state, index) * MIN_ATTESTATION_INCLUSION_DELAY // attestation.inclusion_delay

    # Inactivity penalty
    finality_delay = previous_epoch - state.finalized_epoch
    if finality_delay > MIN_EPOCHS_TO_INACTIVITY_PENALTY:
        matching_target_attesting_indices = get_unslashed_attesting_indices(state, matching_target_attestations)
        for index in eligible_validator_indices:
            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))]
    epoch = get_previous_epoch(state)
    for offset in range(get_epoch_committee_count(state, epoch)):
        shard = (get_epoch_start_shard(state, epoch) + offset) % SHARD_COUNT
        crosslink_committee = get_crosslink_committee(state, epoch, shard)
        winning_crosslink, attesting_indices = get_winning_crosslink_and_attesting_indices(state, epoch, shard)
        attesting_balance = get_total_balance(state, attesting_indices)
        committee_balance = get_total_balance(state, crosslink_committee)
        for index in crosslink_committee:
            base_reward = get_base_reward(state, index)
            if index in attesting_indices:
                rewards[index] += base_reward * attesting_balance // committee_balance
            else:
                penalties[index] += base_reward
    return rewards, penalties

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)]:
        validator = state.validator_registry[index]
        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 + 1) % SLOTS_PER_ETH1_VOTING_PERIOD == 0:
        state.eth1_data_votes = []
    # Update effective balances with hysteresis
    for index, validator in enumerate(state.validator_registry):
        balance = state.balances[index]
        HALF_INCREMENT = EFFECTIVE_BALANCE_INCREMENT // 2
        if balance < validator.effective_balance or validator.effective_balance + 3 * HALF_INCREMENT < balance:
            validator.effective_balance = min(balance - balance % EFFECTIVE_BALANCE_INCREMENT, MAX_EFFECTIVE_BALANCE)
    # Update start shard
    state.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

Note: All functions in this section mutate state.

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.
    """
    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.
    """
    attestation_1 = attester_slashing.attestation_1
    attestation_2 = attester_slashing.attestation_2
    assert is_slashable_attestation_data(attestation_1.data, attestation_2.data)
    assert verify_indexed_attestation(state, attestation_1)
    assert verify_indexed_attestation(state, attestation_2)

    slashed_any = False
    attesting_indices_1 = attestation_1.custody_bit_0_indices + attestation_1.custody_bit_1_indices
    attesting_indices_2 = attestation_2.custody_bit_0_indices + attestation_2.custody_bit_1_indices
    for index in set(attesting_indices_1).intersection(attesting_indices_2):
        if is_slashable_validator(state.validator_registry[index], get_current_epoch(state)):
            slash_validator(state, index)
            slashed_any = True
    assert slashed_any
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.
    """
    attestation_slot = get_attestation_slot(state, attestation)
    assert attestation_slot + MIN_ATTESTATION_INCLUSION_DELAY <= state.slot <= attestation_slot + SLOTS_PER_EPOCH

    data = attestation.data
    pending_attestation = PendingAttestation(
        data=data,
        aggregation_bitfield=attestation.aggregation_bitfield,
        inclusion_delay=state.slot - attestation_slot,
        proposer_index=get_beacon_proposer_index(state),
    )

    assert data.target_epoch in (get_previous_epoch(state), get_current_epoch(state))
    if data.target_epoch == get_previous_epoch(state):
        ffg_data = (state.previous_justified_epoch, state.previous_justified_root, get_previous_epoch(state))
        previous_crosslink = state.previous_crosslinks[data.crosslink.shard]
        state.previous_epoch_attestations.append(pending_attestation)
    if data.target_epoch == get_current_epoch(state):
        ffg_data = (state.current_justified_epoch, state.current_justified_root, get_current_epoch(state))
        previous_crosslink = state.current_crosslinks[data.crosslink.shard]
        state.current_epoch_attestations.append(pending_attestation)

    # Check FFG data, crosslink data, and signature
    assert ffg_data == (data.source_epoch, data.source_root, data.target_epoch)
    assert data.crosslink.epoch == min(data.target_epoch, previous_crosslink.epoch + MAX_CROSSLINK_EPOCHS)
    assert data.crosslink.previous_crosslink_root == hash_tree_root(previous_crosslink)
    assert data.crosslink.crosslink_data_root == ZERO_HASH  # [to be removed in phase 1]
    assert verify_indexed_attestation(state, convert_to_indexed(state, 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.
    """
    # 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=min(amount - amount % EFFECTIVE_BALANCE_INCREMENT, MAX_EFFECTIVE_BALANCE)
        ))
        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.
    """
    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.
    """
    # 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)