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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 Eth 1.0. Activation as a validator happens when Eth 1.0 deposit receipts are processed by the beacon chain, the activation balance is reached, and a queuing process is completed. Exit is either voluntary or done forcibly as a penalty for misbehavior.

The primary source of load on the beacon chain is "attestations". Attestations are simultaneously availability votes for a shard block and proof-of-stake votes for a beacon block. A sufficient number of attestations for the same shard block create a "crosslink", confirming the shard segment up to that shard block into the beacon chain. Crosslinks also serve as infrastructure for asynchronous cross-shard communication.

Notation

Code snippets appearing in this style are to be interpreted as Python code.

Terminology

  • Validator—a registered participant in the beacon chain. You can become one by sending ether into the Eth 1.0 deposit contract.
  • Active validator—an active participant in the Ethereum 2.0 consensus invited to, among other things, propose and attest to blocks and vote for crosslinks.
  • Committee—a (pseudo-) randomly sampled subset of active validators. 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 proof-of-stake chain that is the base of the sharding system.
  • Shard chain—one of the chains on which user transactions take place and account data is stored.
  • Block root—a 32-byte Merkle root of a beacon chain block or shard chain block. Previously called "block hash".
  • Crosslink—a set of signatures from a committee attesting to a block in a shard chain that can be included into the beacon chain. Crosslinks are the main means by which the beacon chain "learns about" the updated state of shard chains.
  • Slot—a period during which one proposer has the ability to create a beacon chain block and some attesters have the ability to make attestations.
  • Epoch—an aligned span of slots during which all validators 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.

Custom types

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

Name SSZ equivalent Description
Slot uint64 a slot number
Epoch uint64 an epoch number
Shard uint64 a shard number
ValidatorIndex uint64 a validator registry index
Gwei uint64 an amount in Gwei
Version Bytes4 a fork version number
Hash Bytes32 a hashed result
BLSPubkey Bytes48 a BLS12-381 public key
BLSSignature Bytes96 a BLS12-381 signature

Constants

The following values are (non-configurable) constants used throughout the specification.

Name Value
FAR_FUTURE_EPOCH Epoch(2**64 - 1)
ZERO_HASH Hash(b'\x00' * 32)
BASE_REWARDS_PER_EPOCH 5
DEPOSIT_CONTRACT_TREE_DEPTH 2**5 (= 32)

Configuration

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

Misc

Name Value
SHARD_COUNT 2**10 (= 1,024)
TARGET_COMMITTEE_SIZE 2**7 (= 128)
MAX_INDICES_PER_ATTESTATION 2**12 (= 4,096)
MIN_PER_EPOCH_CHURN_LIMIT 2**2 (= 4)
CHURN_LIMIT_QUOTIENT 2**16 (= 65,536)
SHUFFLE_ROUND_COUNT 90
GENESIS_ACTIVE_VALIDATOR_COUNT 2**16 (= 65,536)
MIN_GENESIS_TIME 1578009600 (Jan 3, 2020)
  • For the safety of crosslinks, TARGET_COMMITTEE_SIZE exceeds the recommended minimum committee size of 111; with sufficient active validators (at least SLOTS_PER_EPOCH * TARGET_COMMITTEE_SIZE), the shuffling algorithm ensures committee sizes of at least TARGET_COMMITTEE_SIZE. (Unbiasable randomness with a Verifiable Delay Function (VDF) will improve committee robustness and lower the safe minimum committee size.)

Gwei values

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

Initial values

Name Value
GENESIS_SLOT Slot(0)
GENESIS_EPOCH Epoch(0)
BLS_WITHDRAWAL_PREFIX 0

Time parameters

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

State list lengths

Name Value Unit Duration
EPOCHS_PER_HISTORICAL_VECTOR 2**16 (= 65,536) epochs ~0.8 years
EPOCHS_PER_SLASHED_BALANCES_VECTOR 2**13 (= 8,192) epochs ~36 days
HISTORICAL_ROOTS_LIMIT 2**24 (= 16,777,216) historical roots ~26,131 years
VALIDATOR_REGISTRY_LIMIT 2**40 (= 1,099,511,627,776) validator spots

Rewards and penalties

Name Value
BASE_REWARD_FACTOR 2**6 (= 64)
WHISTLEBLOWING_REWARD_QUOTIENT 2**9 (= 512)
PROPOSER_REWARD_QUOTIENT 2**3 (= 8)
INACTIVITY_PENALTY_QUOTIENT 2**25 (= 33,554,432)
MIN_SLASHING_PENALTY_QUOTIENT 2**5 (= 32)
  • The INACTIVITY_PENALTY_QUOTIENT equals INVERSE_SQRT_E_DROP_TIME**2 where INVERSE_SQRT_E_DROP_TIME := 2**12 epochs (about 18 days) is the time it takes the inactivity penalty to reduce the balance of non-participating validators 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

Containers

The following types are SimpleSerialize (SSZ) containers.

Note: The definitions are ordered topologically to facilitate execution of the spec.

Note: Fields missing in container instantiations default to their zero value.

Misc dependencies

Fork

class Fork(Container):
    previous_version: Version
    current_version: Version
    epoch: Epoch  # Epoch of latest fork

Validator

class Validator(Container):
    pubkey: BLSPubkey
    withdrawal_credentials: Hash  # Commitment to pubkey for withdrawals and transfers
    effective_balance: Gwei  # Balance at stake
    slashed: Bool
    # Status epochs
    activation_eligibility_epoch: Epoch  # When criteria for activation were met
    activation_epoch: Epoch
    exit_epoch: Epoch
    withdrawable_epoch: Epoch  # When validator can withdraw or transfer funds

Crosslink

class Crosslink(Container):
    shard: Shard
    parent_root: Hash
    # Crosslinking data
    start_epoch: Epoch
    end_epoch: Epoch
    data_root: Hash

AttestationData

class AttestationData(Container):
    # LMD GHOST vote
    beacon_block_root: Hash
    # FFG vote
    source_epoch: Epoch
    source_root: Hash
    target_epoch: Epoch
    target_root: Hash
    # Crosslink vote
    crosslink: Crosslink

AttestationDataAndCustodyBit

class AttestationDataAndCustodyBit(Container):
    data: AttestationData
    custody_bit: Bit  # Challengeable bit (SSZ-bool, 1 byte) for the custody of crosslink data

IndexedAttestation

class IndexedAttestation(Container):
    custody_bit_0_indices: List[ValidatorIndex, MAX_INDICES_PER_ATTESTATION]  # Indices with custody bit equal to 0
    custody_bit_1_indices: List[ValidatorIndex, MAX_INDICES_PER_ATTESTATION]  # Indices with custody bit equal to 1
    data: AttestationData
    signature: BLSSignature

PendingAttestation

class PendingAttestation(Container):
    aggregation_bitfield: Bytes[MAX_INDICES_PER_ATTESTATION // 8]
    data: AttestationData
    inclusion_delay: Slot
    proposer_index: ValidatorIndex

Eth1Data

class Eth1Data(Container):
    deposit_root: Hash
    deposit_count: uint64
    block_hash: Hash

HistoricalBatch

class HistoricalBatch(Container):
    block_roots: Vector[Hash, SLOTS_PER_HISTORICAL_ROOT]
    state_roots: Vector[Hash, SLOTS_PER_HISTORICAL_ROOT]

DepositData

class DepositData(Container):
    pubkey: BLSPubkey
    withdrawal_credentials: Hash
    amount: Gwei
    signature: BLSSignature

BeaconBlockHeader

class BeaconBlockHeader(Container):
    slot: Slot
    parent_root: Hash
    state_root: Hash
    body_root: Hash
    signature: BLSSignature

Beacon operations

ProposerSlashing

class ProposerSlashing(Container):
    proposer_index: ValidatorIndex
    header_1: BeaconBlockHeader
    header_2: BeaconBlockHeader

AttesterSlashing

class AttesterSlashing(Container):
    attestation_1: IndexedAttestation
    attestation_2: IndexedAttestation

Attestation

class Attestation(Container):
    aggregation_bitfield: Bytes[MAX_INDICES_PER_ATTESTATION // 8]
    data: AttestationData
    custody_bitfield: Bytes[MAX_INDICES_PER_ATTESTATION // 8]
    signature: BLSSignature

Deposit

class Deposit(Container):
    proof: Vector[Hash, DEPOSIT_CONTRACT_TREE_DEPTH]  # Merkle path to deposit root
    data: DepositData

VoluntaryExit

class VoluntaryExit(Container):
    epoch: Epoch  # Earliest epoch when voluntary exit can be processed
    validator_index: ValidatorIndex
    signature: BLSSignature

Transfer

class Transfer(Container):
    sender: ValidatorIndex
    recipient: ValidatorIndex
    amount: Gwei
    fee: Gwei
    slot: Slot  # Slot at which transfer must be processed
    pubkey: BLSPubkey  # Withdrawal pubkey
    signature: BLSSignature  # Signature checked against withdrawal pubkey

Beacon blocks

BeaconBlockBody

class BeaconBlockBody(Container):
    randao_reveal: BLSSignature
    eth1_data: Eth1Data  # Eth1 data vote
    graffiti: Bytes32  # Arbitrary data
    # Operations
    proposer_slashings: List[ProposerSlashing, MAX_PROPOSER_SLASHINGS]
    attester_slashings: List[AttesterSlashing, MAX_ATTESTER_SLASHINGS]
    attestations: List[Attestation, MAX_ATTESTATIONS]
    deposits: List[Deposit, MAX_DEPOSITS]
    voluntary_exits: List[VoluntaryExit, MAX_VOLUNTARY_EXITS]
    transfers: List[Transfer, MAX_TRANSFERS]

BeaconBlock

class BeaconBlock(Container):
    slot: Slot
    parent_root: Hash
    state_root: Hash
    body: BeaconBlockBody
    signature: BLSSignature

Beacon state

BeaconState

class BeaconState(Container):
    # Versioning
    genesis_time: uint64
    slot: Slot
    fork: Fork
    # History
    latest_block_header: BeaconBlockHeader
    block_roots: Vector[Hash, SLOTS_PER_HISTORICAL_ROOT]
    state_roots: Vector[Hash, SLOTS_PER_HISTORICAL_ROOT]
    historical_roots: List[Hash, HISTORICAL_ROOTS_LIMIT]
    # Eth1
    eth1_data: Eth1Data
    eth1_data_votes: List[Eth1Data, SLOTS_PER_ETH1_VOTING_PERIOD]
    eth1_deposit_index: uint64
    # Registry
    validators: List[Validator, VALIDATOR_REGISTRY_LIMIT]
    balances: List[Gwei, VALIDATOR_REGISTRY_LIMIT]
    # Shuffling
    start_shard: Shard
    randao_mixes: Vector[Hash, EPOCHS_PER_HISTORICAL_VECTOR]
    active_index_roots: Vector[Hash, EPOCHS_PER_HISTORICAL_VECTOR]  # Active registry digests for light clients
    # Slashings
    slashed_balances: Vector[Gwei, EPOCHS_PER_SLASHED_BALANCES_VECTOR]  # Sums of slashed effective balances
    # Attestations
    previous_epoch_attestations: List[PendingAttestation, MAX_ATTESTATIONS * SLOTS_PER_EPOCH]
    current_epoch_attestations: List[PendingAttestation, MAX_ATTESTATIONS * SLOTS_PER_EPOCH]
    # Crosslinks
    previous_crosslinks: Vector[Crosslink, SHARD_COUNT]  # Previous epoch snapshot
    current_crosslinks: Vector[Crosslink, SHARD_COUNT]
    # Justification
    previous_justified_epoch: Epoch  # Previous epoch snapshot
    previous_justified_root: Hash  # Previous epoch snapshot
    current_justified_epoch: Epoch
    current_justified_root: Hash
    justification_bitfield: uint64  # Bit set for every recent justified epoch
    # Finality
    finalized_epoch: Epoch
    finalized_root: Hash

Helper functions

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

xor

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

hash

The hash function is SHA256.

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

hash_tree_root

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

signing_root

def signing_root(object: Container) -> Hash is a function for computing signing messages, as defined in the SimpleSerialize spec.

calc_merkle_tree_from_leaves

zerohashes = [ZERO_HASH]
for layer in range(1, 100):
    zerohashes.append(hash(zerohashes[layer - 1] + zerohashes[layer - 1]))
def calc_merkle_tree_from_leaves(values: Sequence[Hash], layer_count: int=32) -> Sequence[Sequence[Hash]]:
    values = list(values)
    tree = [values[::]]
    for h in range(layer_count):
        if len(values) % 2 == 1:
            values.append(zerohashes[h])
        values = [hash(values[i] + values[i + 1]) for i in range(0, len(values), 2)]
        tree.append(values[::])
    return tree

get_merkle_root

def get_merkle_root(values: Sequence[Hash], pad_to: int=1) -> Hash:
    layer_count = int(log2(pad_to))
    if len(values) == 0:
        return zerohashes[layer_count]
    return calc_merkle_tree_from_leaves(values, layer_count)[-1][0]

bls_domain

def bls_domain(domain_type: int, fork_version: bytes=b'\x00\x00\x00\x00') -> int:
    """
    Return the bls domain given by the ``domain_type`` and optional 4 byte ``fork_version`` (defaults to zero).
    """
    return bytes_to_int(int_to_bytes(domain_type, length=4) + fork_version)

slot_to_epoch

def slot_to_epoch(slot: Slot) -> Epoch:
    """
    Return the epoch number of the given ``slot``.
    """
    return Epoch(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 GENESIS_EPOCH if current_epoch == GENESIS_EPOCH else Epoch(current_epoch - 1)

get_current_epoch

def get_current_epoch(state: BeaconState) -> Epoch:
    """
    Return the current epoch of the given ``state``.
    """
    return slot_to_epoch(state.slot)

get_epoch_start_slot

def get_epoch_start_slot(epoch: Epoch) -> Slot:
    """
    Return the starting slot of the given ``epoch``.
    """
    return Slot(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 (not validator.slashed) and (validator.activation_epoch <= epoch < validator.withdrawable_epoch)

get_active_validator_indices

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

increase_balance

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.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 = Epoch(get_current_epoch(state) + 1)
    shard = Shard((state.start_shard + get_shard_delta(state, get_current_epoch(state))) % SHARD_COUNT)
    while check_epoch > epoch:
        check_epoch -= Epoch(1)
        shard = Shard((shard + SHARD_COUNT - get_shard_delta(state, check_epoch)) % SHARD_COUNT)
    return shard

get_attestation_data_slot

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

get_block_root_at_slot

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

get_block_root

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

get_randao_mix

def get_randao_mix(state: BeaconState,
                   epoch: Epoch) -> Hash:
    """
    Return the randao mix at a recent ``epoch``.
    ``epoch`` expected to be between (current_epoch - EPOCHS_PER_HISTORICAL_VECTOR, current_epoch].
    """
    return state.randao_mixes[epoch % EPOCHS_PER_HISTORICAL_VECTOR]

get_active_index_root

def get_active_index_root(state: BeaconState,
                          epoch: Epoch) -> Hash:
    """
    Return the index root at a recent ``epoch``.
    ``epoch`` expected to be between
    (current_epoch - EPOCHS_PER_HISTORICAL_VECTOR + ACTIVATION_EXIT_DELAY, current_epoch + ACTIVATION_EXIT_DELAY].
    """
    return state.active_index_roots[epoch % EPOCHS_PER_HISTORICAL_VECTOR]

generate_seed

def generate_seed(state: BeaconState,
                  epoch: Epoch) -> Hash:
    """
    Generate a seed for the given ``epoch``.
    """
    return hash(
        get_randao_mix(state, Epoch(epoch + EPOCHS_PER_HISTORICAL_VECTOR - MIN_SEED_LOOKAHEAD)) +
        get_active_index_root(state, epoch) +
        int_to_bytes(epoch, length=32)
    )

get_beacon_proposer_index

def get_beacon_proposer_index(state: BeaconState) -> ValidatorIndex:
    """
    Return the current beacon proposer index.
    """
    epoch = get_current_epoch(state)
    committees_per_slot = get_epoch_committee_count(state, epoch) // SLOTS_PER_EPOCH
    offset = committees_per_slot * (state.slot % SLOTS_PER_EPOCH)
    shard = Shard((get_epoch_start_shard(state, epoch) + offset) % SHARD_COUNT)
    first_committee = get_crosslink_committee(state, epoch, shard)
    MAX_RANDOM_BYTE = 2**8 - 1
    seed = generate_seed(state, epoch)
    i = 0
    while True:
        candidate_index = first_committee[(epoch + i) % len(first_committee)]
        random_byte = hash(seed + int_to_bytes(i // 32, length=8))[i % 32]
        effective_balance = state.validators[candidate_index].effective_balance
        if effective_balance * MAX_RANDOM_BYTE >= MAX_EFFECTIVE_BALANCE * random_byte:
            return ValidatorIndex(candidate_index)
        i += 1

verify_merkle_branch

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

get_shuffled_index

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

    # Swap or not (https://link.springer.com/content/pdf/10.1007%2F978-3-642-32009-5_1.pdf)
    # See the 'generalized domain' algorithm on page 3
    for current_round in range(SHUFFLE_ROUND_COUNT):
        pivot = bytes_to_int(hash(seed + int_to_bytes(current_round, length=1))[0:8]) % index_count
        flip = ValidatorIndex((pivot + index_count - index) % index_count)
        position = max(index, flip)
        source = hash(
            seed + int_to_bytes(current_round, length=1) +
            int_to_bytes(position // 256, length=4)
        )
        byte = source[(position % 256) // 8]
        bit = (byte >> (position % 8)) % 2
        index = flip if bit else index

    return ValidatorIndex(index)

compute_committee

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

get_crosslink_committee

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

get_attesting_indices

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

int_to_bytes

def int_to_bytes(integer: int, length: int) -> bytes:
    return integer.to_bytes(length, 'little')

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: Set[ValidatorIndex]) -> Gwei:
    """
    Return the combined effective balance of the ``indices``. (1 Gwei minimum to avoid divisions by zero.)
    """
    return Gwei(max(sum([state.validators[index].effective_balance for index in indices]), 1))

get_domain

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

get_bitfield_bit

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)
    assert set(custody_bit_1_indices).issubset(attesting_indices)
    custody_bit_0_indices = [index for index in attesting_indices if index not in custody_bit_1_indices]

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

validate_indexed_attestation

def validate_indexed_attestation(state: BeaconState, indexed_attestation: IndexedAttestation) -> None:
    """
    Verify validity of ``indexed_attestation``.
    """
    bit_0_indices = indexed_attestation.custody_bit_0_indices
    bit_1_indices = indexed_attestation.custody_bit_1_indices

    # Verify no index has custody bit equal to 1 [to be removed in phase 1]
    assert len(bit_1_indices) == 0
    # Verify max number of indices
    assert len(bit_0_indices) + len(bit_1_indices) <= MAX_INDICES_PER_ATTESTATION
    # Verify index sets are disjoint
    assert len(set(bit_0_indices).intersection(bit_1_indices)) == 0
    # Verify indices are sorted
    assert bit_0_indices == sorted(bit_0_indices) and bit_1_indices == sorted(bit_1_indices)
    # Verify aggregate signature
    assert bls_verify_multiple(
        pubkeys=[
            bls_aggregate_pubkeys([state.validators[i].pubkey for i in bit_0_indices]),
            bls_aggregate_pubkeys([state.validators[i].pubkey for i in bit_1_indices]),
        ],
        message_hashes=[
            hash_tree_root(AttestationDataAndCustodyBit(data=indexed_attestation.data, custody_bit=0b0)),
            hash_tree_root(AttestationDataAndCustodyBit(data=indexed_attestation.data, custody_bit=0b1)),
        ],
        signature=indexed_attestation.signature,
        domain=get_domain(state, DOMAIN_ATTESTATION, indexed_attestation.data.target_epoch),
    )

is_slashable_attestation_data

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(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, as defined in the BLS Signature spec.

bls_verify_multiple

bls_verify_multiple is a function for verifying a BLS signature constructed from multiple messages, as defined in the BLS Signature spec.

bls_aggregate_pubkeys

bls_aggregate_pubkeys is a function for aggregating multiple BLS public keys into a single aggregate key, as defined in the BLS Signature spec.

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 exit of the validator of the given ``index``.
    """
    # Return if validator already initiated exit
    validator = state.validators[index]
    if validator.exit_epoch != FAR_FUTURE_EPOCH:
        return

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

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

slash_validator

def slash_validator(state: BeaconState,
                    slashed_index: ValidatorIndex,
                    whistleblower_index: ValidatorIndex=None) -> None:
    """
    Slash the validator with index ``slashed_index``.
    """
    current_epoch = get_current_epoch(state)
    initiate_validator_exit(state, slashed_index)
    state.validators[slashed_index].slashed = True
    state.validators[slashed_index].withdrawable_epoch = Epoch(current_epoch + EPOCHS_PER_SLASHED_BALANCES_VECTOR)
    slashed_balance = state.validators[slashed_index].effective_balance
    state.slashed_balances[current_epoch % EPOCHS_PER_SLASHED_BALANCES_VECTOR] += slashed_balance

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

Genesis

Genesis trigger

Before genesis has been triggered and for every Eth 1.0 block call is_genesis_trigger(deposits, time) where:

  • deposits is the list of all deposits up to the Eth 1.0 block, ordered chronologically
  • time is the Unix time of the Eth 1.0 block

When is_genesis_trigger(deposits, time) is True for the first time let:

  • genesis_deposits = deposits
  • genesis_time = time - time % SECONDS_PER_DAY + 2 * SECONDS_PER_DAY where SECONDS_PER_DAY = 86400
  • genesis_eth1_block_hash is the Eth 1.0 block hash that emitted the log for the last deposit in deposits

Note: The function is_genesis_trigger has yet to be agreed upon by the community, and can be updated as necessary. We define the following testing placeholder:

def is_genesis_trigger(deposits: List[Deposit, 2**DEPOSIT_CONTRACT_TREE_DEPTH], time: uint64) -> bool:
    SECONDS_PER_DAY = 86400
    # Do not deploy too early
    if time - time % SECONDS_PER_DAY + 2 * SECONDS_PER_DAY < MIN_GENESIS_TIME:
        return False

    # Process deposits
    state = BeaconState()
    leaves = list(map(lambda deposit: hash_tree_root(deposit.data), deposits))
    for deposit_index, deposit in enumerate(deposits):
        state.eth1_data.deposit_root = get_merkle_root(leaves[:deposit_index + 1], 2**DEPOSIT_CONTRACT_TREE_DEPTH)
        state.eth1_data.deposit_count = deposit_index + 1
        state.eth1_deposit_index = deposit_index
        process_deposit(state, deposit)

    # Count active validators at genesis
    active_validator_count = 0
    for validator in state.validators:
        if validator.effective_balance == MAX_EFFECTIVE_BALANCE:
            active_validator_count += 1

    # Check effective balance to trigger genesis
    return active_validator_count >= GENESIS_ACTIVE_VALIDATOR_COUNT

Genesis state

Let genesis_state = get_genesis_beacon_state(genesis_deposits, genesis_time, genesis_eth1_block_hash).

def get_genesis_beacon_state(deposits: Sequence[Deposit], genesis_time: int, genesis_eth1_block_hash: Hash) -> BeaconState:
    state = BeaconState(
        genesis_time=genesis_time,
        eth1_data=Eth1Data(block_hash=genesis_eth1_block_hash),
        latest_block_header=BeaconBlockHeader(body_root=hash_tree_root(BeaconBlockBody())),
    )

    # Process genesis deposits
    leaves = list(map(lambda deposit: hash_tree_root(deposit.data), deposits))
    for deposit_index, deposit in enumerate(deposits):
        state.eth1_data.deposit_root = get_merkle_root(leaves[:deposit_index + 1], 2**DEPOSIT_CONTRACT_TREE_DEPTH)
        state.eth1_data.deposit_count = deposit_index + 1
        state.eth1_deposit_index = deposit_index
        process_deposit(state, deposit)

    # Process genesis activations
    for validator in state.validators:
        if validator.effective_balance == MAX_EFFECTIVE_BALANCE:
            validator.activation_eligibility_epoch = GENESIS_EPOCH
            validator.activation_epoch = GENESIS_EPOCH

    # Populate active_index_roots
    genesis_active_index_root = hash_tree_root(
        List[ValidatorIndex, VALIDATOR_REGISTRY_LIMIT](get_active_validator_indices(state, GENESIS_EPOCH))
    )
    for index in range(EPOCHS_PER_HISTORICAL_VECTOR):
        state.active_index_roots[index] = genesis_active_index_root

    return state

Genesis block

Let genesis_block = BeaconBlock(state_root=hash_tree_root(genesis_state)).

Beacon chain state transition function

The post-state corresponding to a pre-state state and a block block is defined as state_transition(state, block). State transitions that trigger an unhandled excpetion (e.g. a failed assert or an out-of-range list access) are considered invalid.

def state_transition(state: BeaconState, block: BeaconBlock, validate_state_root: bool=False) -> BeaconState:
    # Process slots (including those with no blocks) since block
    process_slots(state, block.slot)
    # Process block
    process_block(state, block)
    # Validate state root (`validate_state_root == True` in production)
    if validate_state_root:
        assert block.state_root == hash_tree_root(state)
    # Return post-state
    return state

def process_slots(state: BeaconState, slot: Slot) -> None:
    assert state.slot <= slot
    while state.slot < slot:
        process_slot(state)
        # Process epoch on the first slot of the next epoch
        if (state.slot + 1) % SLOTS_PER_EPOCH == 0:
            process_epoch(state)
        state.slot += Slot(1)

def process_slot(state: BeaconState) -> None:
    # Cache state root
    previous_state_root = hash_tree_root(state)
    state.state_roots[state.slot % SLOTS_PER_HISTORICAL_ROOT] = previous_state_root

    # Cache latest block header state root
    if state.latest_block_header.state_root == ZERO_HASH:
        state.latest_block_header.state_root = previous_state_root

    # Cache block root
    previous_block_root = signing_root(state.latest_block_header)
    state.block_roots[state.slot % SLOTS_PER_HISTORICAL_ROOT] = previous_block_root

Epoch processing

Note: The # @LabelHere lines below are placeholders to show that code will be inserted here in a future phase.

def process_epoch(state: BeaconState) -> None:
    process_justification_and_finalization(state)
    process_crosslinks(state)
    process_rewards_and_penalties(state)
    process_registry_updates(state)
    # @process_reveal_deadlines
    # @process_challenge_deadlines
    process_slashings(state)
    process_final_updates(state)
    # @after_process_final_updates

Helper functions

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

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

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

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

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

def get_attesting_balance(state: BeaconState, attestations: Sequence[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, Set[ValidatorIndex]]:
    attestations = [a for a in get_matching_source_attestations(state, epoch) if a.data.crosslink.shard == shard]
    crosslinks = list(filter(
        lambda c: hash_tree_root(state.current_crosslinks[shard]) in (c.parent_root, hash_tree_root(c)),
        [a.data.crosslink for a in attestations]
    ))
    # Winning crosslink has the crosslink data root with the most balance voting for it (ties broken lexicographically)
    winning_crosslink = max(crosslinks, key=lambda c: (
        get_attesting_balance(state, [a for a in attestations if a.data.crosslink == c]), c.data_root
    ), default=Crosslink())
    winning_attestations = [a for a in attestations if a.data.crosslink == winning_crosslink]
    return winning_crosslink, get_unslashed_attesting_indices(state, winning_attestations)

Justification and finalization

def process_justification_and_finalization(state: BeaconState) -> None:
    if get_current_epoch(state) <= GENESIS_EPOCH + 1:
        return

    previous_epoch = get_previous_epoch(state)
    current_epoch = get_current_epoch(state)
    old_previous_justified_epoch = state.previous_justified_epoch
    old_current_justified_epoch = state.current_justified_epoch

    # Process justifications
    state.previous_justified_epoch = state.current_justified_epoch
    state.previous_justified_root = state.current_justified_root
    state.justification_bitfield = (state.justification_bitfield << 1) % 2**64
    previous_epoch_matching_target_balance = get_attesting_balance(
        state, get_matching_target_attestations(state, previous_epoch)
    )
    if previous_epoch_matching_target_balance * 3 >= get_total_active_balance(state) * 2:
        state.current_justified_epoch = previous_epoch
        state.current_justified_root = get_block_root(state, state.current_justified_epoch)
        state.justification_bitfield |= (1 << 1)
    current_epoch_matching_target_balance = get_attesting_balance(
        state, get_matching_target_attestations(state, current_epoch)
    )
    if current_epoch_matching_target_balance * 3 >= get_total_active_balance(state) * 2:
        state.current_justified_epoch = current_epoch
        state.current_justified_root = get_block_root(state, state.current_justified_epoch)
        state.justification_bitfield |= (1 << 0)

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

Crosslinks

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

Rewards and penalties

def get_base_reward(state: BeaconState, index: ValidatorIndex) -> Gwei:
    total_balance = get_total_active_balance(state)
    effective_balance = state.validators[index].effective_balance
    return Gwei(effective_balance * BASE_REWARD_FACTOR // integer_squareroot(total_balance) // BASE_REWARDS_PER_EPOCH)

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

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

    # Proposer and inclusion delay micro-rewards
    for index in get_unslashed_attesting_indices(state, matching_source_attestations):
        index = ValidatorIndex(index)
        attestation = min([
            a for a in matching_source_attestations
            if index in get_attesting_indices(state, a.data, a.aggregation_bitfield)
        ], key=lambda a: a.inclusion_delay)
        proposer_reward = Gwei(get_base_reward(state, index) // PROPOSER_REWARD_QUOTIENT)
        rewards[attestation.proposer_index] += proposer_reward
        max_attester_reward = get_base_reward(state, index) - proposer_reward
        rewards[index] += Gwei(max_attester_reward * MIN_ATTESTATION_INCLUSION_DELAY // attestation.inclusion_delay)

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

    return rewards, penalties

def get_crosslink_deltas(state: BeaconState) -> Tuple[Sequence[Gwei], Sequence[Gwei]]:
    rewards = [Gwei(0) for _ in range(len(state.validators))]
    penalties = [Gwei(0) for _ in range(len(state.validators))]
    epoch = get_previous_epoch(state)
    for offset in range(get_epoch_committee_count(state, epoch)):
        shard = Shard((get_epoch_start_shard(state, epoch) + offset) % SHARD_COUNT)
        crosslink_committee = set(get_crosslink_committee(state, epoch, shard))
        winning_crosslink, attesting_indices = get_winning_crosslink_and_attesting_indices(state, epoch, shard)
        attesting_balance = get_total_balance(state, attesting_indices)
        committee_balance = get_total_balance(state, crosslink_committee)
        for index in crosslink_committee:
            base_reward = get_base_reward(state, index)
            if index in attesting_indices:
                rewards[index] += base_reward * attesting_balance // committee_balance
            else:
                penalties[index] += base_reward
    return rewards, penalties

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

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

Registry updates

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

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

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

Slashings

def process_slashings(state: BeaconState) -> None:
    epoch = get_current_epoch(state)
    total_balance = get_total_active_balance(state)

    # Compute slashed balances in the current epoch
    total_at_start = state.slashed_balances[(epoch + 1) % EPOCHS_PER_SLASHED_BALANCES_VECTOR]
    total_at_end = state.slashed_balances[epoch % EPOCHS_PER_SLASHED_BALANCES_VECTOR]
    total_penalties = total_at_end - total_at_start

    for index, validator in enumerate(state.validators):
        if validator.slashed and epoch + EPOCHS_PER_SLASHED_BALANCES_VECTOR // 2 == validator.withdrawable_epoch:
            penalty = max(
                validator.effective_balance * min(total_penalties * 3, total_balance) // total_balance,
                validator.effective_balance // MIN_SLASHING_PENALTY_QUOTIENT
            )
            decrease_balance(state, ValidatorIndex(index), penalty)

Final updates

def process_final_updates(state: BeaconState) -> None:
    current_epoch = get_current_epoch(state)
    next_epoch = current_epoch + 1
    # Reset eth1 data votes
    if (state.slot + 1) % SLOTS_PER_ETH1_VOTING_PERIOD == 0:
        state.eth1_data_votes = []
    # Update effective balances with hysteresis
    for index, validator in enumerate(state.validators):
        balance = state.balances[index]
        HALF_INCREMENT = EFFECTIVE_BALANCE_INCREMENT // 2
        if balance < validator.effective_balance or validator.effective_balance + 3 * HALF_INCREMENT < balance:
            validator.effective_balance = min(balance - balance % EFFECTIVE_BALANCE_INCREMENT, MAX_EFFECTIVE_BALANCE)
    # Update start shard
    state.start_shard = Shard((state.start_shard + get_shard_delta(state, current_epoch)) % SHARD_COUNT)
    # Set active index root
    index_root_position = (next_epoch + ACTIVATION_EXIT_DELAY) % EPOCHS_PER_HISTORICAL_VECTOR
    state.active_index_roots[index_root_position] = hash_tree_root(
        List[ValidatorIndex, VALIDATOR_REGISTRY_LIMIT](
            get_active_validator_indices(state, Epoch(next_epoch + ACTIVATION_EXIT_DELAY))
        )
    )
    # Set total slashed balances
    state.slashed_balances[next_epoch % EPOCHS_PER_SLASHED_BALANCES_VECTOR] = (
        state.slashed_balances[current_epoch % EPOCHS_PER_SLASHED_BALANCES_VECTOR]
    )
    # Set randao mix
    state.randao_mixes[next_epoch % EPOCHS_PER_HISTORICAL_VECTOR] = get_randao_mix(state, current_epoch)
    # Set historical root accumulator
    if next_epoch % (SLOTS_PER_HISTORICAL_ROOT // SLOTS_PER_EPOCH) == 0:
        historical_batch = HistoricalBatch(
            block_roots=state.block_roots,
            state_roots=state.state_roots,
        )
        state.historical_roots.append(hash_tree_root(historical_batch))
    # Rotate current/previous epoch attestations
    state.previous_epoch_attestations = state.current_epoch_attestations
    state.current_epoch_attestations = []

Block processing

def process_block(state: BeaconState, block: BeaconBlock) -> None:
    process_block_header(state, block)
    process_randao(state, block.body)
    process_eth1_data(state, block.body)
    process_operations(state, block.body)

Block header

def process_block_header(state: BeaconState, block: BeaconBlock) -> None:
    # Verify that the slots match
    assert block.slot == state.slot
    # Verify that the parent matches
    assert block.parent_root == signing_root(state.latest_block_header)
    # Save current block as the new latest block
    state.latest_block_header = BeaconBlockHeader(
        slot=block.slot,
        parent_root=block.parent_root,
        state_root=ZERO_HASH,  # Overwritten in next `process_slot` call
        body_root=hash_tree_root(block.body),
    )
    # Verify proposer is not slashed
    proposer = state.validators[get_beacon_proposer_index(state)]
    assert not proposer.slashed
    # Verify proposer signature
    assert bls_verify(proposer.pubkey, signing_root(block), block.signature, get_domain(state, DOMAIN_BEACON_PROPOSER))

RANDAO

def process_randao(state: BeaconState, body: BeaconBlockBody) -> None:
    epoch = get_current_epoch(state)
    # Verify RANDAO reveal
    proposer = state.validators[get_beacon_proposer_index(state)]
    assert bls_verify(proposer.pubkey, hash_tree_root(epoch), body.randao_reveal, get_domain(state, DOMAIN_RANDAO))
    # Mix in RANDAO reveal
    mix = xor(get_randao_mix(state, epoch), hash(body.randao_reveal))
    state.randao_mixes[epoch % EPOCHS_PER_HISTORICAL_VECTOR] = mix

Eth1 data

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

Operations

def process_operations(state: BeaconState, body: BeaconBlockBody) -> None:
    # Verify that outstanding deposits are processed up to the maximum number of deposits
    assert len(body.deposits) == min(MAX_DEPOSITS, state.eth1_data.deposit_count - state.eth1_deposit_index)
    # Verify that there are no duplicate transfers
    assert len(body.transfers) == len(set(body.transfers))

    all_operations = (
        (body.proposer_slashings, process_proposer_slashing),
        (body.attester_slashings, process_attester_slashing),
        (body.attestations, process_attestation),
        (body.deposits, process_deposit),
        (body.voluntary_exits, process_voluntary_exit),
        (body.transfers, process_transfer),
    )  # type: Sequence[Tuple[List, Callable]]
    for operations, function in all_operations:
        for operation in operations:
            function(state, operation)
Proposer slashings
def process_proposer_slashing(state: BeaconState, proposer_slashing: ProposerSlashing) -> None:
    """
    Process ``ProposerSlashing`` operation.
    """
    proposer = state.validators[proposer_slashing.proposer_index]
    # Verify that the epoch is the same
    assert slot_to_epoch(proposer_slashing.header_1.slot) == slot_to_epoch(proposer_slashing.header_2.slot)
    # But the headers are different
    assert proposer_slashing.header_1 != proposer_slashing.header_2
    # Check proposer is slashable
    assert is_slashable_validator(proposer, get_current_epoch(state))
    # Signatures are valid
    for header in (proposer_slashing.header_1, proposer_slashing.header_2):
        domain = get_domain(state, DOMAIN_BEACON_PROPOSER, slot_to_epoch(header.slot))
        assert bls_verify(proposer.pubkey, signing_root(header), header.signature, domain)

    slash_validator(state, proposer_slashing.proposer_index)
Attester slashings
def process_attester_slashing(state: BeaconState, attester_slashing: AttesterSlashing) -> None:
    """
    Process ``AttesterSlashing`` operation.
    """
    attestation_1 = attester_slashing.attestation_1
    attestation_2 = attester_slashing.attestation_2
    assert is_slashable_attestation_data(attestation_1.data, attestation_2.data)
    validate_indexed_attestation(state, attestation_1)
    validate_indexed_attestation(state, attestation_2)

    slashed_any = False
    attesting_indices_1 = attestation_1.custody_bit_0_indices + attestation_1.custody_bit_1_indices
    attesting_indices_2 = attestation_2.custody_bit_0_indices + attestation_2.custody_bit_1_indices
    for index in sorted(set(attesting_indices_1).intersection(attesting_indices_2)):
        if is_slashable_validator(state.validators[index], get_current_epoch(state)):
            slash_validator(state, index)
            slashed_any = True
    assert slashed_any
Attestations
def process_attestation(state: BeaconState, attestation: Attestation) -> None:
    """
    Process ``Attestation`` operation.
    """
    data = attestation.data

    assert data.crosslink.shard < SHARD_COUNT
    assert data.target_epoch in (get_previous_epoch(state), get_current_epoch(state))

    attestation_slot = get_attestation_data_slot(state, data)
    assert attestation_slot + MIN_ATTESTATION_INCLUSION_DELAY <= state.slot <= attestation_slot + SLOTS_PER_EPOCH

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

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

    # Check FFG data, crosslink data, and signature
    assert ffg_data == (data.source_epoch, data.source_root, data.target_epoch)
    assert data.crosslink.start_epoch == parent_crosslink.end_epoch
    assert data.crosslink.end_epoch == min(data.target_epoch, parent_crosslink.end_epoch + MAX_EPOCHS_PER_CROSSLINK)
    assert data.crosslink.parent_root == hash_tree_root(parent_crosslink)
    assert data.crosslink.data_root == ZERO_HASH  # [to be removed in phase 1]
    validate_indexed_attestation(state, convert_to_indexed(state, attestation))
Deposits
def process_deposit(state: BeaconState, deposit: Deposit) -> None:
    """
    Process an Eth1 deposit, registering a validator or increasing its balance.
    """
    # Verify the Merkle branch
    assert verify_merkle_branch(
        leaf=hash_tree_root(deposit.data),
        proof=deposit.proof,
        depth=DEPOSIT_CONTRACT_TREE_DEPTH,
        index=state.eth1_deposit_index,
        root=state.eth1_data.deposit_root,
    )

    # Deposits must be processed in order
    state.eth1_deposit_index += 1

    pubkey = deposit.data.pubkey
    amount = deposit.data.amount
    validator_pubkeys = [v.pubkey for v in state.validators]
    if pubkey not in validator_pubkeys:
        # Verify the deposit signature (proof of possession).
        # Invalid signatures are allowed by the deposit contract,
        # and hence included on-chain, but must not be processed.
        # Note: Deposits are valid across forks, hence the deposit domain is retrieved directly from `bls_domain`
        if not bls_verify(
            pubkey, signing_root(deposit.data), deposit.data.signature, bls_domain(DOMAIN_DEPOSIT)
        ):
            return

        # Add validator and balance entries
        state.validators.append(Validator(
            pubkey=pubkey,
            withdrawal_credentials=deposit.data.withdrawal_credentials,
            activation_eligibility_epoch=FAR_FUTURE_EPOCH,
            activation_epoch=FAR_FUTURE_EPOCH,
            exit_epoch=FAR_FUTURE_EPOCH,
            withdrawable_epoch=FAR_FUTURE_EPOCH,
            effective_balance=min(amount - amount % EFFECTIVE_BALANCE_INCREMENT, MAX_EFFECTIVE_BALANCE)
        ))
        state.balances.append(amount)
    else:
        # Increase balance by deposit amount
        index = ValidatorIndex(validator_pubkeys.index(pubkey))
        increase_balance(state, index, amount)
Voluntary exits
def process_voluntary_exit(state: BeaconState, exit: VoluntaryExit) -> None:
    """
    Process ``VoluntaryExit`` operation.
    """
    validator = state.validators[exit.validator_index]
    # Verify the validator is active
    assert is_active_validator(validator, get_current_epoch(state))
    # Verify the validator has not yet exited
    assert validator.exit_epoch == FAR_FUTURE_EPOCH
    # Exits must specify an epoch when they become valid; they are not valid before then
    assert get_current_epoch(state) >= exit.epoch
    # Verify the validator has been active long enough
    assert get_current_epoch(state) >= validator.activation_epoch + PERSISTENT_COMMITTEE_PERIOD
    # Verify signature
    domain = get_domain(state, DOMAIN_VOLUNTARY_EXIT, exit.epoch)
    assert bls_verify(validator.pubkey, signing_root(exit), exit.signature, domain)
    # Initiate exit
    initiate_validator_exit(state, exit.validator_index)
Transfers
def process_transfer(state: BeaconState, transfer: Transfer) -> None:
    """
    Process ``Transfer`` operation.
    """
    # Verify the balance the covers amount and fee (with overflow protection)
    assert state.balances[transfer.sender] >= max(transfer.amount + transfer.fee, transfer.amount, transfer.fee)
    # A transfer is valid in only one slot
    assert state.slot == transfer.slot
    # Sender must satisfy at least one of the following conditions in the parenthesis:
    assert (
        # * Has not been activated
        state.validators[transfer.sender].activation_eligibility_epoch == FAR_FUTURE_EPOCH or
        # * Is withdrawable
        get_current_epoch(state) >= state.validators[transfer.sender].withdrawable_epoch or
        # * Balance after transfer is more than the effective balance threshold
        transfer.amount + transfer.fee + MAX_EFFECTIVE_BALANCE <= state.balances[transfer.sender]
    )
    # Verify that the pubkey is valid
    assert (
        state.validators[transfer.sender].withdrawal_credentials ==
        int_to_bytes(BLS_WITHDRAWAL_PREFIX, length=1) + hash(transfer.pubkey)[1:]
    )
    # Verify that the signature is valid
    assert bls_verify(transfer.pubkey, signing_root(transfer), transfer.signature, get_domain(state, DOMAIN_TRANSFER))
    # Process the transfer
    decrease_balance(state, transfer.sender, transfer.amount + transfer.fee)
    increase_balance(state, transfer.recipient, transfer.amount)
    increase_balance(state, get_beacon_proposer_index(state), transfer.fee)
    # Verify balances are not dust
    assert not (0 < state.balances[transfer.sender] < MIN_DEPOSIT_AMOUNT)
    assert not (0 < state.balances[transfer.recipient] < MIN_DEPOSIT_AMOUNT)