Ethereum 2.0 Phase 0 -- The Beacon Chain

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

Introduction

This document represents the specification for Phase 0 of Ethereum 2.0 -- The Beacon Chain.

At the core of Ethereum 2.0 is a system chain called the "beacon chain". The beacon chain stores and manages the registry of validators. In the initial deployment phases of Ethereum 2.0 the only mechanism to become a validator is to make a one-way ETH transaction to a deposit contract on Ethereum 1.0. Activation as a validator happens when Ethereum 1.0 deposit receipts are processed by the beacon chain, the activation balance is reached, and after a queuing process. Exit is either voluntary or done forcibly as a penalty for misbehavior.

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

Notation

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

Terminology

Validator - a registered participant in the beacon chain. You can become one by sending Ether into the Ethereum 1.0 deposit contract.
Active validator - an active participant in the Ethereum 2.0 consensus invited to, among other things, propose and attest to blocks and vote for crosslinks.
Committee - a (pseudo-) randomly sampled subset of active validators. When a committee is referred to collectively, as in "this committee attests to X", this is assumed to mean "some subset of that committee that contains enough validators that the protocol recognizes it as representing the committee".
Proposer - the validator that creates a beacon chain block
Attester - a validator that is part of a committee that needs to sign off on a beacon chain block while simultaneously creating a link (crosslink) to a recent shard block on a particular shard chain.
Beacon chain - the central PoS chain that is the base of the sharding system.
Shard chain - one of the chains on which user transactions take place and account data is stored.
Block root - a 32-byte Merkle root of a beacon chain block or shard chain block. Previously called "block hash".
Crosslink - a set of signatures from a committee attesting to a block in a shard chain, which can be included into the beacon chain. Crosslinks are the main means by which the beacon chain "learns about" the updated state of shard chains.
Slot - a period during which one proposer has the ability to create a beacon chain block and some attesters have the ability to make attestations
Epoch - an aligned span of slots during which all validators get exactly one chance to make an attestation
Finalized, justified - see Casper FFG finalization [casper-ffg]
Withdrawal period - the number of slots between a validator exit and the validator balance being withdrawable
Genesis time - the Unix time of the genesis beacon chain block at slot 0

Constants

Misc

Name	Value
`SHARD_COUNT`	`2**10` (= 1,024)
`TARGET_COMMITTEE_SIZE`	`2**7` (= 128)
`MAX_BALANCE_CHURN_QUOTIENT`	`2**5` (= 32)
`MAX_INDICES_PER_SLASHABLE_VOTE`	`2**12` (= 4,096)
`MAX_EXIT_DEQUEUES_PER_EPOCH`	`2**2` (= 4)
`SHUFFLE_ROUND_COUNT`	90

For the safety of crosslinks TARGET_COMMITTEE_SIZE exceeds the recommended minimum committee size of 111; with sufficient active validators (at least SLOTS_PER_EPOCH * TARGET_COMMITTEE_SIZE), the shuffling algorithm ensures committee sizes of at least TARGET_COMMITTEE_SIZE. (Unbiasable randomness with a Verifiable Delay Function (VDF) will improve committee robustness and lower the safe minimum committee size.)

Deposit contract

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

Gwei values

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

Initial values

Name	Value
`GENESIS_FORK_VERSION`	`int_to_bytes4(0)`
`GENESIS_SLOT`	`2**32`
`GENESIS_EPOCH`	`slot_to_epoch(GENESIS_SLOT)`
`GENESIS_START_SHARD`	`0`
`FAR_FUTURE_EPOCH`	`2**64 - 1`
`ZERO_HASH`	`int_to_bytes32(0)`
`EMPTY_SIGNATURE`	`int_to_bytes96(0)`
`BLS_WITHDRAWAL_PREFIX_BYTE`	`int_to_bytes1(0)`

GENESIS_SLOT should be at least as large in terms of time as the largest of the time parameters or state list lengths below (ie. it should be at least as large as any value measured in slots, and at least SLOTS_PER_EPOCH times as large as any value measured in epochs).

Time parameters

Name	Value	Unit	Duration
`SECONDS_PER_SLOT`	`6`	seconds	6 seconds
`MIN_ATTESTATION_INCLUSION_DELAY`	`2**2` (= 4)	slots	24 seconds
`SLOTS_PER_EPOCH`	`2**6` (= 64)	slots	6.4 minutes
`MIN_SEED_LOOKAHEAD`	`2**0` (= 1)	epochs	6.4 minutes
`ACTIVATION_EXIT_DELAY`	`2**2` (= 4)	epochs	25.6 minutes
`EPOCHS_PER_ETH1_VOTING_PERIOD`	`2**4` (= 16)	epochs	~1.7 hours
`SLOTS_PER_HISTORICAL_ROOT`	`2**13` (= 8,192)	slots	~13 hours
`MIN_VALIDATOR_WITHDRAWABILITY_DELAY`	`2**8` (= 256)	epochs	~27 hours
`PERSISTENT_COMMITTEE_PERIOD`	`2**11` (= 2,048)	epochs	9 days
`MAX_CROSSLINK_EPOCHS`	`2**6` (= 64)

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)
`WHISTLEBLOWER_REWARD_QUOTIENT`	`2**9` (= 512)
`ATTESTATION_INCLUSION_REWARD_QUOTIENT`	`2**3` (= 8)
`INACTIVITY_PENALTY_QUOTIENT`	`2**24` (= 16,777,216)
`MIN_PENALTY_QUOTIENT`	`2**5` (= 32)

The BASE_REWARD_QUOTIENT parameter dictates the per-epoch reward. It corresponds to ~2.54% annual interest assuming 10 million participating ETH in every epoch.
The INACTIVITY_PENALTY_QUOTIENT equals INVERSE_SQRT_E_DROP_TIME**2 where INVERSE_SQRT_E_DROP_TIME := 2**12 epochs (~18 days) is the time it takes the inactivity penalty to reduce the balance of non-participating validators to about 1/sqrt(e) ~= 60.6%. Indeed, the balance retained by offline validators after n epochs is about (1-1/INACTIVITY_PENALTY_QUOTIENT)**(n**2/2) so after INVERSE_SQRT_E_DROP_TIME epochs it is roughly (1-1/INACTIVITY_PENALTY_QUOTIENT)**(INACTIVITY_PENALTY_QUOTIENT/2) ~= 1/sqrt(e).

Max transactions per block

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

Signature domains

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

Data structures

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

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

Misc dependencies

`Fork`

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

`Crosslink`

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

`Eth1Data`

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

`Eth1DataVote`

{
    # Data being voted for
    'eth1_data': Eth1Data,
    # Vote count
    'vote_count': 'uint64',
}

`AttestationData`

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

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

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

`AttestationDataAndCustodyBit`

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

`SlashableAttestation`

{
    # Validator indices
    'validator_indices': ['uint64'],
    # Attestation data
    'data': AttestationData,
    # Custody bitfield
    'custody_bitfield': 'bytes',
    # Aggregate signature
    'aggregate_signature': 'bytes96',
}

`DepositInput`

{
    # BLS pubkey
    'pubkey': 'bytes48',
    # Withdrawal credentials
    'withdrawal_credentials': 'bytes32',
    # A BLS signature of this `DepositInput`
    'proof_of_possession': 'bytes96',
}

`DepositData`

{
    # Amount in Gwei
    'amount': 'uint64',
    # Timestamp from deposit contract
    'timestamp': 'uint64',
    # Deposit input
    'deposit_input': DepositInput,
}

`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 validator activated
    'activation_epoch': 'uint64',
    # Epoch when validator exited
    'exit_epoch': 'uint64',
    # Epoch when validator is eligible to withdraw
    'withdrawable_epoch': 'uint64',
    # Did the validator initiate an exit
    'initiated_exit': 'bool',
    # Was the validator slashed
    'slashed': 'bool',
}

`PendingAttestation`

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

`HistoricalBatch`

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

Beacon transactions

`ProposerSlashing`

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

`AttesterSlashing`

{
    # First slashable attestation
    'slashable_attestation_1': SlashableAttestation,
    # Second slashable attestation
    'slashable_attestation_2': SlashableAttestation,
}

`Attestation`

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

`Deposit`

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

`VoluntaryExit`

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

`Transfer`

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

Beacon blocks

`BeaconBlockBody`

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

`BeaconBlock`

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

Beacon state

`BeaconState`

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

    # Validator registry
    'validator_registry': [Validator],
    'validator_balances': ['uint64'],
    'validator_registry_update_epoch': '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
    'latest_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': [Eth1DataVote],
    '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 denoted by hash. In Phase 0 the beacon chain is deployed with the same hash function as Ethereum 1.0, i.e. Keccak-256 (also incorrectly known as SHA3).

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

`hash_tree_root`

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

`signed_root`

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

`get_temporary_block_header`

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

`slot_to_epoch`

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

`get_previous_epoch`

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

`get_current_epoch`

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

`get_epoch_start_slot`

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

`is_active_validator`

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

`get_active_validator_indices`

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

`get_permuted_index`

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

    Utilizes 'swap or not' shuffling found in
    https://link.springer.com/content/pdf/10.1007%2F978-3-642-32009-5_1.pdf
    See the 'generalized domain' algorithm on page 3.
    """
    assert index < list_size
    assert list_size <= 2**40
    
    for round in range(SHUFFLE_ROUND_COUNT):
        pivot = bytes_to_int(hash(seed + int_to_bytes1(round))[0:8]) % list_size
        flip = (pivot - index) % list_size
        position = max(index, flip)
        source = hash(seed + int_to_bytes1(round) + int_to_bytes4(position // 256))
        byte = source[(position % 256) // 8]
        bit = (byte >> (position % 8)) % 2
        index = flip if bit else index

    return index

`split`

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

`get_epoch_committee_count`

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

`get_shuffling`

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

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

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

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

`get_current_epoch_committee_count`

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

`get_crosslink_committees_at_slot`

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

    Note: There are two possible shufflings for crosslink committees for a
    ``slot`` in the next epoch -- with and without a `registry_change`
    """
    epoch = slot_to_epoch(slot)
    current_epoch = get_current_epoch(state)
    previous_epoch = get_previous_epoch(state)
    next_epoch = current_epoch + 1

    assert previous_epoch <= epoch <= next_epoch
    committees_per_epoch = get_epoch_committee_count(get_active_validator_indices(
        state.validator_registry,
        epoch,
    ))

    if epoch == current_epoch:
        start_shard = state.latest_start_shard
    elif epoch == previous_epoch:
        start_shard = (state.latest_start_shard - SLOTS_PER_EPOCH * committees_per_epoch) % SHARD_COUNT
    elif epoch == next_epoch:
        current_epoch_committees = get_current_epoch_committee_count(state)
        start_shard = (state.latest_start_shard + EPOCH_LENGTH * current_epoch_committees) % SHARD_COUNT

    shuffling = get_shuffling(
        generate_seed(state, epoch),
        state.validator_registry,
        epoch,
    )
    offset = slot % SLOTS_PER_EPOCH
    committees_per_slot = committees_per_epoch // SLOTS_PER_EPOCH
    slot_start_shard = (start_shard + committees_per_slot * offset) % SHARD_COUNT

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

`get_block_root`

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

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

`get_state_root`

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

`get_randao_mix`

def get_randao_mix(state: BeaconState,
                   epoch: Epoch) -> Bytes32:
    """
    Return the randao mix at a recent ``epoch``.
    """
    assert get_current_epoch(state) - LATEST_RANDAO_MIXES_LENGTH < epoch <= get_current_epoch(state)
    return state.latest_randao_mixes[epoch % LATEST_RANDAO_MIXES_LENGTH]

`get_active_index_root`

def get_active_index_root(state: BeaconState,
                          epoch: Epoch) -> Bytes32:
    """
    Return the index root at a recent ``epoch``.
    """
    assert get_current_epoch(state) - LATEST_ACTIVE_INDEX_ROOTS_LENGTH + ACTIVATION_EXIT_DELAY < epoch <= get_current_epoch(state) + ACTIVATION_EXIT_DELAY
    return state.latest_active_index_roots[epoch % LATEST_ACTIVE_INDEX_ROOTS_LENGTH]

`generate_seed`

def generate_seed(state: BeaconState,
                  epoch: Epoch) -> Bytes32:
    """
    Generate a seed for the given ``epoch``.
    """
    return hash(
        get_randao_mix(state, epoch - MIN_SEED_LOOKAHEAD) +
        get_active_index_root(state, epoch) +
        int_to_bytes32(epoch)
    )

`get_beacon_proposer_index`

def get_beacon_proposer_index(state: BeaconState,
                              slot: Slot,
                              registry_change: bool=False) -> ValidatorIndex:
    """
    Return the beacon proposer index for the ``slot``.
    """
    epoch = slot_to_epoch(slot)
    current_epoch = get_current_epoch(state)
    previous_epoch = get_previous_epoch(state)
    next_epoch = current_epoch + 1

    assert previous_epoch <= epoch <= next_epoch

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

`verify_merkle_branch`

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

`get_attestation_participants`

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

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

    assert verify_bitfield(bitfield, len(crosslink_committee))

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

`int_to_bytes1`, `int_to_bytes2`, ...

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

`bytes_to_int`

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

`get_effective_balance`

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

`get_total_balance`

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

`get_fork_version`

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

`get_domain`

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

`verify_slashable_attestation`

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

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

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

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

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

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

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

`is_double_vote`

def is_double_vote(attestation_data_1: AttestationData,
                   attestation_data_2: AttestationData) -> bool:
    """
    Check if ``attestation_data_1`` and ``attestation_data_2`` have the same target.
    """
    target_epoch_1 = slot_to_epoch(attestation_data_1.slot)
    target_epoch_2 = slot_to_epoch(attestation_data_2.slot)
    return target_epoch_1 == target_epoch_2

`is_surround_vote`

def is_surround_vote(attestation_data_1: AttestationData,
                     attestation_data_2: AttestationData) -> bool:
    """
    Check if ``attestation_data_1`` surrounds ``attestation_data_2``.
    """
    source_epoch_1 = attestation_data_1.source_epoch
    source_epoch_2 = attestation_data_2.source_epoch
    target_epoch_1 = slot_to_epoch(attestation_data_1.slot)
    target_epoch_2 = slot_to_epoch(attestation_data_2.slot)

    return source_epoch_1 < source_epoch_2 and target_epoch_2 < target_epoch_1

`integer_squareroot`

def integer_squareroot(n: int) -> int:
    """
    The largest integer ``x`` such that ``x**2`` is less than or equal to ``n``.
    """
    assert n >= 0
    x = n
    y = (x + 1) // 2
    while y < x:
        x = y
        y = (x + n // x) // 2
    return x

`get_delayed_activation_exit_epoch`

def get_delayed_activation_exit_epoch(epoch: Epoch) -> Epoch:
    """
    Return the epoch at which an activation or exit triggered in ``epoch`` takes effect.
    """
    return epoch + 1 + ACTIVATION_EXIT_DELAY

`bls_verify`

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

`bls_verify_multiple`

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

`bls_aggregate_pubkeys`

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

`process_deposit`

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

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

    # Should equal 8 bytes for deposit_data.amount +
    #              8 bytes for deposit_data.timestamp +
    #              176 bytes for deposit_data.deposit_input
    # It should match the deposit_data in the eth1.0 deposit contract
    serialized_deposit_data = serialize(deposit.deposit_data)
    # Deposits must be processed in order
    assert deposit.index == state.deposit_index

    # Verify the Merkle branch
    merkle_branch_is_valid = verify_merkle_branch(
        leaf=hash(serialized_deposit_data),
        proof=deposit.proof,
        depth=DEPOSIT_CONTRACT_TREE_DEPTH,
        index=deposit.index,
        root=state.latest_eth1_data.deposit_root,
    )
    assert merkle_branch_is_valid

    # Increment the next deposit index we are expecting. Note that this
    # needs to be done here because while the deposit contract will never
    # create an invalid Merkle branch, it may admit an invalid deposit
    # object, and we need to be able to skip over it
    state.deposit_index += 1

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

    if pubkey not in validator_pubkeys:
        # Verify the proof of possession
        proof_is_valid = bls_verify(
            pubkey=deposit_input.pubkey,
            message_hash=signed_root(deposit_input),
            signature=deposit_input.proof_of_possession,
            domain=get_domain(
                state.fork,
                get_current_epoch(state),
                DOMAIN_DEPOSIT,
            )
        )
        if not proof_is_valid:
            return

        # Add new validator
        validator = Validator(
            pubkey=pubkey,
            withdrawal_credentials=withdrawal_credentials,
            activation_epoch=FAR_FUTURE_EPOCH,
            exit_epoch=FAR_FUTURE_EPOCH,
            withdrawable_epoch=FAR_FUTURE_EPOCH,
            initiated_exit=False,
            slashed=False,
        )

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

Routines for updating validator status

Note: All functions in this section mutate state.

`activate_validator`

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

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

`initiate_validator_exit`

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

`exit_validator`

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

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

`slash_validator`

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

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

`prepare_validator_for_withdrawal`

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

Ethereum 1.0 deposit contract

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

Deposit arguments

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

Withdrawal credentials

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

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

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

`Deposit` logs

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

`Eth2Genesis` log

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

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

Vyper code

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

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

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

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

On genesis

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

Let genesis_validator_deposits be the list of deposits, ordered chronologically, up to and including the deposit that triggered the Eth2Genesis log.
Let genesis_time be the timestamp specified in the Eth2Genesis log.
Let genesis_eth1_data be the Eth1Data object where:
- genesis_eth1_data.deposit_root is the deposit_root contained in the Eth2Genesis log.
- genesis_eth1_data.block_hash is the hash of the Ethereum 1.0 block that emitted the Eth2Genesis log.
Let genesis_state = get_genesis_beacon_state(genesis_validator_deposits, genesis_time, genesis_eth1_data).
Let genesis_block = get_empty_block().
Set genesis_block.state_root = hash_tree_root(genesis_state).

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

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

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

        # Randomness and committees
        latest_randao_mixes=Vector([ZERO_HASH for _ in range(LATEST_RANDAO_MIXES_LENGTH)]),
        latest_start_shard=GENESIS_START_SHARD,

        # Finality
        previous_epoch_attestations=[],
        current_epoch_attestations=[],
        previous_justified_epoch=GENESIS_EPOCH,
        current_justified_epoch=GENESIS_EPOCH,
        previous_justified_root=ZERO_HASH,
        current_justified_root=ZERO_HASH,
        justification_bitfield=0,
        finalized_epoch=GENESIS_EPOCH,
        finalized_root=ZERO_HASH,

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

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

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

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

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

    return state

Beacon chain processing

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

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

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

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

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

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

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

Beacon chain fork choice rule

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

Abstractly define Store as the type of storage object for the chain data and store be the set of attestations and blocks that the validator v has observed and verified (in particular, block ancestors must be recursively verified). Attestations not yet included in any chain are still included in store.
Let finalized_head be the finalized block with the highest epoch. (A block B is finalized if there is a descendant of B in store the processing of which sets B as finalized.)
Let justified_head be the descendant of finalized_head with the highest epoch that has been justified for at least 1 epoch. (A block B is justified if there is a descendant of B in store the processing of which sets B as justified.) If no such descendant exists set justified_head to finalized_head.
Let get_ancestor(store: Store, block: BeaconBlock, slot: Slot) -> BeaconBlock be the ancestor of block with slot number slot. The get_ancestor function can be defined recursively as:

def get_ancestor(store: Store, block: BeaconBlock, slot: Slot) -> BeaconBlock:
    """
    Get the ancestor of ``block`` with slot number ``slot``; return ``None`` if not found.
    """
    if block.slot == slot:
        return block
    elif block.slot < slot:
        return None
    else:
        return get_ancestor(store, store.get_parent(block), slot)

Let get_latest_attestation(store: Store, validator_index: ValidatorIndex) -> Attestation be the attestation with the highest slot number in store from the validator with the given validator_index. If several such attestations exist, use the one the validator v observed first.
Let get_latest_attestation_target(store: Store, validator_index: ValidatorIndex) -> BeaconBlock be the target block in the attestation get_latest_attestation(store, validator_index).
Let get_children(store: Store, block: BeaconBlock) -> List[BeaconBlock] returns the child blocks of the given block.
Let justified_head_state be the resulting BeaconState object from processing the chain up to the justified_head.
The head is lmd_ghost(store, justified_head_state, justified_head) where the function lmd_ghost is defined below. Note that the implementation below is suboptimal; there are implementations that compute the head in time logarithmic in slot count.

def lmd_ghost(store: Store, start_state: BeaconState, start_block: BeaconBlock) -> BeaconBlock:
    """
    Execute the LMD-GHOST algorithm to find the head ``BeaconBlock``.
    """
    validators = start_state.validator_registry
    active_validator_indices = get_active_validator_indices(validators, slot_to_epoch(start_state.slot))
    attestation_targets = [
        (validator_index, get_latest_attestation_target(store, validator_index))
        for validator_index in active_validator_indices
    ]

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

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

Beacon chain state transition function

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

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

Transition section notes:

The state caching, caches the state root of the previous slot.
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 and block roots records updates.
The per-block transitions generally focus on verifying aggregate signatures and saving temporary records relating to the per-block activity in the BeaconState.

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:
    previous_slot_state_root = hash_tree_root(state)

    # store the previous slot's post state transition root
    state.latest_state_roots[state.slot % SLOTS_PER_HISTORICAL_ROOT] = previous_slot_state_root

    # cache state root in stored latest_block_header if empty
    if state.latest_block_header.state_root == ZERO_HASH:
        state.latest_block_header.state_root = previous_slot_state_root

    # store latest known block for previous slot
    state.latest_block_roots[state.slot % SLOTS_PER_HISTORICAL_ROOT] = hash_tree_root(state.latest_block_header)

Per-epoch processing

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

Helper functions

We define some helper functions utilized when processing an epoch transition:

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

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

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

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

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

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

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

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

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

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

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

    # Winning crosslink root is the root with the most votes for it, ties broken in favor of
    # lexicographically higher hash
    winning_root = max(all_roots, key=lambda r: (get_attesting_balance(state, get_attestations_for(r)), r))

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

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

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

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

Justification

Run the following function:

def update_justification_and_finalization(state: BeaconState) -> None:
    new_justified_epoch = state.current_justified_epoch
    new_finalized_epoch = state.finalized_epoch

    # Rotate the justification bitfield up one epoch to make room for the current epoch
    state.justification_bitfield <<= 1
    # If the previous epoch gets justified, fill the second last bit
    previous_boundary_attesting_balance = get_attesting_balance(state, get_previous_epoch_boundary_attestations(state))
    if previous_boundary_attesting_balance * 3 >= get_previous_total_balance(state) * 2:
        new_justified_epoch = get_current_epoch(state) - 1
        state.justification_bitfield |= 2
    # If the current epoch gets justified, fill the last bit
    current_boundary_attesting_balance = get_attesting_balance(state, get_current_epoch_boundary_attestations(state))
    if current_boundary_attesting_balance * 3 >= get_current_total_balance(state) * 2:
        new_justified_epoch = get_current_epoch(state)
        state.justification_bitfield |= 1

    # Process finalizations
    bitfield = state.justification_bitfield
    current_epoch = get_current_epoch(state)
    # The 2nd/3rd/4th most recent epochs are all justified, the 2nd using the 4th as source
    if (bitfield >> 1) % 8 == 0b111 and state.previous_justified_epoch == current_epoch - 3:
        new_finalized_epoch = state.previous_justified_epoch
    # The 2nd/3rd most recent epochs are both justified, the 2nd using the 3rd as source
    if (bitfield >> 1) % 4 == 0b11 and state.previous_justified_epoch == current_epoch - 2:
        new_finalized_epoch = state.previous_justified_epoch
    # The 1st/2nd/3rd most recent epochs are all justified, the 1st using the 3rd as source
    if (bitfield >> 0) % 8 == 0b111 and state.current_justified_epoch == current_epoch - 2:
        new_finalized_epoch = state.current_justified_epoch
    # The 1st/2nd most recent epochs are both justified, the 1st using the 2nd as source
    if (bitfield >> 0) % 4 == 0b11 and state.current_justified_epoch == current_epoch - 1:
        new_finalized_epoch = state.current_justified_epoch

    # Update state jusification/finality fields
    state.previous_justified_epoch = state.current_justified_epoch
    state.previous_justified_root = state.current_justified_root
    if new_justified_epoch != state.current_justified_epoch:
        state.current_justified_epoch = new_justified_epoch
        state.current_justified_root = get_block_root(state, get_epoch_start_slot(new_justified_epoch))
    if new_finalized_epoch != state.finalized_epoch:
        state.finalized_epoch = new_finalized_epoch
        state.finalized_root = get_block_root(state, get_epoch_start_slot(new_finalized_epoch))

Crosslinks

Run the following function:

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

Eth1 data

Run the following function:

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

Rewards and penalties

First, we define some additional helpers:

def get_base_reward(state: BeaconState, index: ValidatorIndex) -> Gwei:
    if get_previous_total_balance(state) == 0:
        return 0

    adjusted_quotient = integer_squareroot(get_previous_total_balance(state)) // BASE_REWARD_QUOTIENT
    return get_effective_balance(state, index) // adjusted_quotient // 5

def get_inactivity_penalty(state: BeaconState, index: ValidatorIndex, epochs_since_finality: int) -> Gwei:
    return (
        get_base_reward(state, index) +
        get_effective_balance(state, index) * epochs_since_finality // INACTIVITY_PENALTY_QUOTIENT // 2
    )

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

Justification and finalization

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

When blocks are finalizing normally...

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

When blocks are not finalizing normally...

def compute_inactivity_leak_deltas(state: BeaconState) -> Tuple[List[Gwei], List[Gwei]]:
    # deltas[0] for rewards
    # deltas[1] for penalties
    deltas = [
        [0 for index in range(len(state.validator_registry))],
        [0 for index in range(len(state.validator_registry))]
    ]
    boundary_attestations = get_previous_epoch_boundary_attestations(state)
    matching_head_attestations = get_previous_epoch_matching_head_attestations(state)
    active_validator_indices = get_active_validator_indices(state.validator_registry, get_previous_epoch(state))
    epochs_since_finality = get_current_epoch(state) + 1 - state.finalized_epoch
    for index in active_validator_indices:
        if index not in get_attesting_indices(state, state.previous_epoch_attestations):
            deltas[1][index] += get_inactivity_penalty(state, index, epochs_since_finality)
        else:
            # If a validator did attest, apply a small penalty for getting attestations included late
            deltas[0][index] += (
                get_base_reward(state, index) * MIN_ATTESTATION_INCLUSION_DELAY //
                inclusion_distance(state, index)
            )
            deltas[1][index] += get_base_reward(state, index)
        if index not in get_attesting_indices(state, boundary_attestations):
            deltas[1][index] += get_inactivity_penalty(state, index, epochs_since_finality)
        if index not in get_attesting_indices(state, matching_head_attestations):
            deltas[1][index] += get_base_reward(state, index)
    # Penalize slashed-but-inactive validators as though they were active but offline
    for index in range(len(state.validator_registry)):
        eligible = (
            index not in active_validator_indices and
            state.validator_registry[index].slashed and
            get_current_epoch(state) < state.validator_registry[index].withdrawable_epoch
        )
        if eligible:
            deltas[1][index] += (
                2 * get_inactivity_penalty(state, index, epochs_since_finality) +
                get_base_reward(state, index)
            )
    return deltas

Crosslinks

def get_crosslink_deltas(state: BeaconState) -> Tuple[List[Gwei], List[Gwei]]:
    # deltas[0] for rewards
    # deltas[1] for penalties
    deltas = [
        [0 for index in range(len(state.validator_registry))],
        [0 for index in range(len(state.validator_registry))]
    ]
    previous_epoch_start_slot = get_epoch_start_slot(get_previous_epoch(state))
    current_epoch_start_slot = get_epoch_start_slot(get_current_epoch(state))
    for slot in range(previous_epoch_start_slot, current_epoch_start_slot):
        for crosslink_committee, shard in get_crosslink_committees_at_slot(state, slot):
            winning_root, participants = get_winning_root_and_participants(state, shard)
            participating_balance = get_total_balance(state, participants)
            total_balance = get_total_balance(state, crosslink_committee)
            for index in crosslink_committee:
                if index in participants:
                    deltas[0][index] += get_base_reward(state, index) * participating_balance // total_balance
                else:
                    deltas[1][index] += get_base_reward(state, index)
    return deltas

Apply rewards

Run the following:

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

Ejections

Run process_ejections(state).

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

Validator registry and shuffling seed data

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

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

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

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

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

            # Exit validator
            exit_validator(state, index)

    state.validator_registry_update_epoch = current_epoch

Run the following function:

def update_registry(state: BeaconState) -> None:
    # Check if we should update, and if so, update
    if state.finalized_epoch > state.validator_registry_update_epoch:
        update_validator_registry(state)
    state.latest_start_shard = (
        state.latest_start_shard +
        get_current_epoch_committee_count(state)
    ) % SHARD_COUNT

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

Slashings and exit queue

Run process_slashings(state) and process_exit_queue(state):

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

    # Compute `total_penalties`
    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(
                get_effective_balance(state, index) * min(total_penalties * 3, total_balance) // total_balance,
                get_effective_balance(state, index) // MIN_PENALTY_QUOTIENT
            )
            state.validator_balances[index] -= penalty

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

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

Final updates

Run the following function:

def finish_epoch_update(state: BeaconState) -> None:
    current_epoch = get_current_epoch(state)
    next_epoch = current_epoch + 1
    # Set active index root
    index_root_position = (next_epoch + ACTIVATION_EXIT_DELAY) % LATEST_ACTIVE_INDEX_ROOTS_LENGTH
    state.latest_active_index_roots[index_root_position] = hash_tree_root(
        get_active_validator_indices(state.validator_registry, next_epoch + ACTIVATION_EXIT_DELAY)
    )
    # Set total slashed balances
    state.latest_slashed_balances[next_epoch % LATEST_SLASHED_EXIT_LENGTH] = (
        state.latest_slashed_balances[current_epoch % LATEST_SLASHED_EXIT_LENGTH]
    )
    # Set randao mix
    state.latest_randao_mixes[next_epoch % LATEST_RANDAO_MIXES_LENGTH] = get_randao_mix(state, current_epoch)
    # Set historical root accumulator
    if next_epoch % (SLOTS_PER_HISTORICAL_ROOT // SLOTS_PER_EPOCH) == 0:
        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 == hash_tree_root(state.latest_block_header)
    # Save current block as the new latest block
    state.latest_block_header = get_temporary_block_header(block)
    # Verify proposer signature
    proposer = state.validator_registry[get_beacon_proposer_index(state, state.slot)]
    assert bls_verify(
        pubkey=proposer.pubkey,
        message_hash=signed_root(block),
        signature=block.signature,
        domain=get_domain(state.fork, get_current_epoch(state), DOMAIN_BEACON_BLOCK)
    )

RANDAO

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

Eth1 data

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

Transactions

Proposer slashings

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

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

def process_proposer_slashing(state: BeaconState,
                              proposer_slashing: ProposerSlashing) -> None:
    """
    Process ``ProposerSlashing`` transaction.
    Note that this function mutates ``state``.
    """
    proposer = state.validator_registry[proposer_slashing.proposer_index]
    # Verify that the 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
    # Proposer is not yet slashed
    assert proposer.slashed is False
    # Signatures are valid
    for header in (proposer_slashing.header_1, proposer_slashing.header_2):
        assert bls_verify(
            pubkey=proposer.pubkey,
            message_hash=signed_root(header),
            signature=header.signature,
            domain=get_domain(state.fork, slot_to_epoch(header.slot), DOMAIN_BEACON_BLOCK)
        )
    slash_validator(state, proposer_slashing.proposer_index)

Attester slashings

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

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

def process_attester_slashing(state: BeaconState,
                              attester_slashing: AttesterSlashing) -> None:
    """
    Process ``AttesterSlashing`` transaction.
    Note that this function mutates ``state``.
    """
    attestation1 = attester_slashing.slashable_attestation_1
    attestation2 = attester_slashing.slashable_attestation_2
    # Check that the attestations are conflicting
    assert attestation1.data != attestation2.data
    assert (
        is_double_vote(attestation1.data, attestation2.data) or
        is_surround_vote(attestation1.data, attestation2.data)
    )
    assert verify_slashable_attestation(state, attestation1)
    assert verify_slashable_attestation(state, attestation2)
    slashable_indices = [
        index for index in attestation1.validator_indices
        if (
            index in attestation2.validator_indices and
            state.validator_registry[index].slashed is False
        )
    ]
    assert len(slashable_indices) >= 1
    for index in slashable_indices:
        slash_validator(state, index)

Attestations

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

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

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

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

Deposits

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

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

Voluntary exits

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

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

def process_voluntary_exit(state: BeaconState, exit: VoluntaryExit) -> None:
    """
    Process ``VoluntaryExit`` transaction.
    Note that this function mutates ``state``.
    """
    validator = state.validator_registry[exit.validator_index]
    # Verify the validator has not yet exited
    assert validator.exit_epoch == FAR_FUTURE_EPOCH
    # Verify the validator has not initiated an exit
    assert validator.initiated_exit is False
    # Exits must specify an epoch when they become valid; they are not valid before then
    assert get_current_epoch(state) >= exit.epoch
    # Must have been in the validator set long enough
    assert get_current_epoch(state) - validator.activation_epoch >= PERSISTENT_COMMITTEE_PERIOD
    # Verify signature
    assert bls_verify(
        pubkey=validator.pubkey,
        message_hash=signed_root(exit),
        signature=exit.signature,
        domain=get_domain(state.fork, exit.epoch, DOMAIN_VOLUNTARY_EXIT)
    )
    # Run the exit
    initiate_validator_exit(state, exit.validator_index)

Transfers

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

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

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

def process_transfer(state: BeaconState, transfer: Transfer) -> None:
    """
    Process ``Transfer`` transaction.
    Note that this function mutates ``state``.
    """
    # Verify the amount and fee aren't individually too big (for anti-overflow purposes)
    assert state.validator_balances[transfer.sender] >= max(transfer.amount, transfer.fee)
    # Verify that we have enough ETH to send, and that after the transfer the balance will be either
    # exactly zero or at least MIN_DEPOSIT_AMOUNT
    assert (
        state.validator_balances[transfer.sender] == transfer.amount + transfer.fee or
        state.validator_balances[transfer.sender] >= transfer.amount + transfer.fee + MIN_DEPOSIT_AMOUNT
    )
    # A transfer is valid in only one slot
    assert state.slot == transfer.slot
    # Only withdrawn or not-yet-deposited accounts can transfer
    assert (
        get_current_epoch(state) >= state.validator_registry[transfer.sender].withdrawable_epoch or
        state.validator_registry[transfer.sender].activation_epoch == FAR_FUTURE_EPOCH
    )
    # Verify that the pubkey is valid
    assert (
        state.validator_registry[transfer.sender].withdrawal_credentials ==
        BLS_WITHDRAWAL_PREFIX_BYTE + hash(transfer.pubkey)[1:]
    )
    # Verify that the signature is valid
    assert bls_verify(
        pubkey=transfer.pubkey,
        message_hash=signed_root(transfer),
        signature=transfer.signature,
        domain=get_domain(state.fork, slot_to_epoch(transfer.slot), DOMAIN_TRANSFER)
    )
    # Process the transfer
    state.validator_balances[transfer.sender] -= transfer.amount + transfer.fee
    state.validator_balances[transfer.recipient] += transfer.amount
    state.validator_balances[get_beacon_proposer_index(state, state.slot)] += transfer.fee

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)

References

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

Normative

Informative

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

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

Copyright

Copyright and related rights waived via CC0.

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Ethereum 2.0 Phase 0 -- The Beacon Chain

Table of contents

Introduction

Notation

Terminology

Constants

Misc

Deposit contract

Gwei values

Initial values

Time parameters

State list lengths

Reward and penalty quotients

Max transactions per block

Signature domains

Data structures

Misc dependencies

Fork

Crosslink

Eth1Data

Eth1DataVote

AttestationData

AttestationDataAndCustodyBit

SlashableAttestation

DepositInput

DepositData

BeaconBlockHeader

Validator

PendingAttestation

HistoricalBatch

Beacon transactions

ProposerSlashing

AttesterSlashing

Attestation

Deposit

VoluntaryExit

Transfer

Beacon blocks

BeaconBlockBody

BeaconBlock

Beacon state

BeaconState

Custom Types

Helper functions

xor

hash

hash_tree_root

signed_root

get_temporary_block_header

slot_to_epoch

get_previous_epoch

get_current_epoch

get_epoch_start_slot

is_active_validator

get_active_validator_indices

get_permuted_index

split

get_epoch_committee_count

get_shuffling

get_current_epoch_committee_count

get_crosslink_committees_at_slot

get_block_root

get_state_root

get_randao_mix

get_active_index_root

generate_seed

get_beacon_proposer_index

verify_merkle_branch

get_attestation_participants

int_to_bytes1, int_to_bytes2, ...

bytes_to_int

get_effective_balance

get_total_balance

get_fork_version

`Fork`

`Crosslink`

`Eth1Data`

`Eth1DataVote`

`AttestationData`

`AttestationDataAndCustodyBit`

`SlashableAttestation`

`DepositInput`

`DepositData`

`BeaconBlockHeader`

`Validator`

`PendingAttestation`

`HistoricalBatch`

`ProposerSlashing`

`AttesterSlashing`

`Attestation`

`Deposit`

`VoluntaryExit`

`Transfer`

`BeaconBlockBody`

`BeaconBlock`

`BeaconState`

`xor`

`hash`

`hash_tree_root`

`signed_root`

`get_temporary_block_header`

`slot_to_epoch`

`get_previous_epoch`

`get_current_epoch`

`get_epoch_start_slot`

`is_active_validator`

`get_active_validator_indices`

`get_permuted_index`

`split`

`get_epoch_committee_count`

`get_shuffling`

`get_current_epoch_committee_count`

`get_crosslink_committees_at_slot`

`get_block_root`

`get_state_root`

`get_randao_mix`

`get_active_index_root`

`generate_seed`

`get_beacon_proposer_index`

`verify_merkle_branch`

`get_attestation_participants`

`int_to_bytes1`, `int_to_bytes2`, ...

`bytes_to_int`

`get_effective_balance`

`get_total_balance`

`get_fork_version`

`get_domain`

`get_bitfield_bit`

`verify_bitfield`

`verify_slashable_attestation`

`is_double_vote`

`is_surround_vote`

`integer_squareroot`

`get_delayed_activation_exit_epoch`

`bls_verify`

`bls_verify_multiple`

`bls_aggregate_pubkeys`

`process_deposit`

`activate_validator`

`initiate_validator_exit`

`exit_validator`

`slash_validator`

`prepare_validator_for_withdrawal`

`Deposit` logs

`Eth2Genesis` log