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Ethereum 2.0 Phase 0 -- The Beacon Chain

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

Table of contents

Introduction

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

At the core of Ethereum 2.0 is a system chain called the "beacon chain". The beacon chain stores and manages the registry of validators. In the initial deployment phases of Ethereum 2.0 the only mechanism to become a validator is to make a one-way ETH transaction to a deposit contract on Ethereum 1.0. Activation as a validator happens when deposit transaction 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 chain 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

Unless otherwise indicated, code appearing in this style is to be interpreted as an algorithm defined in Python. Implementations may implement such algorithms using any code and programming language desired as long as the behavior is identical to that of the algorithm provided.

Terminology

  • Validator - a participant in the Casper/sharding consensus system. You can become one by depositing 32 ETH into the Casper mechanism.
  • Active validator - a validator currently participating in the protocol which the Casper mechanism looks to produce and attest to blocks, crosslinks and other consensus objects.
  • 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 of SLOT_DURATION seconds, 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 Unit
SHARD_COUNT 2**10 (= 1,024) shards
TARGET_COMMITTEE_SIZE 2**7 (= 128) validators
EJECTION_BALANCE 2**4 (= 16) ETH
MAX_BALANCE_CHURN_QUOTIENT 2**5 (= 32) -
GWEI_PER_ETH 10**9 Gwei/ETH
BEACON_CHAIN_SHARD_NUMBER 2**64 - 1 -
BLS_WITHDRAWAL_PREFIX_BYTE 0x00 -
MAX_CASPER_VOTES 2**10 (= 1,024) votes
LATEST_BLOCK_ROOTS_LENGTH 2**13 (= 8,192) block roots
LATEST_RANDAO_MIXES_LENGTH 2**13 (= 8,192) randao mixes
EMPTY_SIGNATURE [bytes48(0), bytes48(0)] -
  • For the safety of crosslinks TARGET_COMMITTEE_SIZE exceeds the recommended minimum committee size of 111; with sufficient active validators (at least EPOCH_LENGTH * TARGET_COMMITTEE_SIZE), the shuffling algorithm ensures committee sizes at least TARGET_COMMITTEE_SIZE. (Unbiasable randomness with a Verifiable Delay Function (VDF) will improve committee robustness and lower the safe minimum committee size.)

Deposit contract

Name Value Unit
DEPOSIT_CONTRACT_ADDRESS TBD
DEPOSIT_CONTRACT_TREE_DEPTH 2**5 (= 32) -
MIN_DEPOSIT 2**0 (= 1) ETH
MAX_DEPOSIT 2**5 (= 32) ETH

Initial values

Name Value
INITIAL_FORK_VERSION 0
INITIAL_SLOT_NUMBER 0
ZERO_HASH bytes([0] * 32)

Time parameters

Name Value Unit Duration
SLOT_DURATION 6 seconds 6 seconds
MIN_ATTESTATION_INCLUSION_DELAY 2**2 (= 4) slots 24 seconds
EPOCH_LENGTH 2**6 (= 64) slots 6.4 minutes
MIN_VALIDATOR_REGISTRY_CHANGE_INTERVAL 2**8 (= 256) slots 25.6 minutes
POW_RECEIPT_ROOT_VOTING_PERIOD 2**10 (= 1,024) slots ~1.7 hours
SHARD_PERSISTENT_COMMITTEE_CHANGE_PERIOD 2**17 (= 131,072) slots ~9 days
COLLECTIVE_PENALTY_CALCULATION_PERIOD 2**20 (= 1,048,576) slots ~73 days
ZERO_BALANCE_VALIDATOR_TTL 2**22 (= 4,194,304) slots ~291 days

Reward and penalty quotients

Name Value
BASE_REWARD_QUOTIENT 2**10 (= 1,024)
WHISTLEBLOWER_REWARD_QUOTIENT 2**9 (= 512)
INCLUDER_REWARD_QUOTIENT 2**3 (= 8)
INACTIVITY_PENALTY_QUOTIENT 2**34 (= 17,179,869,184)
  • 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**17 slots (~9 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 slots is about (1-1/INACTIVITY_PENALTY_QUOTIENT)**(n**2/2) so after INVERSE_SQRT_E_DROP_TIME slots it is roughly (1-1/INACTIVITY_PENALTY_QUOTIENT)**(INACTIVITY_PENALTY_QUOTIENT/2) ~= 1/sqrt(e).

Status codes

Name Value
PENDING_ACTIVATION 0
ACTIVE 1
ACTIVE_PENDING_EXIT 2
EXITED_WITHOUT_PENALTY 3
EXITED_WITH_PENALTY 4

Max operations per block

Name Value
MAX_PROPOSER_SLASHINGS 2**4 (= 16)
MAX_CASPER_SLASHINGS 2**4 (= 16)
MAX_ATTESTATIONS 2**7 (= 128)
MAX_DEPOSITS 2**4 (= 16)
MAX_EXITS 2**4 (= 16)

Validator registry delta flags

Name Value
ACTIVATION 0
EXIT 1

Signature domains

Name Value
DOMAIN_DEPOSIT 0
DOMAIN_ATTESTATION 1
DOMAIN_PROPOSAL 2
DOMAIN_EXIT 3

Data structures

Beacon chain operations

Proposer slashings

ProposerSlashing
{
    # Proposer index
    'proposer_index': 'uint24',
    # First proposal data
    'proposal_data_1': ProposalSignedData,
    # First proposal signature
    'proposal_signature_1': '[uint384]',
    # Second proposal data
    'proposal_data_2': ProposalSignedData,
    # Second proposal signature
    'proposal_signature_2': '[uint384]',
}

Casper slashings

CasperSlashing
{
    # First batch of votes
    'slashable_vote_data_1': SlashableVoteData,
    # Second batch of votes
    'slashable_vote_data_2': SlashableVoteData,
}
SlashableVoteData
{
    # Proof-of-custody indices (0 bits)
    'aggregate_signature_poc_0_indices': '[uint24]',
    # Proof-of-custody indices (1 bits)
    'aggregate_signature_poc_1_indices': '[uint24]',
    # Attestation data
    'data': AttestationData,
    # Aggregate signature
    'aggregate_signature': '[uint384]',
}

Attestations

Attestation
{
    # Attestation data
    'data': AttestationData,
    # Attester participation bitfield
    'participation_bitfield': 'bytes',
    # Proof of custody bitfield
    'custody_bitfield': 'bytes',
    # BLS aggregate signature
    'aggregate_signature': ['uint384'],
}
AttestationData
{
    # Slot number
    'slot': 'uint64',
    # Shard number
    'shard': 'uint64',
    # Hash of root of the signed beacon block
    'beacon_block_root': 'hash32',
    # Hash of root of the ancestor at the epoch boundary
    'epoch_boundary_root': 'hash32',
    # Shard block's hash of root
    'shard_block_root': 'hash32',
    # Last crosslink's hash of root
    'latest_crosslink_root': 'hash32',
    # Slot of the last justified beacon block
    'justified_slot': 'uint64',
    # Hash of the last justified beacon block
    'justified_block_root': 'hash32',
}

Deposits

Deposit
{
    # Receipt Merkle branch
    'merkle_branch': '[hash32]',
    # Merkle tree index
    'merkle_tree_index': 'uint64',
    # Deposit data
    'deposit_data': DepositData,
}
DepositData
{
    # Deposit parameters
    'deposit_input': DepositInput,
    # Value in Gwei
    'value': 'uint64',
    # Timestamp from deposit contract
    'timestamp': 'uint64',
}
DepositInput
{
    # BLS pubkey
    'pubkey': 'uint384',
    # Withdrawal credentials
    'withdrawal_credentials': 'hash32',
    # Initial RANDAO commitment
    'randao_commitment': 'hash32',
    # a BLS signature of this ``DepositInput``
    'proof_of_possession': ['uint384'],
}

Exits

Exit
{
    # Minimum slot for processing exit
    'slot': 'uint64',
    # Index of the exiting validator
    'validator_index': 'uint24',
    # Validator signature
    'signature': '[uint384]',
}

Beacon chain blocks

BeaconBlock

{
    ## Header ##
    'slot': 'uint64',
    'parent_root': 'hash32',
    'state_root': 'hash32',
    'randao_reveal': 'hash32',
    'candidate_pow_receipt_root': 'hash32',
    'signature': ['uint384'],

    ## Body ##
    'body': BeaconBlockBody,
}

BeaconBlockBody

{
    'proposer_slashings': [ProposerSlashing],
    'casper_slashings': [CasperSlashing],
    'attestations': [Attestation],
    'deposits': [Deposit],
    'exits': [Exit],
}

ProposalSignedData

{
    # Slot number
    'slot': 'uint64',
    # Shard number (`BEACON_CHAIN_SHARD_NUMBER` for beacon chain)
    'shard': 'uint64',
    # Block's hash of root
    'block_root': 'hash32',
}

Beacon chain state

BeaconState

{
    # Misc
    'slot': 'uint64',
    'genesis_time': 'uint64',
    'fork_data': ForkData,  # For versioning hard forks

    # Validator registry
    'validator_registry': [ValidatorRecord],
    'validator_balances': ['uint64'],
    'validator_registry_latest_change_slot': 'uint64',
    'validator_registry_exit_count': 'uint64',
    'validator_registry_delta_chain_tip': 'hash32',  # For light clients to track deltas

    # Randomness and committees
    'latest_randao_mixes': ['hash32'],
    'shard_committees_at_slots': [[ShardCommittee]],
    'persistent_committees': [['uint24']],
    'persistent_committee_reassignments': [ShardReassignmentRecord],

    # Finality
    'previous_justified_slot': 'uint64',
    'justified_slot': 'uint64',
    'justification_bitfield': 'uint64',
    'finalized_slot': 'uint64',

    # Recent state
    'latest_crosslinks': [CrosslinkRecord],
    'latest_block_roots': ['hash32'],  # Needed to process attestations, older to newer
    'latest_penalized_exit_balances': ['uint64'],  # Balances penalized at every withdrawal period
    'latest_attestations': [PendingAttestationRecord],
    'batched_block_roots': ['hash32'],

    # PoW receipt root
    'processed_pow_receipt_root': 'hash32',
    'candidate_pow_receipt_roots': [CandidatePoWReceiptRootRecord],
}

ValidatorRecord

{
    # BLS public key
    'pubkey': 'uint384',
    # Withdrawal credentials
    'withdrawal_credentials': 'hash32',
    # RANDAO commitment
    'randao_commitment': 'hash32',
    # Slots the proposer has skipped (i.e. layers of RANDAO expected)
    'randao_layers': 'uint64',
    # Status code
    'status': 'uint64',
    # Slot when validator last changed status (or 0)
    'latest_status_change_slot': 'uint64',
    # Exit counter when validator exited (or 0)
    'exit_count': 'uint64',
}

CrosslinkRecord

{
    # Slot number
    'slot': 'uint64',
    # Shard block root
    'shard_block_root': 'hash32',
}

ShardCommittee

{
    # Shard number
    'shard': 'uint64',
    # Validator indices
    'committee': ['uint24'],
    # Total validator count (for proofs of custody)
    'total_validator_count': 'uint64',
}

ShardReassignmentRecord

{
    # Which validator to reassign
    'validator_index': 'uint24',
    # To which shard
    'shard': 'uint64',
    # When
    'slot': 'uint64',
}

CandidatePoWReceiptRootRecord

{
    # Candidate PoW receipt root
    'candidate_pow_receipt_root': 'hash32',
    # Vote count
    'vote_count': 'uint64',
}

PendingAttestationRecord

{
    # Signed data
    'data': AttestationData,
    # Attester participation bitfield
    'participation_bitfield': 'bytes',
    # Proof of custody bitfield
    'custody_bitfield': 'bytes',
    # Slot in which it was included
    'slot_included': 'uint64',
}

ForkData

{
    # Previous fork version
    'pre_fork_version': 'uint64',
    # Post fork version
    'post_fork_version': 'uint64',
    # Fork slot number
    'fork_slot': 'uint64',
}

ValidatorRegistryDeltaBlock

{
    latest_registry_delta_root: 'hash32',
    validator_index: 'uint24',
    pubkey: 'uint384',
    flag: 'uint64',
}

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 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. One of the DepositInput fields is withdrawal_credentials which must satisfy:

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

We recommend the private key corresponding to withdrawal_pubkey be stored in cold storage until a withdrawal is required.

Eth1Deposit logs

Every deposit, of size between MIN_DEPOSIT and MAX_DEPOSIT, emits an Eth1Deposit log for consumption by the beacon chain. The deposit contract does little validation, pushing most of the validator onboarding logic to the beacon chain. In particular, the proof of possession (a BLS12-381 signature) is not verified by the deposit contract.

ChainStart log

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

  • genesis_time equals time in the ChainStart log
  • processed_pow_receipt_root equals receipt_root in the ChainStart log
  • initial_validator_deposits is a list of Deposit objects built according to the Eth1Deposit logs up to the deposit that triggered the ChainStart log, processed in the order in which they were emitted (oldest to newest)

Vyper code

## compiled with v0.1.0-beta.4 ##

MIN_DEPOSIT: constant(uint256) = 1  # ETH
MAX_DEPOSIT: constant(uint256) = 32  # ETH
GWEI_PER_ETH: constant(uint256) = 1000000000  # 10**9
CHAIN_START_FULL_DEPOSIT_THRESHOLD: constant(uint256) = 16384  # 2**14
DEPOSIT_CONTRACT_TREE_DEPTH: constant(uint256) = 32
SECONDS_PER_DAY: constant(uint256) = 86400

Eth1Deposit: event({previous_receipt_root: bytes32, data: bytes[2064], deposit_count: uint256})
ChainStart: event({receipt_root: bytes32, time: bytes[8]})

receipt_tree: bytes32[uint256]
deposit_count: uint256
full_deposit_count: uint256

@payable
@public
def deposit(deposit_input: bytes[2048]):
    assert msg.value >= as_wei_value(MIN_DEPOSIT, "ether")
    assert msg.value <= as_wei_value(MAX_DEPOSIT, "ether")

    index: uint256 = self.deposit_count + 2**DEPOSIT_CONTRACT_TREE_DEPTH
    msg_gwei_bytes8: bytes[8] = slice(concat("", convert(msg.value / GWEI_PER_ETH, bytes32)), start=24, len=8)
    timestamp_bytes8: bytes[8] = slice(concat("", convert(block.timestamp, bytes32)), start=24, len=8)
    deposit_data: bytes[2064] = concat(msg_gwei_bytes8, timestamp_bytes8, deposit_input)

    log.Eth1Deposit(self.receipt_tree[1], deposit_data, self.deposit_count)

    # add deposit to merkle tree
    self.receipt_tree[index] = sha3(deposit_data)
    for i in range(32):  # DEPOSIT_CONTRACT_TREE_DEPTH (range of constant var not yet supported)
        index /= 2
        self.receipt_tree[index] = sha3(concat(self.receipt_tree[index * 2], self.receipt_tree[index * 2 + 1]))

    self.deposit_count += 1
    if msg.value == as_wei_value(MAX_DEPOSIT, "ether"):
        self.full_deposit_count += 1
        if self.full_deposit_count == CHAIN_START_FULL_DEPOSIT_THRESHOLD:
            timestamp_day_boundary: uint256 = as_unitless_number(block.timestamp) - as_unitless_number(block.timestamp) % SECONDS_PER_DAY + SECONDS_PER_DAY
            timestamp_day_boundary_bytes8: bytes[8] = slice(concat("", convert(timestamp_day_boundary, bytes32)), start=24, len=8)
            log.ChainStart(self.receipt_tree[1], timestamp_day_boundary_bytes8)

@public
@constant
def get_receipt_root() -> bytes32:
    return self.receipt_tree[1]

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-slot 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.parent_root has been processed and accepted.
  • The node has processed its state up to slot, block.slot - 1.
  • The Ethereum 1.0 block pointed to by the state.processed_pow_receipt_root has been processed and accepted.
  • The node's local clock time is greater than or equal to state.genesis_time + block.slot * SLOT_DURATION.

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 SLOT_DURATION 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.

  • Let 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 part of any chain are still included in store.
  • Let finalized_head be the finalized block with the highest slot number. (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 slot number that has been justified for at least EPOCH_LENGTH slots. (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, block, slot) be the ancestor of block with slot number slot. The get_ancestor function can be defined recursively as def get_ancestor(store, block, slot): return block if block.slot == slot else get_ancestor(store, store.get_parent(block), slot).
  • Let get_latest_attestation(store, validator) be the attestation with the highest slot number in store from validator. If several such attestations exist, use the one the validator v observed first.
  • Let get_latest_attestation_target(store, validator) be the target block in the attestation get_latest_attestation(store, validator).
  • The head is lmd_ghost(store, 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, start):
    validators = start.state.validator_registry
    active_validators = [validators[i] for i in
                         get_active_validator_indices(validators)]
    attestation_targets = [get_latest_attestation_target(store, validator)
                           for validator in active_validators]
    def get_vote_count(block):
        return len([target for target in attestation_targets if
                    get_ancestor(store, target, block.slot) == block])

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

Beacon chain state transition function

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

  1. The per-slot transitions, which happens every slot, and only affects a parts of the state.
  2. The per-epoch transitions, which happens at every epoch boundary (i.e. state.slot % EPOCH_LENGTH == 0), and affects the entire state.

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

Helper functions

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

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

hash_tree_root is a function for hashing objects into a single root utilizing a hash tree structure. hash_tree_root is defined in the SimpleSerialize spec.

is_active_validator

def is_active_validator(validator: ValidatorRecord) -> bool:
    """
    Checks if ``validator`` is active.
    """
    return validator.status in [ACTIVE, ACTIVE_PENDING_EXIT]

get_active_validator_indices

def get_active_validator_indices(validators: [ValidatorRecord]) -> List[int]:
    """
    Gets indices of active validators from ``validators``.
    """
    return [i for i, v in enumerate(validators) if is_active_validator(v)]

shuffle

def shuffle(values: List[Any], seed: Hash32) -> List[Any]:
    """
    Returns the shuffled ``values`` with ``seed`` as entropy.
    """
    values_count = len(values)

    # Entropy is consumed from the seed in 3-byte (24 bit) chunks.
    rand_bytes = 3
    # The highest possible result of the RNG.
    rand_max = 2 ** (rand_bytes * 8) - 1

    # The range of the RNG places an upper-bound on the size of the list that
    # may be shuffled. It is a logic error to supply an oversized list.
    assert values_count < rand_max

    output = [x for x in values]
    source = seed
    index = 0
    while index < values_count - 1:
        # Re-hash the `source` to obtain a new pattern of bytes.
        source = hash(source)
        # Iterate through the `source` bytes in 3-byte chunks.
        for position in range(0, 32 - (32 % rand_bytes), rand_bytes):
            # Determine the number of indices remaining in `values` and exit
            # once the last index is reached.
            remaining = values_count - index
            if remaining == 1:
                break

            # Read 3-bytes of `source` as a 24-bit big-endian integer.
            sample_from_source = int.from_bytes(source[position:position + rand_bytes], 'big')

            # Sample values greater than or equal to `sample_max` will cause
            # modulo bias when mapped into the `remaining` range.
            sample_max = rand_max - rand_max % remaining

            # Perform a swap if the consumed entropy will not cause modulo bias.
            if sample_from_source < sample_max:
                # Select a replacement index for the current index.
                replacement_position = (sample_from_source % remaining) + index
                # Swap the current index with the replacement index.
                output[index], output[replacement_position] = output[replacement_position], output[index]
                index += 1
            else:
                # The sample causes modulo bias. A new sample should be read.
                pass

    return output

split

def split(values: List[Any], split_count: int) -> 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_new_shuffling

def get_new_shuffling(seed: Hash32,
                      validators: List[ValidatorRecord],
                      crosslinking_start_shard: int) -> List[List[ShardCommittee]]:
    """
    Shuffles ``validators`` into shard committees using ``seed`` as entropy.
    """
    active_validator_indices = get_active_validator_indices(validators)

    committees_per_slot = max(
        1,
        min(
            SHARD_COUNT // EPOCH_LENGTH,
            len(active_validator_indices) // EPOCH_LENGTH // TARGET_COMMITTEE_SIZE,   
        )
    )

    # Shuffle with seed
    shuffled_active_validator_indices = shuffle(active_validator_indices, seed)

    # Split the shuffled list into epoch_length pieces
    validators_per_slot = split(shuffled_active_validator_indices, EPOCH_LENGTH)

    output = []
    for slot, slot_indices in enumerate(validators_per_slot):
        # Split the shuffled list into committees_per_slot pieces
        shard_indices = split(slot_indices, committees_per_slot)

        shard_id_start = crosslinking_start_shard + slot * committees_per_slot

        shard_committees = [
            ShardCommittee(
                shard=(shard_id_start + shard_position) % SHARD_COUNT,
                committee=indices,
                total_validator_count=len(active_validator_indices),
            )
            for shard_position, indices in enumerate(shard_indices)
        ]
        output.append(shards_and_committees_for_slot)

    return output

Here's a diagram of what is going on:

get_shard_committees_at_slot

def get_shard_committees_at_slot(state: BeaconState,
                                 slot: int) -> List[ShardCommittee]:
    """
    Returns the ``ShardCommittee`` for the ``slot``.
    """
    earliest_slot_in_array = state.slot - (state.slot % EPOCH_LENGTH) - EPOCH_LENGTH
    assert earliest_slot_in_array <= slot < earliest_slot_in_array + EPOCH_LENGTH * 2
    return state.shard_committees_at_slots[slot - earliest_slot_in_array]

get_block_root

def get_block_root(state: BeaconState,
                   slot: int) -> Hash32:
    """
    Returns the block root at a recent ``slot``.
    """
    assert state.slot <= slot + LATEST_BLOCK_ROOTS_LENGTH
    assert slot < state.slot
    return state.latest_block_roots[slot % LATEST_BLOCK_ROOTS_LENGTH]

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

get_beacon_proposer_index

def get_beacon_proposer_index(state: BeaconState,
                              slot: int) -> int:
    """
    Returns the beacon proposer index for the ``slot``.
    """
    first_committee = get_shard_committees_at_slot(state, slot)[0].committee
    return first_committee[slot % len(first_committee)]

merkle_root

def merkle_root(values):
    o = [0] * len(values) + values
    for i in range(len(values)-1, 0, -1):
        o[i] = hash(o[i*2] + o[i*2+1])
    return o[1]

get_attestation_participants

def get_attestation_participants(state: BeaconState,
                                 attestation_data: AttestationData,
                                 participation_bitfield: bytes) -> List[int]:
    """
    Returns the participant indices at for the ``attestation_data`` and ``participation_bitfield``.
    """

    # Find the relevant committee
    shard_committees = get_shard_committees_at_slot(state, attestation_data.slot)
    shard_committee = [x for x in shard_committees if x.shard == attestation_data.shard][0]
    assert len(participation_bitfield) == ceil_div8(len(shard_committee.committee))

    # Find the participating attesters in the committee
    participants = []
    for i, validator_index in enumerate(shard_committee.committee):
        participation_bit = (participation_bitfield[i//8] >> (7 - (i % 8))) % 2
        if participation_bit == 1:
            participants.append(validator_index)
    return participants

bytes1, bytes2, ...

bytes1(x): return x.to_bytes(1, 'big'), bytes2(x): return x.to_bytes(2, 'big'), and so on for all integers, particularly 1, 2, 3, 4, 8, 32.

get_effective_balance

def get_effective_balance(state: State, index: int) -> int:
    """
    Returns the effective balance (also known as "balance at stake") for a ``validator`` with the given ``index``.
    """
    return min(state.validator_balances[index], MAX_DEPOSIT * GWEI_PER_ETH)

get_new_validator_registry_delta_chain_tip

def get_new_validator_registry_delta_chain_tip(current_validator_registry_delta_chain_tip: Hash32,
                                               validator_index: int,
                                               pubkey: int,
                                               flag: int) -> Hash32:
    """
    Compute the next root in the validator registry delta chain.
    """
    return hash_tree_root(
        ValidatorRegistryDeltaBlock(
            latest_registry_delta_root=current_validator_registry_delta_chain_tip,
            validator_index=validator_index,
            pubkey=pubkey,
            flag=flag,
        )
    )

get_fork_version

def get_fork_version(fork_data: ForkData,
                     slot: int) -> int:
    if slot < fork_data.fork_slot:
        return fork_data.pre_fork_version
    else:
        return fork_data.post_fork_version

get_domain

def get_domain(fork_data: ForkData,
               slot: int,
               domain_type: int) -> int:
    return get_fork_version(
        fork_data,
        slot
    ) * 2**32 + domain_type

verify_slashable_vote_data

def verify_slashable_vote_data(state: BeaconState, vote_data: SlashableVoteData) -> bool:
    if len(vote_data.aggregate_signature_poc_0_indices) + len(vote_data.aggregate_signature_poc_1_indices) > MAX_CASPER_VOTES:
        return False

    pubs = [
        aggregate_pubkey([state.validators[i].pubkey for i in vote_data.aggregate_signature_poc_0_indices]),
        aggregate_pubkey([state.validators[i].pubkey for i in vote_data.aggregate_signature_poc_1_indices])
    ]
    vote_data_root = hash_tree_root(vote_data)
    messages = [
        vote_data_root + bytes1(0),
        vote_data_root + bytes1(1)
    ]
    return bls_verify_multiple(
        pubkeys=pubs,
        messages=messages,
        signature=vote_data.aggregate_signature,
        domain=get_domain(
            state.fork_data,
            state.slot,
            DOMAIN_ATTESTATION,
        ),
    )

is_double_vote

def is_double_vote(attestation_data_1: AttestationData,
                   attestation_data_2: AttestationData) -> bool
    """
    Assumes ``attestation_data_1`` is distinct from ``attestation_data_2``.
    Returns True if the provided ``AttestationData`` are slashable
    due to a 'double vote'.
    """
    return attestation_data_1.slot == attestation_data_2.slot

is_surround_vote

def is_surround_vote(attestation_data_1: AttestationData,
                     attestation_data_2: AttestationData) -> bool:
    """
    Assumes ``attestation_data_1`` is distinct from ``attestation_data_2``.
    Returns True if the provided ``AttestationData`` are slashable
    due to a 'surround vote'.
    Note: parameter order matters as this function only checks
    that ``attestation_data_1`` surrounds ``attestation_data_2``.
    """
    return (
        (attestation_data_1.justified_slot < attestation_data_2.justified_slot) and
        (attestation_data_2.justified_slot + 1 == attestation_data_2.slot) and
        (attestation_data_2.slot < attestation_data_1.slot)
    )

integer_squareroot

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

bls_verify

bls_verify is a function for verifying a BLS12-381 signature, defined in the BLS Signature spec.

bls_verify_multiple

bls_verify_multiple is a function for verifying a BLS12-381 signature constructed from multiple messages, defined in the BLS Signature spec.

bls_aggregate_pubkeys

bls_aggregate_pubkeys is a function for aggregating a BLS12-381 public keys into a single aggregate key, defined in the BLS Signature spec.

On startup

A valid block with slot INITIAL_SLOT_NUMBER (a "genesis block") has the following values. Other validity rules (e.g. requiring a signature) do not apply.

{
    slot=INITIAL_SLOT_NUMBER,
    parent_root=ZERO_HASH,
    state_root=STARTUP_STATE_ROOT,
    randao_reveal=ZERO_HASH,
    candidate_pow_receipt_root=ZERO_HASH,
    signature=EMPTY_SIGNATURE,
    body=BeaconBlockBody(
        proposer_slashings=[],
        casper_slashings=[],
        attestations=[],
        deposits=[],
        exits=[]
    ),
}

STARTUP_STATE_ROOT (in the above "genesis block") is generated from the get_initial_beacon_state function below. When enough full deposits have been made to the deposit contract and the ChainStart log has been emitted, get_initial_beacon_state will execute to compute the hash_tree_root of BeaconState.

def get_initial_beacon_state(initial_validator_deposits: List[Deposit],
                             genesis_time: int,
                             processed_pow_receipt_root: Hash32) -> BeaconState:
    state = BeaconState(
        # Misc
        slot=INITIAL_SLOT_NUMBER,
        genesis_time=genesis_time,
        fork_data=ForkData(
            pre_fork_version=INITIAL_FORK_VERSION,
            post_fork_version=INITIAL_FORK_VERSION,
            fork_slot=INITIAL_SLOT_NUMBER,
        ),

        # Validator registry
        validator_registry=[],
        validator_balances=[],
        validator_registry_latest_change_slot=INITIAL_SLOT_NUMBER,
        validator_registry_exit_count=0,
        validator_registry_delta_chain_tip=ZERO_HASH,

        # Randomness and committees
        latest_randao_mixes=[ZERO_HASH for _ in range(LATEST_RANDAO_MIXES_LENGTH)],
        shard_committees_at_slots=[],
        persistent_committees=[],
        persistent_committee_reassignments=[],

        # Finality
        previous_justified_slot=INITIAL_SLOT_NUMBER,
        justified_slot=INITIAL_SLOT_NUMBER,
        justification_bitfield=0,
        finalized_slot=INITIAL_SLOT_NUMBER,

        # Recent state
        latest_crosslinks=[CrosslinkRecord(slot=INITIAL_SLOT_NUMBER, shard_block_root=ZERO_HASH) for _ in range(SHARD_COUNT)],
        latest_block_roots=[ZERO_HASH for _ in range(LATEST_BLOCK_ROOTS_LENGTH)],
        latest_penalized_exit_balances=[],
        latest_attestations=[],
        batched_block_roots=[],

        # PoW receipt root
        processed_pow_receipt_root=processed_pow_receipt_root,
        candidate_pow_receipt_roots=[],
    )

    # handle initial deposits and activations
    for deposit in initial_validator_deposits:
        validator_index = process_deposit(
            state=state,
            pubkey=deposit.deposit_data.deposit_input.pubkey,
            deposit=deposit.deposit_data.value,
            proof_of_possession=deposit.deposit_data.deposit_input.proof_of_possession,
            withdrawal_credentials=deposit.deposit_data.deposit_input.withdrawal_credentials,
            randao_commitment=deposit.deposit_data.deposit_input.randao_commitment
        )
        if get_effective_balance(state, validator_index) == MAX_DEPOSIT * GWEI_PER_ETH:
            update_validator_status(state, validator_index, ACTIVE)

    # set initial committee shuffling
    initial_shuffling = get_new_shuffling(ZERO_HASH, state.validator_registry, 0)
    state.shard_committees_at_slots = initial_shuffling + initial_shuffling

    # set initial persistent shuffling
    active_validator_indices = get_active_validator_indices(state.validator_registry)
    state.persistent_committees = split(shuffle(active_validator_indices, ZERO_HASH), SHARD_COUNT)

    return state

Routine for processing deposits

First, two helper functions:

def min_empty_validator_index(validators: List[ValidatorRecord],
                              validator_balances: List[int],
                              current_slot: int) -> int:
    for i, (v, vbal) in enumerate(zip(validators, validator_balances)):
        if vbal == 0 and v.latest_status_change_slot + ZERO_BALANCE_VALIDATOR_TTL <= current_slot:
            return i
    return None

def validate_proof_of_possession(state: BeaconState,
                                 pubkey: int,
                                 proof_of_possession: bytes,
                                 withdrawal_credentials: Hash32,
                                 randao_commitment: Hash32) -> bool:
    proof_of_possession_data = DepositInput(
        pubkey=pubkey,
        withdrawal_credentials=withdrawal_credentials,
        randao_commitment=randao_commitment,
        proof_of_possession=EMPTY_SIGNATURE,
    )

    return bls_verify(
        pubkey=pubkey,
        message=hash_tree_root(proof_of_possession_data),
        signature=proof_of_possession,
        domain=get_domain(
            state.fork_data,
            state.slot,
            DOMAIN_DEPOSIT,
        )
    )

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

def process_deposit(state: BeaconState,
                    pubkey: int,
                    deposit: int,
                    proof_of_possession: bytes,
                    withdrawal_credentials: Hash32,
                    randao_commitment: Hash32) -> int:
    """
    Process a deposit from Ethereum 1.0.
    Note that this function mutates ``state``.
    """
    # Validate the given `proof_of_possession`
    assert validate_proof_of_possession(
        state,
        pubkey,
        proof_of_possession,
        withdrawal_credentials,
        randao_commitment,
    )

    validator_pubkeys = [v.pubkey for v in state.validator_registry]

    if pubkey not in validator_pubkeys:
        # Add new validator
        validator = ValidatorRecord(
            pubkey=pubkey,
            withdrawal_credentials=withdrawal_credentials,
            randao_commitment=randao_commitment,
            randao_layers=0,
            status=PENDING_ACTIVATION,
            latest_status_change_slot=state.slot,
            exit_count=0
        )

        index = min_empty_validator_index(state.validator_registry, state.validator_balances, state.slot)
        if index is None:
            state.validator_registry.append(validator)
            state.validator_balances.append(deposit)
            index = len(state.validator_registry) - 1
        else:
            state.validator_registry[index] = validator
            state.validator_balances[index] = deposit
    else:
        # Increase balance by deposit
        index = validator_pubkeys.index(pubkey)
        assert state.validator_registry[index].withdrawal_credentials == withdrawal_credentials

        state.validator_balances[index] += deposit

    return index

Routine for updating validator status

def update_validator_status(state: BeaconState,
                            index: int,
                            new_status: int) -> None:
    """
    Update the validator status with the given ``index`` to ``new_status``.
    Handle other general accounting related to this status update.
    Note that this function mutates ``state``.
    """
    if new_status == ACTIVE:
        activate_validator(state, index)
    if new_status == ACTIVE_PENDING_EXIT:
        initiate_validator_exit(state, index)
    if new_status in [EXITED_WITH_PENALTY, EXITED_WITHOUT_PENALTY]:
        exit_validator(state, index, new_status)

The following are helpers and should only be called via update_validator_status:

def activate_validator(state: BeaconState,
                       index: int) -> None:
    """
    Activate the validator with the given ``index``.
    Note that this function mutates ``state``.
    """
    validator = state.validator_registry[index]
    if validator.status != PENDING_ACTIVATION:
        return

    validator = state.validator_registry[index]
    validator.status = ACTIVE
    validator.latest_status_change_slot = state.slot
    state.validator_registry_delta_chain_tip = get_new_validator_registry_delta_chain_tip(
        current_validator_registry_delta_chain_tip=state.validator_registry_delta_chain_tip,
        validator_index=index,
        pubkey=validator.pubkey,
        flag=ACTIVATION,
    )

def initiate_validator_exit(state: BeaconState,
                            index: int) -> None:
    """
    Initiate exit for the validator with the given ``index``.
    Note that this function mutates ``state``.
    """
    validator = state.validator_registry[index]
    if validator.status != ACTIVE:
        return

    validator = state.validator_registry[index]
    validator.status = ACTIVE_PENDING_EXIT
    validator.latest_status_change_slot = state.slot

def exit_validator(state: BeaconState,
                   index: int,
                   new_status: int) -> None:
    """
    Exit the validator with the given ``index``.
    Note that this function mutates ``state``.
    """
    validator = state.validator_registry[index]
    prev_status = validator.status

    if prev_status == EXITED_WITH_PENALTY:
        return

    validator.status = new_status
    validator.latest_status_change_slot = state.slot

    if new_status == EXITED_WITH_PENALTY:
        state.latest_penalized_exit_balances[state.slot // COLLECTIVE_PENALTY_CALCULATION_PERIOD] += 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

    if prev_status == EXITED_WITHOUT_PENALTY:
        return

    # The following updates only occur if not previous exited
    state.validator_registry_exit_count += 1
    validator.exit_count = state.validator_registry_exit_count
    state.validator_registry_delta_chain_tip = get_new_validator_registry_delta_chain_tip(
        current_validator_registry_delta_chain_tip=state.validator_registry_delta_chain_tip,
        validator_index=index,
        pubkey=validator.pubkey,
        flag=EXIT
    )

    # Remove validator from persistent committees
    for committee in state.persistent_committees:
        for i, validator_index in committee:
            if validator_index == index:
                committee.pop(i)
                break

Per-slot processing

Below are the processing steps that happen at every slot.

Misc counters

  • Set state.slot += 1.
  • Set state.validator_registry[get_beacon_proposer_index(state, state.slot)].randao_layers += 1.
  • Set state.latest_randao_mixes[state.slot % LATEST_RANDAO_MIXES_LENGTH] = state.latest_randao_mixes[(state.slot - 1) % LATEST_RANDAO_MIXES_LENGTH]

Block roots

  • Let previous_block_root be the tree_hash_root of the previous beacon block processed in the chain.
  • Set state.latest_block_roots[(state.slot - 1) % LATEST_BLOCK_ROOTS_LENGTH] = previous_block_root.
  • If state.slot % LATEST_BLOCK_ROOTS_LENGTH == 0 append merkle_root(state.latest_block_roots) to state.batched_block_roots.

Per-block processing

Below are the processing steps that happen at every block.

Slot

  • Verify that block.slot == state.slot.

Proposer signature

  • Let block_without_signature_root be the hash_tree_root of block where block.signature is set to EMPTY_SIGNATURE.
  • Let proposal_root = hash_tree_root(ProposalSignedData(state.slot, BEACON_CHAIN_SHARD_NUMBER, block_without_signature_root)).
  • Verify that bls_verify(pubkey=state.validator_registry[get_beacon_proposer_index(state, state.slot)].pubkey, data=proposal_root, signature=block.signature, domain=get_domain(state.fork_data, state.slot, DOMAIN_PROPOSAL)).

RANDAO

  • Let repeat_hash(x, n) = x if n == 0 else repeat_hash(hash(x), n-1).
  • Let proposer = state.validator_registry[get_beacon_proposer_index(state, state.slot)].
  • Verify that repeat_hash(block.randao_reveal, proposer.randao_layers) == proposer.randao_commitment.
  • Set state.latest_randao_mixes[state.slot % LATEST_RANDAO_MIXES_LENGTH] = xor(state.latest_randao_mixes[state.slot % LATEST_RANDAO_MIXES_LENGTH], block.randao_reveal)
  • Set proposer.randao_commitment = block.randao_reveal.
  • Set proposer.randao_layers = 0.

PoW receipt root

  • If block.candidate_pow_receipt_root is x.candidate_pow_receipt_root for some x in state.candidate_pow_receipt_roots, set x.vote_count += 1.
  • Otherwise, append to state.candidate_pow_receipt_roots a new CandidatePoWReceiptRootRecord(candidate_pow_receipt_root=block.candidate_pow_receipt_root, vote_count=1).

Operations

Proposer slashings

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

For each proposer_slashing in block.body.proposer_slashings:

  • Let proposer = state.validator_registry[proposer_slashing.proposer_index].
  • Verify that proposer_slashing.proposal_data_1.slot == proposer_slashing.proposal_data_2.slot.
  • Verify that proposer_slashing.proposal_data_1.shard == proposer_slashing.proposal_data_2.shard.
  • Verify that proposer_slashing.proposal_data_1.block_root != proposer_slashing.proposal_data_2.block_root.
  • Verify that proposer.status != EXITED_WITH_PENALTY.
  • Verify that bls_verify(pubkey=proposer.pubkey, message=hash_tree_root(proposer_slashing.proposal_data_1), signature=proposer_slashing.proposal_signature_1, domain=get_domain(state.fork_data, proposer_slashing.proposal_data_1.slot, DOMAIN_PROPOSAL)).
  • Verify that bls_verify(pubkey=proposer.pubkey, message=hash_tree_root(proposer_slashing.proposal_data_2), signature=proposer_slashing.proposal_signature_2, domain=get_domain(state.fork_data, proposer_slashing.proposal_data_2.slot, DOMAIN_PROPOSAL)).
  • Run update_validator_status(state, proposer_slashing.proposer_index, new_status=EXITED_WITH_PENALTY).

Casper slashings

Verify that len(block.body.casper_slashings) <= MAX_CASPER_SLASHINGS.

For each casper_slashing in block.body.casper_slashings:

  • Let slashable_vote_data_1 = casper_slashing.slashable_vote_data_1.
  • Let slashable_vote_data_2 = casper_slashing.slashable_vote_data_2.
  • Let indices(slashable_vote_data) = slashable_vote_data.aggregate_signature_poc_0_indices + slashable_vote_data.aggregate_signature_poc_1_indices.
  • Let intersection = [x for x in indices(slashable_vote_data_1) if x in indices(slashable_vote_data_2)].
  • Verify that len(intersection) >= 1.
  • Verify that slashable_vote_data_1.data != slashable_vote_data_2.data.
  • Verify that is_double_vote(slashable_vote_data_1.data, slashable_vote_data_2.data) or is_surround_vote(slashable_vote_data_1.data, slashable_vote_data_2.data).
  • Verify that verify_slashable_vote_data(state, slashable_vote_data_1).
  • Verify that verify_slashable_vote_data(state, slashable_vote_data_2).
  • For each validator index i in intersection, if state.validator_registry[i].status does not equal EXITED_WITH_PENALTY, then run update_validator_status(state, i, new_status=EXITED_WITH_PENALTY)

Attestations

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

For each attestation in block.body.attestations:

  • Verify that attestation.data.slot + MIN_ATTESTATION_INCLUSION_DELAY <= state.slot.
  • Verify that attestation.data.slot + EPOCH_LENGTH >= state.slot.
  • Verify that attestation.data.justified_slot is equal to state.justified_slot if attestation.data.slot >= state.slot - (state.slot % EPOCH_LENGTH) else state.previous_justified_slot.
  • Verify that attestation.data.justified_block_root is equal to get_block_root(state, attestation.data.justified_slot).
  • Verify that either attestation.data.latest_crosslink_root or attestation.data.shard_block_root equals state.latest_crosslinks[shard].shard_block_root.
  • aggregate_signature verification:
    • Let participants = get_attestation_participants(state, attestation.data, attestation.participation_bitfield).
    • Let group_public_key = bls_aggregate_pubkeys([state.validator_registry[v].pubkey for v in participants]).
    • Verify that bls_verify(pubkey=group_public_key, message=hash_tree_root(attestation.data) + bytes1(0), signature=attestation.aggregate_signature, domain=get_domain(state.fork_data, attestation.data.slot, DOMAIN_ATTESTATION)).
  • [TO BE REMOVED IN PHASE 1] Verify that attestation.data.shard_block_root == ZERO_HASH.
  • Append PendingAttestationRecord(data=attestation.data, participation_bitfield=attestation.participation_bitfield, custody_bitfield=attestation.custody_bitfield, slot_included=state.slot) to state.latest_attestations.

Deposits

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

[TODO: add logic to ensure that deposits from 1.0 chain are processed in order]

For each deposit in block.body.deposits:

  • Let serialized_deposit_data be the serialized form of deposit.deposit_data. It should be the DepositInput followed by 8 bytes for deposit_data.value and 8 bytes for deposit_data.timestamp. That is, it should match deposit_data in the Ethereum 1.0 deposit contract of which the hash was placed into the Merkle tree.
  • Use the following procedure to verify deposit.merkle_branch, setting leaf=serialized_deposit_data, depth=DEPOSIT_CONTRACT_TREE_DEPTH and root=state.processed_pow_receipt_root:
def verify_merkle_branch(leaf: Hash32, branch: [Hash32], depth: int, index: int, root: Hash32) -> bool:
    value = leaf
    for i in range(depth):
        if index % 2:
            value = hash(branch[i], value)
        else:
            value = hash(value, branch[i])
    return value == root
  • Verify that state.slot - (deposit.deposit_data.timestamp - state.genesis_time) // SLOT_DURATION < ZERO_BALANCE_VALIDATOR_TTL.
  • Run the following:
process_deposit(
    state=state,
    pubkey=deposit.deposit_data.deposit_input.pubkey,
    deposit=deposit.deposit_data.value,
    proof_of_possession=deposit.deposit_data.deposit_input.proof_of_possession,
    withdrawal_credentials=deposit.deposit_data.deposit_input.withdrawal_credentials,
    randao_commitment=deposit.deposit_data.deposit_input.randao_commitment
)

Exits

Verify that len(block.body.exits) <= MAX_EXITS.

For each exit in block.body.exits:

  • Let validator = state.validator_registry[exit.validator_index].
  • Verify that validator.status == ACTIVE.
  • Verify that state.slot >= exit.slot.
  • Verify that state.slot >= validator.latest_status_change_slot + SHARD_PERSISTENT_COMMITTEE_CHANGE_PERIOD.
  • Verify that bls_verify(pubkey=validator.pubkey, message=ZERO_HASH, signature=exit.signature, domain=get_domain(state.fork_data, exit.slot, DOMAIN_EXIT)).
  • Run update_validator_status(state, validator_index, new_status=ACTIVE_PENDING_EXIT).

Per-epoch processing

The steps below happen when state.slot % EPOCH_LENGTH == 0.

Helpers

All validators:

  • Let active_validator_indices = get_active_validator_indices(state.validator_registry).
  • Let total_balance = sum([get_effective_balance(state, i) for i in active_validator_indices]).

Validators attesting during the current epoch:

  • Let this_epoch_attestations = [a for a in state.latest_attestations if state.slot - EPOCH_LENGTH <= a.data.slot < state.slot]. (Note: this is the set of attestations of slots in the epoch state.slot-EPOCH_LENGTH...state.slot-1, not attestations that got included in the chain during the epoch state.slot-EPOCH_LENGTH...state.slot-1.)
  • Validators justifying the epoch boundary block at the start of the current epoch:
    • Let this_epoch_boundary_attestations = [a for a in this_epoch_attestations if a.data.epoch_boundary_root == get_block_root(state, state.slot-EPOCH_LENGTH) and a.justified_slot == state.justified_slot].
    • Let this_epoch_boundary_attester_indices be the union of the validator index sets given by [get_attestation_participants(state, a.data, a.participation_bitfield) for a in this_epoch_boundary_attestations].
    • Let this_epoch_boundary_attesting_balance = sum([get_effective_balance(state, i) for i in this_epoch_boundary_attester_indices]).

Validators attesting during the previous epoch:

  • Validators that made an attestation during the previous epoch:
    • Let previous_epoch_attestations = [a for a in state.latest_attestations if state.slot - 2 * EPOCH_LENGTH <= a.slot < state.slot - EPOCH_LENGTH].
    • Let previous_epoch_attester_indices be the union of the validator index sets given by [get_attestation_participants(state, a.data, a.participation_bitfield) for a in previous_epoch_attestations].
  • Validators targeting the previous justified hash:
    • Let previous_epoch_justified_attestations = [a for a in this_epoch_attestations + previous_epoch_attestations if a.justified_slot == state.previous_justified_slot].
    • Let previous_epoch_justified_attester_indices be the union of the validator index sets given by [get_attestation_participants(state, a.data, a.participation_bitfield) for a in previous_epoch_justified_attestations].
    • Let previous_epoch_justified_attesting_balance = sum([get_effective_balance(state, i) for i in previous_epoch_justified_attester_indices]).
  • Validators justifying the epoch boundary block at the start of the previous epoch:
    • Let previous_epoch_boundary_attestations = [a for a in previous_epoch_justified_attestations if a.epoch_boundary_root == get_block_root(state, state.slot - 2 * EPOCH_LENGTH)].
    • Let previous_epoch_boundary_attester_indices be the union of the validator index sets given by [get_attestation_participants(state, a.data, a.participation_bitfield) for a in previous_epoch_boundary_attestations].
    • Let previous_epoch_boundary_attesting_balance = sum([get_effective_balance(state, i) for i in previous_epoch_boundary_attester_indices]).
  • Validators attesting to the expected beacon chain head during the previous epoch:
    • Let previous_epoch_head_attestations = [a for a in previous_epoch_attestations if a.beacon_block_root == get_block_root(state, a.slot)].
    • Let previous_epoch_head_attester_indices be the union of the validator index sets given by [get_attestation_participants(state, a.data, a.participation_bitfield) for a in previous_epoch_head_attestations].
    • Let previous_epoch_head_attesting_balance = sum([get_effective_balance(state, i) for i in previous_epoch_head_attester_indices]).

Note: previous_epoch_boundary_attesting_balance balance might be marginally different than this_epoch_boundary_attesting_balance 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.

For every shard_committee in state.shard_committees_at_slots:

  • Let attesting_validator_indices(shard_committee, shard_block_root) be the union of the validator index sets given by [get_attestation_participants(state, a.data, a.participation_bitfield) for a in this_epoch_attestations + previous_epoch_attestations if a.shard == shard_committee.shard and a.shard_block_root == shard_block_root].
  • Let winning_root(shard_committee) be equal to the value of shard_block_root such that sum([get_effective_balance(state, i) for i in attesting_validator_indices(shard_committee, shard_block_root)]) is maximized (ties broken by favoring lower shard_block_root values).
  • Let attesting_validators(shard_committee) be equal to attesting_validators(shard_committee, winning_root(shard_committee)) for convenience.
  • Let total_attesting_balance(shard_committee) be the sum of the balances-at-stake of attesting_validators(shard_committee).
  • Let total_balance(shard_committee) = sum([get_effective_balance(state, i) for i in shard_committee.committee]).
  • Let inclusion_slot(state, index) = a.slot_included for the attestation a where index is in get_attestation_participants(state, a.data, a.participation_bitfield).
  • Let inclusion_distance(state, index) = a.slot_included - a.data.slot where a is the above attestation.
  • Let adjust_for_inclusion_distance(magnitude, distance) be the function below.
def adjust_for_inclusion_distance(magnitude: int, distance: int) -> int:
    """
    Adjusts the reward of an attestation based on how long it took to get included (the longer, the lower the reward).
    Returns a value between ``0`` and ``magnitude``.
    ""
    return magnitude // 2 + (magnitude // 2) * MIN_ATTESTATION_INCLUSION_DELAY // distance

Receipt roots

If state.slot % POW_RECEIPT_ROOT_VOTING_PERIOD == 0:

  • Set state.processed_pow_receipt_root = x.receipt_root if x.vote_count * 2 > POW_RECEIPT_ROOT_VOTING_PERIOD for some x in state.candidate_pow_receipt_root.
  • Set state.candidate_pow_receipt_roots = [].

Justification

  • Set state.previous_justified_slot = state.justified_slot.
  • Set state.justification_bitfield = (state.justification_bitfield * 2) % 2**64.
  • Set state.justification_bitfield |= 2 and state.justified_slot = state.slot - 2 * EPOCH_LENGTH if 3 * previous_epoch_boundary_attesting_balance >= 2 * total_balance.
  • Set state.justification_bitfield |= 1 and state.justified_slot = state.slot - 1 * EPOCH_LENGTH if 3 * this_epoch_boundary_attesting_balance >= 2 * total_balance.

Finalization

Set state.finalized_slot = state.previous_justified_slot if any of the following are true:

  • state.previous_justified_slot == state.slot - 2 * EPOCH_LENGTH and state.justification_bitfield % 4 == 3
  • state.previous_justified_slot == state.slot - 3 * EPOCH_LENGTH and state.justification_bitfield % 8 == 7
  • state.previous_justified_slot == state.slot - 4 * EPOCH_LENGTH and state.justification_bitfield % 16 in (15, 14)

Crosslinks

For every shard_committee in state.shard_committees_at_slots:

  • Set state.latest_crosslinks[shard] = CrosslinkRecord(slot=state.slot, block_root=winning_root(shard_committee)) if 3 * total_attesting_balance(shard_committee) >= 2 * total_balance(shard_committee).

Rewards and penalties

First, we define some additional helpers:

  • Let base_reward_quotient = BASE_REWARD_QUOTIENT * integer_squareroot(total_balance // GWEI_PER_ETH).
  • Let base_reward(state, index) = get_effective_balance(state, index) // base_reward_quotient // 4 for any validator with the given index.
  • Let inactivity_penalty(state, index, slots_since_finality) = base_reward(state, index) + get_effective_balance(state, index) * slots_since_finality // INACTIVITY_PENALTY_QUOTIENT for any validator with the given index.

Justification and finalization

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

  • Let slots_since_finality = state.slot - state.finalized_slot.

Case 1: slots_since_finality <= 4 * EPOCH_LENGTH:

  • Expected FFG source:
    • Any validator index in previous_epoch_justified_attester_indices gains adjust_for_inclusion_distance(base_reward(state, index) * previous_epoch_justified_attesting_balance // total_balance, inclusion_distance(state, index)).
    • Any active validator v not in previous_epoch_justified_attester_indices loses base_reward(state, index).
  • Expected FFG target:
    • Any validator index in previous_epoch_boundary_attester_indices gains adjust_for_inclusion_distance(base_reward(state, index) * previous_epoch_boundary_attesting_balance // total_balance, inclusion_distance(state, index)).
    • Any active validator index not in previous_epoch_boundary_attester_indices loses base_reward(state, index).
  • Expected beacon chain head:
    • Any validator index in previous_epoch_head_attester_indices gains adjust_for_inclusion_distance(base_reward(state, index) * previous_epoch_head_attesting_balance // total_balance, inclusion_distance(state, index)).
    • Any active validator index not in previous_epoch_head_attester_indices loses base_reward(state, index).

Case 2: slots_since_finality > 4 * EPOCH_LENGTH:

  • Any active validator index not in previous_epoch_justified_attester_indices, loses inactivity_penalty(state, index, slots_since_finality).
  • Any active validator index not in previous_epoch_boundary_attester_indices, loses inactivity_penalty(state, index, slots_since_finality).
  • Any active validator index not in previous_epoch_head_attester_indices, loses inactivity_penalty(state, index, slots_since_finality).
  • Any validator index with status == EXITED_WITH_PENALTY, loses 3 * inactivity_penalty(state, index, slots_since_finality).

Attestation inclusion

For each index in previous_epoch_attester_indices, we determine the proposer proposer_index = get_beacon_proposer_index(state, inclusion_slot(state, index)) and set state.validator_balances[proposer_index] += base_reward(state, index) // INCLUDER_REWARD_QUOTIENT.

Crosslinks

For every shard_committee in state.shard_committees_at_slots[:EPOCH_LENGTH] (i.e. the objects corresponding to the epoch before the current one), for each index in shard_committee.committee, adjust balances as follows:

  • If index in attesting_validators(shard_committee), state.validator_balances[index] += adjust_for_inclusion_distance(base_reward(state, index) * total_attesting_balance(shard_committee) // total_balance(shard_committee)), inclusion_distance(state, index)).
  • If index not in attesting_validators(shard_committee), state.validator_balances[index] -= base_reward(state, index).

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 active_validator_indices(state.validator_registry):
        if state.validator_balances[index] < EJECTION_BALANCE:
            update_validator_status(state, index, new_status=EXITED_WITHOUT_PENALTY)

Validator registry

If the following are satisfied:

  • state.finalized_slot > state.validator_registry_latest_change_slot
  • state.latest_crosslinks[shard].slot > state.validator_registry_latest_change_slot for every shard number shard in state.shard_committees_at_slots

update the validator registry and associated fields by running

def update_validator_registry(state: BeaconState) -> None:
    """
    Update validator registry.
    Note that this function mutates ``state``.
    """
    # The active validators
    active_validator_indices = get_active_validator_indices(state.validator_registry)
    # The total effective balance of active validators
    total_balance = sum([get_effective_balance(state, i) for i in active_validator_indices])

    # The maximum balance churn in Gwei (for deposits and exits separately)
    max_balance_churn = max(
        MAX_DEPOSIT * GWEI_PER_ETH,
        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.status == PENDING_ACTIVATION and state.validator_balances[index] >= MAX_DEPOSIT * GWEI_PER_ETH:
            # 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
            update_validator_status(state, index, new_status=ACTIVE)

    # Exit validators within the allowable balance churn
    balance_churn = 0
    for index, validator in enumerate(state.validator_registry):
        if validator.status == ACTIVE_PENDING_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
            update_validator_status(state, index, new_status=EXITED_WITHOUT_PENALTY)


    # Calculate the total ETH that has been penalized in the last ~2-3 withdrawal periods
    period_index = current_slot // COLLECTIVE_PENALTY_CALCULATION_PERIOD
    total_penalties = (
        (latest_penalized_exit_balances[period_index]) +
        (latest_penalized_exit_balances[period_index - 1] if period_index >= 1 else 0) +
        (latest_penalized_exit_balances[period_index - 2] if period_index >= 2 else 0)
    )

    # Calculate penalties for slashed validators
    def to_penalize(index):
        return state.validator_registry[index].status == EXITED_WITH_PENALTY
    validators_to_penalize = filter(to_penalize, range(len(validator_registry)))
    for index in validators_to_penalize:
        state.validator_balances[index] -= get_effective_balance(state, index) * min(total_penalties * 3, total_balance) // total_balance

    return validator_registry, latest_penalized_exit_balances, validator_registry_delta_chain_tip

Also perform the following updates:

  • Set state.validator_registry_latest_change_slot = state.slot.
  • Set state.shard_committees_at_slots[:EPOCH_LENGTH] = state.shard_committees_at_slots[EPOCH_LENGTH:].
  • Set state.shard_committees_at_slots[EPOCH_LENGTH:] = get_new_shuffling(state.latest_randao_mixes[(state.slot - EPOCH_LENGTH) % LATEST_RANDAO_MIXES_LENGTH], state.validator_registry, next_start_shard) where next_start_shard = (state.shard_committees_at_slots[-1][-1].shard + 1) % SHARD_COUNT.

If a validator registry update does not happen do the following:

  • Set state.shard_committees_at_slots[:EPOCH_LENGTH] = state.shard_committees_at_slots[EPOCH_LENGTH:].
  • Let slots_since_finality = state.slot - state.validator_registry_latest_change_slot.
  • Let start_shard = state.shard_committees_at_slots[0][0].shard.
  • If slots_since_finality * EPOCH_LENGTH <= MIN_VALIDATOR_REGISTRY_CHANGE_INTERVAL or slots_since_finality is an exact power of 2, set state.shard_committees_at_slots[EPOCH_LENGTH:] = get_new_shuffling(state.latest_randao_mixes[(state.slot - CYCLE_LENGTH) % LATEST_RANDAO_MIXES_LENGTH], state.validator_registry, start_shard). Note that start_shard is not changed from the last epoch.

Proposer reshuffling

Run the following code to update the shard proposer set:

active_validator_indices = get_active_validator_indices(state.validator_registry)
num_validators_to_reshuffle = len(active_validator_indices) // SHARD_PERSISTENT_COMMITTEE_CHANGE_PERIOD
for i in range(num_validators_to_reshuffle):
    # Multiplying i to 2 to ensure we have different input to all the required hashes in the shuffling
    # and none of the hashes used for entropy in this loop will be the same
    validator_index = active_validator_indices[hash(state.latest_randao_mixes[state.slot % LATEST_RANDAO_MIXES_LENGTH] + bytes8(i * 2)) % len(active_validator_indices)]
    new_shard = hash(state.latest_randao_mixes[state.slot % LATEST_RANDAO_MIXES_LENGTH] + bytes8(i * 2 + 1)) % SHARD_COUNT
    shard_reassignment_record = ShardReassignmentRecord(
        validator_index=validator_index,
        shard=new_shard,
        slot=s + SHARD_PERSISTENT_COMMITTEE_CHANGE_PERIOD
    )
    state.persistent_committee_reassignments.append(shard_reassignment_record)

while len(state.persistent_committee_reassignments) > 0 and state.persistent_committee_reassignments[0].slot <= s:
    reassignment = state.persistent_committee_reassignments.pop(0)
    for committee in state.persistent_committees:
        if reassignment.validator_index in committee:
            committee.pop(committee.index(reassignment.validator_index))
    state.persistent_committees[reassignment.shard].append(reassignment.validator_index)

Final updates

  • Remove any attestation in state.latest_attestations such that attestation.data.slot < state.slot - EPOCH_LENGTH.

State root processing

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

References

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

Normative

Informative

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

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

Copyright

Copyright and related rights waived via CC0.